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Berdyugin1, Vilppu Piirola1, Stefano Bagnulo2, John D. Landstreet2, 3, and Svetlana V. Berdyugina4, 5, 6 +1 Department of Physics and Astronomy, FI-20014 University of Turku, Finland; e-mail: andber@utu.fi +2 Armagh Observatory & Planetarium, College Hill, Armagh BT61 9DG, UK; +3 Department of Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada; +4 Leibniz-Institut für Sonnenphysik (KIS), Schöneckstr 6, Freibirg, Germany; +5 IRSOL Istituto Ricerche Solari “Aldo e Cele Daccò", Faculty of Informatics, Università della Svizzera italiana, Via Patocci 57, +Locarno, Switzerland; +6 Euler Institute, Faculty of Informatics, Università della Svizzera italiana, Via la Santa 1, 6962 Lugano, Switzerland +Received October 7, 2022; accepted October 27, 2022 +ABSTRACT +About half of white dwarfs (WDs) evolve to the DC state as they cool; the others become DQ or (temporarily?) DZ WDs. The recent +magnetic survey of the local 20 pc volume has established a high frequency of magnetic fields among WDs older than 2–3 Gyr, +demonstrating that in low- and average-mass WDs, the effects of magnetism become more common as they age, and the fields on +average become stronger. However, the available statistics of WDs older than about 5 Gyr do not clearly establish how fields evolve +beyond this age. We are carrying out a survey to clarify the occurrence of magnetism in DC-type WDs in order to better understand this +late evolution. We use broadband filter polarimetry, arguably the most efficient way to detect magnetic fields in featureless WDs via +continuum circular polarization. Here we report the discovery of a magnetic field in five DC WDs (of 23 observed), almost doubling +the total sample of known magnetic WDs belonging to the DC spectral class. +Key words. White dwarfs – Stars: magnetic fields – polarization +1. Introduction +Single stars of M <∼ 8M⊙ evolve to become white dwarfs (WDs). +The descendants of these single stars of intermediate mass pro- +vide most of the population of WDs, concentrated around the +mean mass of 0.6M⊙. A smaller fraction of current WDs were +also formed from close binary systems. Some of these systems +eventually merged to form a single collapsed remnant, frequently +of a significantly larger mass; others ended their nuclear lifetimes +as double WD binaries. +Once formed, the evolution of a WD is normally to cool +slowly over several gigayears. Cooling is a fairly complex process +even for single-star evolution, both to observe and to understand. +Observationally, young hot WDs usually show strong spectra of +H (DA WDs), He (DB WDs), or sometimes C (DQs). As they +cool, spectral lines of the dominant elements H or He become +weaker: He lines vanish at about 11000 K, H lines around 5000 K. +In parallel with this general evolution, WDs may (temporarily) +show lines of metals such as Mg, Si, Ca, and/or Fe (DZ, DAZ, +and DZA stars), and some have spectra dominated by C. Below +about 5000 K, about one-quarter of WDs have very weak Hα, +another quarter show spectral lines of metals (especially Ca ii) or +of C2, and the remaining half show essentially featureless spectra +(DC WDs; Bagnulo & Landstreet 2021, Table 1). It appears that +the dominant element(s) in the atmosphere can change as cooling +occurs, for example due to gravitational diffusion, development +of convection, and accretion of circumstellar planetary debris. +One of the physical effects adding complexity to our ef- +forts to understand WD evolution is that a significant frac- +tion, about 20-25%, of WDs in the local volume near the Sun +(Bagnulo & Landstreet 2021) possess detectable surface mag- +netic fields (this high frequency was already suggested on the +basis of literature reports of fields in nine WDs in the 13 pc vol- +ume by Kawka et al. 2007). The fields observed at the surface +range in strength, measured by the mean surface field ⟨|B|⟩, from +tens of kG to hundreds of MG. Such fields can significantly affect +WD evolution by altering or suppressing surface convection and +internal shear, and by transferring angular momentum between +internal layers or during accretion or mass loss (see for example +Tremblay et al. 2015). The fields may also introduce additional +forces into envelope and atmosphere layers, altering their hy- +drostatic structure from that expected when magnetic effects are +absent (Landstreet 1987). +For WDs formed by single-star evolution, which generates +most of the large populationsof WDs with masses around 0.6M⊙, +it has become clear that recently formed WDs (with cooling +ages of less than, say, 1 Gyr) are very rarely detectably mag- +netic, and when they are magnetic, the fields are usually very +weak (Bagnulo & Landstreet 2022). As WDs cool, fields begin +to appear more frequently and usually become stronger. In WDs +older than 3 or 4 Gyr, megagauss-scale fields are not uncommon +(Bagnulo & Landstreet 2021). +The observed evolution in magnetic field frequency and +strength of normal-mass WDs for the first few gigayears of cool- +ing may be understood as a slow emergence – as a result of field +relaxation to the stellar surface – of the internal fields present +in the degenerate cores of the WD precursors. An additional +contribution to observed surface fields may be due to magnetic +fields generated during cooling by a dynamo that acts during +the period when the core of the WD is crystallising (Isern et al. +2017; Gentile Fusillo et al. 2018). Beyond the end of crystalli- +Article number, page 1 of 6 + +A&A proofs: manuscript no. 45149corr +sation, the only identified evolution mechanisms are continued +field relaxation and Ohmic decay. +Observationally, however, after about 5 Gyr of WD cooling, +we have very limited information with which to guide and con- +front theory. Only small survey samples constrain observed field +evolution on cool WDs, such as DQ WDs, where C2 bands show +no polarization in strong fields (Berdyugina et al. 2007). Partic- +ularly little is known about the magnetic fields of DC WDs, in +which no spectral features are seen at all, leading to the ques- +tions of whether field strength begins to decay Ohmically and +whether the frequency of surface fields continues to increase. +Data that could help us answer these questions are very limited. +For WDs within 20 pc of the Sun (the 20 pc volume sample), +Bagnulo & Landstreet (2021) showed that of 31 DC WDs, only 4 +are magnetic white dwarfs (MWDs), and that only 4 of 24 WDs +of any spectral class older than ∼ 6 Gyr are magnetic. These +data are obviously too limited to clearly describe the evolution of +fields in these old WDs. +Previous surveys have provided almost no information about +magnetic fields in DC WDs. Fields in such stars cannot be de- +tected through the magnetic splitting of spectral lines. They can +only be detected via the observation of continuum circular po- +larisation (CCP; Kemp 1970), a method of observation hardly +employed since the 1970s (Angel et al. 1981). Remarkably, most +CCP observations have led to the discovery of magnetic fields +in stars that are not featureless but in which the magnetic field +is strong enough to shift and broaden spectral lines in a such +a way as to make their intensity spectra unrecognisable. Only +seven featureless DC WDs are presently known to be mag- +netic. Five of them were discovered only in the last couple of +years (Bagnulo & Landstreet 2020; Berdyugin et al. 2022). Be- +fore these results, the only known magnetic DC stars were G195- +19 and G111-49, discovered respectively by Angel & Landstreet +(1971) and Putney (1995). +To improve our knowledge of the magnetic fields in the latest +stages of stellar evolution, we have started a volume-limited sur- +vey of DC stars in the local 33 pc volume,which is about4.5 times +larger than the previously explored 20 pc volume and should have +a correspondingly larger sample of DCs and DC MWDs. With +this sample we expect to find enough DC MWDs to delineate the +evolution of their magnetic fields, both in the WDs with He-rich +atmospheres that become DCs as soon as their effective tempera- +tures reach about 11 000 K (‘young’ DCs), and in DC WDs with +Teff below about 5000 K, with cooling ages of around 4 Gyr or +more (‘old’ DCs). +2. Observations +Almost all known MWDs have been discovered via the mag- +netic (Zeeman) splitting of spectral lines, observed in stellar +flux spectra, or via the Zeeman polarisation of spectral features +(Ferrario et al. 2015). Using these methods, fields of a few kG +up to 1 GG can be reliably detected. However, these techniques +cannot be used to measure fields in WDs that lack spectral lines. +For such stars, it is necessary to rely on continuum polarisation, +which Kemp (1970) showed should occur in radiation from a +magnetized emitter. The value of this effect was confirmed by the +discovery of a very strong field in the bright WD Grw+70 8247 += WD 1900+705 through the detection of broadband circular po- +larisation (BBCP) by Kemp et al. (1970). +Broadband circular polarisation is a relatively weak effect. +Bagnulo & Landstreet(2020) have estimated that a field of ⟨Bz⟩ ∼ +15 MG is required to produce BBCP of order 1 % in optical +radiation from a cool WD. However, with a sensitive polarimeter, +especially one with a very stable and well-established zero point, +it is possible in principle to detect polarisation of 10−4 or less, +corresponding to ∼ 100 kG fields in ‘sufficiently bright’ WDs. +To detect and measure broadband continuum polarisation, +one uses either spectropolarimetry or filter polarimetry with +broad, photometry-like filters. It is very difficult to establish the +zero point with sufficient accuracy below polarisation levels of +the order of 10−3 in spectropolarimetric measurements ofthe con- +tinuum (Fossati et al. 2007; Siebenmorgen et al. 2014); therefore, +in DC WDs, only megagauss-scale fields can be detected in this +way (Bagnulo & Landstreet 2020). In contrast, broadband filter +polarimeters can be very stable, and instrumental polarisation +can be calibrated at the 10−5 level, so detections with such instru- +ments of fields of hundreds of kilogauss are in practice limited +by the telescope aperture and WD brightness (Berdyugin et al. +2022). +The search for magnetic fields of a fraction of 1 MG or +stronger in DC WDs that is reported here was carried out with +the DIPol-UF broadband filter polarimeter (Piirola et al. 2020) +mounted on the 2.5 m Nordic Optical Telescope (NOT) at the +Observatorio del Roque de los Muchachos on the island of La +Palma, in the Canaries. This instrument obtains simultaneous +circular polarisation (normalized Stokes V/I) measurements in +three filter bands isolated by dichroic mirrors. The passbands are +centred at about 4450 Å (the B′ band), 5400 Å (the V′ band), and +6400Å (the R′ band) with full widths at half maximum (FWHMs) +of 1140, 750, and 960 Å , respectively. With this instrument on +the NOT, we can detect a polarisation degree at the 3σ level of +∼ 10−4 for the Gaia G-band magnitude G ∼ 12, down to a degree +of ∼ 10−3 at G ∼ 17. This instrument and the filter system are +discussed in more detail in our previous paper, which reports the +results of the first part of our search for magnetic fields in DC +WDs (Berdyugin et al. 2022). +Here we report observations of 23 DC WDs and discovery +of 5 new DC MWDs. We note that our survey includes nine +young DCs of He-rich atmospheres with 11000 <∼ Teff <∼ 5000 K +and 14 old WDs with Teff <∼ 5000 K and ages τ >∼ 4 Gyr. The +stars are selected from available classifications and with help +from Gentile Fusillo et al. (2021). Our new observations were +obtained between June 27 and July 5, 2022. +2.1. Instrumental polarisation and alignment of the +polarimetric optics +During our observing run we obtained seven observations of +seven different bright nearby stars, which are believed to have +zero circular polarisation, to check for instrumental polarisation. +These observations are reported in Table 1. +As in our previous run in July 2021, the high S/N measure- +ments of non-polarised stars yield the instrumental polarisation +to a precision better than 10−5. In the B′V′R′ bands, the values +of Stokes V/I are 0.0121 ± 0.0004 %, 0.0109 ± 0.0005 %, and +0.0084 ± 0.0004 %, respectively. These are very close to the +values obtained in 2021. The instrumental polarisation was sub- +tracted from the observed polarisation of all targets, including +the measurements of the standard stars reported in Table 1. +In addition, we obtained one measurement of the well-known +MWD WD 1900+705, which appears to show a signal of cir- +cular polarisation that is nearly constant with time (see e.g. +Bagnulo & Landstreet2019). Our new measurementis compared +in Table 1 to one of the same star that we made during the July +2021 run. The agreement is very satisfactory and demonstrates +that we can obtain measurements that are precise at the 0.02 % +Article number, page 2 of 6 + +Andrei V. Berdyugin et al.: Discovery of magnetic fields in five DC white dwarfs +Table 1. Observing log of bright non-polarised standard stars and the highly polarised MWD WD 1900+705. Polarisation values are given assuming +as instrumental polarisation the values of 0.0121±0.0004 %, 0.0109±0.0005 %, and 0.0084±0.0004 % in the B′, V′, and R′ filters, respectively. For +comparison, we report the polarisation values of WD 1900+705 measured in our 2021 and 2022 runs. +STAR +G +DATE +UT +JD – +Exp. +VI (%) +yyyy-mm-dd +hh:mm +2400000 +(s) +B′ +V′ +R′ +HD 107146 +6.9 +2022-06-27 +21:23 +59758.391 +1680 +0.0005±0.0007 +−0.0006±0.0007 +−0.0001±0.0006 +HD 115043 +6.7 +2022-06-28 +21:21 +59759.390 +1520 +0.0005±0.0008 +−0.0001±0.0012 +−0.0004±0.0004 +HD 122652 +7.0 +2022-06-29 +21:18 +59760.387 +1520 +−0.0012±0.0009 +−0.0015±0.0015 +−0.0019±0.0014 +HD 122676 +7.1 +2022-06-30 +21:17 +59761.387 +1520 +0.0000±0.0011 +0.0003±0.0010 +0.0018±0.0008 +HD 124694 +7.0 +2022-07-01 +21:16 +59762.386 +1520 +−0.0012±0.0008 +0.0014±0.0011 +0.0002±0.0007 +HD 135891 +6.9 +2022-07-02 +21:18 +59763.387 +1520 +0.0014±0.0008 +0.0006±0.0010 +−0.0004±0.0007 +HD 117860 +7.2 +2022-07-03 +21:16 +59764.386 +1520 +−0.0004±0.0010 +−0.0002±0.0012 +0.0005±0.0006 +WD 1900+705 +13.2 +2021-07-02 +22:22 +59398.432 +640 +3.756±0.016 +3.604±0.016 +3.827±0.019 +2022-07-02 +00:25 +59762.518 +3.789±0.016 +3.602±0.019 +3.838±0.018 +Table 2. Programme stars and their main physical features. Star names in boldface identify WDs in which fields were discovered during the +observations reported in this paper (see Table 3). +STAR +G +d +Teff +log g +M +Age +Atmosphere and ref. +(pc) +(K) +c.g.s. +(M⊙) +(Gyr) +WD 0005+395 +LP 240-30 +16.6 +34.4 +4680 +6.77 +0.08 +1.40 +DC, H ProbWD 0.70 (1,3) +WD 0010+543 +LSR J0013+5437 +18.0 +32.3 +4123 +7.77 +0.46 +7.08 +DC, (2, H assumed) +WD 0028+035 +PB 6002 +16.1 +27.8 +6548 +8.14 +0.68 +2.40 +DC, (2, H assumed) +WD 1251+366 +LP 267-311 +17.2 +28.5 +4445 +7.62 +0.37 +3.78 +DC, He (1) +WD 1315+222 +LP 378-956 +16.7 +31.8 +6235 +8.21 +0.71 +3.61 +DCH, He (1) +WD 1346+121 +LP 498-66 +17.8 +28.3 +4150 +7.88 +0.50 +6.58 +DCH, He (1) +WD 1425+495 +CSO 649 +16.7 +33.9 +6895 +8.41 +0.85 +3.77 +DC, (2, H assumed) +WD 1427−238 +LP 857-45 +17.4 +32.6 +4866 +7.90 +0.52 +5.40 +DC, (2, H assumed) +WD 1434+437 +LP 221-217 +17.2 +27.2 +4685 +7.93 +0.54 +6.30 +DC, H-He (1) +WD 1533+469 +LP 176-60 +17.8 +30.8 +4310 +7.83 +0.48 +6.45 +DC?, H (1) +WD 1601−073 +LP 684-16 +17.9 +26.9 +4920 +8.55 +0.94 +9.83 +DCH, (2, H assumed) +WD 1612+092 +LSPM J1614+0906 +17.2 +27.9 +4775 +7.90 +0.52 +5.57 +DC, H (1) +WD 1702−016 +LP 626-29 +17.3 +28.3 +4700 +7.94 +0.54 +6.50 +DC, (2, H assumed) +WD 1737+798 +LP 24-66 +16.9 +26.8 +5535 +8.28 +0.75 +5.72 +DC, He (1) +WD 1746+450 +GD 366 +15.5 +29.9 +9331 +8.47 +0.90 +1.72 +DC, (2, H assumed) +WD 1800+508 +LP 139-38 +17.4 +31.0 +4635 +7.85 +0.48 +5.12 +DC, He-H (1) +WD 1853+775 +LP 25-7 +17.0 +30.5 +4850 +7.74 +0.43 +3.63 +DCH, He (1) +WD 2058+550 +LSR J2059+5517 +17.1 +22.7 +4415 +7.93 +0.53 +7.15 +DC, H-He (1) +WD 2109−295 +EC 21096-2934 +15.1 +32.8 +9260 +7.98 +0.57 +0.78 +DC, He-H (3) +WD 2152−280 +LP930-61 +16.3 +23.5 +5220 +7.85 +0.48 +3.68 +DC, He (1) +WD 2211+372 +LP 287-35 +16.8 +29.2 +6345 +8.47 +0.88 +4.56 +DC?H, He (1) +WD 2215+368 +LP 287-39 +16.8 +20.3 +4485 +7.92 +0.53 +6.80 +DC, H (1) +WD 2311−068 +G 157-34 +15.3 +25.9 +7360 +7.97 +0.56 +1.31 +DC, He (1) +Key to references: 1: Blouin et al. (2019); 2: Gentile Fusillo et al. (2021); 3: Bergeron et al. (2021). Where not found in these +references, ages have been interpolated using the tables from Bédard et al. (2020). +level for a G = 13.2 star with about 10 minutes of exposure time. +This shows that the alignment of our polarimetric optics is stable +over a few years. +2.2. Results +The WDs observed during our 2022 June-July run are listed in +Table 2, with their G magnitudes, distances, physical parameters, +cooling ages, and some comments. Physical parameters were +obtained from various studies, cited in the table’s notes; cooling +ages are interpolated from the online cooling data provided by +the Montreal group (Bédard et al. 2020). +The observations are described in the log in Table 3, which +gives dates, integration times, and the polarisation data in the +three filter bands for each WD observation. We list measured +BBCP values in boldface if non-zero polarisation is detected at +above the 3σ level. We consider that real polarisation has been +detected if a consistent picture of detection is found across the +bands, and we highlight star names of WDs in which polarisation +is convincingly detected in boldface in Tables2 and 3. Of the 23 +stars observed, BBCP has been definitely detected in 5 WDs. The +data for these stars are plotted in Fig. 1. +We observed three of the five WDs in which fields were de- +tected in order to fully confirm the weak field detections and to +check for possible variability. No variability is detected with con- +fidence. There are in addition two further WDs, WD 1434+437 +and WD 1533+469, in which marginal polarisation detections +have been obtained; these WDs await further observation to con- +Article number, page 3 of 6 + +A&A proofs: manuscript no. 45149corr +Table 3. Observing log of WDs. Detections are marked in boldface. +STAR +DATE +UT +JD – +Exp. +VI (%) +yyyy-mm-dd hh:mm +2400000 +(s) +B′ +V′ +R′ +WD 0005+395 +2022-07-06 +04:38 +59766.693 3900 −0.017±0.063 −0.082±0.056 +0.028±0.044 +WD 0010+543 +2022-07-05 +04:12 +59765.675 7100 −0.028±0.140 −0.011±0.129 +0.094±0.067 +WD 0028+035 +2002-07-06 +03:39 +59766.652 3300 −0.084±0.045 −0.051±0.050 −0.010±0.058 +WD 1251+366 +2022-06-27 +22:41 +59758.445 5200 −0.155±0.065 +0.006±0.063 +0.039±0.038 +WD 1315+222 +2022-06-28 +22:25 +59759.435 4200 +0.104±0.045 +0.182±0.052 +0.215±0.064 +2022-07-01 +22:24 +59762.434 4200 +0.084±0.040 +0.253±0.061 +0.199±0.044 +WD 1346+121 +2022-07-02 +22:43 +59763.446 6500 −0.508±0.093 +0.044±0.091 −1.074±0.054 +2022-07-04 +22:34 +59765.44 +6500 −0.691±0.109 +0.222±0.098 −1.256±0.062 +WD 1425+495 +2022-06-29 +22:22 +59760.432 4200 +0.012±0.045 −0.002±0.058 +0.018±0.038 +WD 1427-238 +2022-06-30 +23:11 +59761.466 5600 −0.038±0.114 +0.102±0.078 −0.044±0.061 +WD 1434+437 +2022-06-30 +00:04 +59760.503 5200 +0.231±0.077 −0.019±0.067 −0.019±0.059 +WD 1533+469 +2022-07-01 +01:02 +59761.543 6600 −0.065±0.127 −0.273±0.110 −0.195±0.052 +2022-07-03 +00:42 +59763.529 6600 +0.139±0.098 −0.300±0.090 −0.019±0.042 +WD 1601-073 +2022-07-05 +22:52 +59766.453 6800 −0.484±0.111 −0.386±0.106 +1.597±0.054 +WD 1612+092 +2022-06-28 +00:59 +59758.541 5100 +0.087±0.069 +0.033±0.052 +0.097±0.046 +WD 1702-016 +2022-06-30 +01:59 +59760.582 5300 +0.215±0.091 +0.087±0.086 −0.104±0.053 +WD 1737+798 +2022-06-29 +01:34 +59759.566 4500 −0.014±0.082 −0.085±0.093 −0.071±0.056 +WD 1746+450 +2022-06-28 +02:12 +59758.592 2600 +0.012±0.035 −0.049±0.032 −0.074±0.035 +WD 1800+508 +2022-07-05 +00:59 +59765.541 5600 −0.157±0.073 −0.047±0.061 +0.051±0.053 +WD 1853+775 +2002-07-06 +01:15 +59766.552 4800 −0.098±0.066 −0.680±0.068 −0.492±0.039 +WD 2058+550 +2022-07-02 +04:15 +59762.677 5000 +0.068±0.075 −0.052±0.070 +0.064±0.045 +WD 2109-295 +2022-07-01 +03:51 +59761.660 2200 +0.011±0.020 +0.036±0.034 −0.039±0.037 +WD 2152-280 +2022-06-28 +03:59 +59758.666 3600 +0.007±0.040 −0.055±0.041 −0.043±0.022 +WD 2211+372 +2022-07-02 +02:10 +59762.590 4400 +1.254±0.041 +0.703±0.054 +0.446±0.044 +2022-07-03 +04:24 +59763.683 4400 +1.333±0.038 +0.623±0.044 +0.285±0.040 +WD 2215+368 +2022-06-30 +04:12 +59760.675 4400 +0.187±0.083 +0.133±0.068 +0.101±0.044 +WD 2311-068 +2022-06-29 +04:38 +59759.693 2400 −0.011±0.028 +0.011±0.039 +0.032±0.032 +firm (or not) the fields that may have been detected. However, +a single pair of measurements does not probe all the possible +timescales of variation; in particular, our measurements require +integration of the order of one hour and so cannot probe all the +rotation periods that might result from the formation of a MWD +from a close binary. +With these new discoveries, we almost double the number of +DC WDs in which magnetic fields have been detected. One of +the new MWDs discovered, LP 684-16 = WD 1601–073, is quite +massive compared to most of the rest of the DC WDs observed. +Therefore, because of its relatively small radius, it has cooled +quite slowly, reaching only Teff = 4920 K, but has a computed +cooling time of 9.8 Gyr. It is probably the oldest magnetic WD of +any spectral type discovered so far. For comparison, according to +the parameters listed by Bagnulo & Landstreet (2021), the oldest +MWD in the 20 pc volume, in which such old MWDs are most +likely to be discovered, is WD 1008+290 = LHS 2229. It is a +DQpec star with an age of about 7.9 Gyr, almost 2 Gyr younger +than LP 684-16. +We note that no really large polarisation signals, such as that +exhibited by WD 1900+705 (see Table 1), are found. However, +the observed level of polarisation in three of the five definite +detections reaches the range 1.2 to 1.6%, so some of the fields +detected are probably quite strong. +3. Discussion and conclusions +We continue to detect MWDs in roughly one-fifth of the DC +sample observed. Considering that only relatively strong fields +can be detected in featureless stars, our results suggest that the +frequency of the occurrence of magnetic fields in older WDs +may be as high as 25 or even 30 %, consistent with the frequency +suggested by Bagnulo & Landstreet (2021). +Polarisation levels in the seven DC MWDs discovered +by Berdyugin et al. (2022) and in this paper range from +about 0.1 to 1.6 %. Using the order-of-magnitude estimator of +Bagnulo & Landstreet (2020) of a longitudinal field of 15 MG, +which leads to BBCP of the order of 1%, inferred fields ⟨Bz⟩ +are thus estimated to lie between perhaps 1 and 30 MG. From +this result, the fields ⟨|B|⟩ that we detect likely lie in the range of +roughly 3 to 200 MG. +Some of the fields produce a polarisation with the same sign +in the three filter bands, while in other stars we detect a polar- +isation that reverses sign between one filter band and another +(Fig. 1). Similar behaviour was found in our earlier survey data +(Berdyugin et al. 2022) as well as in other strongly magnetic old +WDs (Angel & Landstreet 1971; Angel et al. 1974, 1975; Putney +1995). +Article number, page 4 of 6 + +Andrei V. Berdyugin et al.: Discovery of magnetic fields in five DC white dwarfs +Fig. 1. Wavelength dependence of circular polarisation detected for five +targets (> 3σ confidence level). A wide variety of polarisation behaviour +is observed. Horizontal bars in the bottom panel show the FWHM of the +B′V′R′ filter passbands. +We carried out a second observation for three of our five new +discoveries and for one suspected candidate. The confirming ob- +servations were obtained between one and three days after the +discovery observations. For each of these four stars, the repeated +observation confirms the presence of the magnetic field first de- +tected, except for one R′ observationof WD 1533+469.In no case +do we detect clearly significant variability; we note, however, that +our repeated measurements probe only a very limited range of +timescales. +We find that, in practice, a magnetic field can be reliably +detected from BBCP at the polarization levels of approximately +0.2% with our broadband filter polarimeter on a 2.5 m telescope +in a DC WD of G ∼ 17. We would gain about a factor of 3 in +precision, or a 2.5 magnitude increase in limiting magnitude, by +going to an 8 m telescope. +To summarise the current statistical situation, we com- +bine the results from the 20 pc volume magnetic field survey +(Bagnulo & Landstreet 2021), our exploratory observing run +(Berdyugin et al. 2022), and this work. We have collected lit- +erature data on and surveyed 30 young DC stars, of which 7 have +been found to host magnetic fields, and 43 old DCs, of which 6 +are magnetic. +As more detections of DC MWDs are made, especially in +the context of volume-limited surveys such as ours, comparisons +with magnetic data for other types of WDs will at first be ham- +pered by our current inability to assign precise field strength +values to magnetic DC WDs. However, a first comparison can be +made between the overall level of polarisation observed in young +and old DC MWDs, which may be taken as an indicator of the +evolution of the overall field strength with cooling age between +these two populations. In the currently small sample, it appears +that larger polarisation levels (say, above Stokes V/I >∼ 1 %) ap- +pear to be at least as common in the old DC MWD population +as in the young group. There is no clear signal of Ohmic field +strength decay. However, the available sample is still very small, +and as our sample increases we can hope to obtain a statistically +more significant constraint and carry out modelling to provide +more accurate field strength estimates corresponding to observed +polarisation. Such surveys need to be carried on until a clearer +statistical view of the magnetism in DC WDs is obtained. +Acknowledgements. Based on observations made with the Nordic Optical Tele- +scope, owned in collaboration by the University of Turku and Aarhus University, +and operated jointly by Aarhus University, the University of Turku and the Uni- +versity of Oslo, representing Denmark, Finland and Norway, the University of +Iceland and Stockholm University at the Observatorio del Roque de los Mucha- +chos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. DIPol-UF +is a joint effort between University of Turku (Finland) and Leibniz Institute for +Solar Physics (Germany). We acknowledge support from the Magnus Ehrnrooth +foundation and the ERC Advanced Grant HotMol ERC-2011-AdG-291659. JDL +acknowledges the financial support of the Natural Sciences and Engineering +Research Council of Canada (NSERC), funding reference number 6377-2016. +4. Data Availability +All raw data and calibrations are available on request from the +authors. +References +Angel, J. R. P., Borra, E. F., & Landstreet, J. D. 1981, ApJS, 45, 457 +Angel, J. R. P., Hintzen, P., & Landstreet, J. D. 1975, ApJ, 196, L27 +Angel, J. R. P., Hintzen, P., Strittmatter, P. A., & Martin, P. G. 1974, ApJ, 190, +L71 +Angel, J. R. P. & Landstreet, J. D. 1971, ApJ, 164, L15 +Bagnulo, S. & Landstreet, J. D. 2019, MNRAS, 486, 4655 +Bagnulo, S. & Landstreet, J. D. 2020, A&A, 643, A134 +Bagnulo, S. & Landstreet, J. D. 2021, MNRAS, 507, 5902 +Bagnulo, S. & Landstreet, J. D. 2022, ApJ, 935, L12 +Bédard, A., Bergeron, P., Brassard, P., & Fontaine, G. 2020, ApJ, 901, 93 +Berdyugin, A. V., Piirola, V., Bagnulo, S., Landstreet, J. D., & Berdyugina, S. V. +2022, A&A, 657, A105 +Berdyugina, S. V., Berdyugin, A. V., & Piirola, V. 2007, Phys. Rev. Lett., 99, +091101 +Bergeron, P., Wesemael, F., Fontaine, G., et al. 2021, AJ, 162, 188 +Blouin, S., Dufour, P., Thibeault, C., & Allard, N. F. 2019, ApJ, 878, 63 +Ferrario, L., de Martino, D., & Gänsicke, B. T. 2015, Space Sci. Rev., 191, 111 +Article number, page 5 of 6 + +A&A proofs: manuscript no. 45149corr +Fossati, L., Bagnulo, S., Mason, E., & Landi Degl’Innocenti, E. 2007, in As- +tronomical Society of the Pacific Conference Series, Vol. 364, The Future +of Photometric, Spectrophotometric and Polarimetric Standardization, ed. +C. Sterken, 503 +Gentile Fusillo, N. P., Tremblay, P. E., Cukanovaite, E., et al. 2021, MNRAS, +508, 3877 +Gentile Fusillo, N. P., Tremblay, P. E., Jordan, S., et al. 2018, MNRAS, 473, 3693 +Isern, J., García-Berro, E., Külebi, B., & Lorén-Aguilar, P. 2017, ApJ, 836, L28 +Kawka, A., Vennes, S., Schmidt, G. D., Wickramasinghe, D. T., & Koch, R. 2007, +ApJ, 654, 499 +Kemp, J. C. 1970, ApJ, 162, 169 +Kemp, J. C., Swedlund, J. B., Landstreet, J. D., & Angel, J. R. P. 1970, ApJ, 161, +L77 +Landstreet, J. D. 1987, MNRAS, 225, 437 +Piirola, V., Kosenkov, I. A., Berdyugin, A. V., Berdyugina, S. V., & Poutanen, J. +2020, The Astronomical Journal, 161, 20 +Putney, A. 1995, ApJ, 451, L67 +Siebenmorgen, R., Voshchinnikov, N. V., & Bagnulo, S. 2014, A&A, 561, A82 +Tremblay, P. E., Fontaine, G., Freytag, B., et al. 2015, ApJ, 812, 19 +Article number, page 6 of 6 + diff --git a/1NE2T4oBgHgl3EQfigeU/content/tmp_files/load_file.txt b/1NE2T4oBgHgl3EQfigeU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d991ad6b9c14abb066794cad1c42001bb5c64bdc --- /dev/null +++ b/1NE2T4oBgHgl3EQfigeU/content/tmp_files/load_file.txt @@ -0,0 +1,756 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf,len=755 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='03959v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='SR] 10 Jan 2023 Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 45149corr ©ESO 2023 January 11, 2023 Discovery of magnetic fields in five DC white dwarfs Andrei V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Berdyugin1, Vilppu Piirola1, Stefano Bagnulo2, John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Landstreet2, 3, and Svetlana V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Berdyugina4, 5, 6 1 Department of Physics and Astronomy, FI-20014 University of Turku, Finland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' e-mail: andber@utu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='fi 2 Armagh Observatory & Planetarium, College Hill, Armagh BT61 9DG, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 3 Department of Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 4 Leibniz-Institut für Sonnenphysik (KIS), Schöneckstr 6, Freibirg, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 5 IRSOL Istituto Ricerche Solari “Aldo e Cele Daccò", Faculty of Informatics, Università della Svizzera italiana, Via Patocci 57, Locarno, Switzerland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 6 Euler Institute, Faculty of Informatics, Università della Svizzera italiana, Via la Santa 1, 6962 Lugano, Switzerland Received October 7, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' accepted October 27, 2022 ABSTRACT About half of white dwarfs (WDs) evolve to the DC state as they cool;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' the others become DQ or (temporarily?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=') DZ WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The recent magnetic survey of the local 20 pc volume has established a high frequency of magnetic fields among WDs older than 2–3 Gyr, demonstrating that in low- and average-mass WDs, the effects of magnetism become more common as they age, and the fields on average become stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, the available statistics of WDs older than about 5 Gyr do not clearly establish how fields evolve beyond this age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We are carrying out a survey to clarify the occurrence of magnetism in DC-type WDs in order to better understand this late evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We use broadband filter polarimetry, arguably the most efficient way to detect magnetic fields in featureless WDs via continuum circular polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Here we report the discovery of a magnetic field in five DC WDs (of 23 observed), almost doubling the total sample of known magnetic WDs belonging to the DC spectral class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' White dwarfs – Stars: magnetic fields – polarization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Introduction Single stars of M <∼ 8M⊙ evolve to become white dwarfs (WDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The descendants of these single stars of intermediate mass pro- vide most of the population of WDs, concentrated around the mean mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='6M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' A smaller fraction of current WDs were also formed from close binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Some of these systems eventually merged to form a single collapsed remnant, frequently of a significantly larger mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' others ended their nuclear lifetimes as double WD binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Once formed, the evolution of a WD is normally to cool slowly over several gigayears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Cooling is a fairly complex process even for single-star evolution, both to observe and to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Observationally, young hot WDs usually show strong spectra of H (DA WDs), He (DB WDs), or sometimes C (DQs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' As they cool, spectral lines of the dominant elements H or He become weaker: He lines vanish at about 11000 K, H lines around 5000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' In parallel with this general evolution, WDs may (temporarily) show lines of metals such as Mg, Si, Ca, and/or Fe (DZ, DAZ, and DZA stars), and some have spectra dominated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Below about 5000 K, about one-quarter of WDs have very weak Hα, another quarter show spectral lines of metals (especially Ca ii) or of C2, and the remaining half show essentially featureless spectra (DC WDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Bagnulo & Landstreet 2021, Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' It appears that the dominant element(s) in the atmosphere can change as cooling occurs, for example due to gravitational diffusion, development of convection, and accretion of circumstellar planetary debris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' One of the physical effects adding complexity to our ef- forts to understand WD evolution is that a significant frac- tion, about 20-25%, of WDs in the local volume near the Sun (Bagnulo & Landstreet 2021) possess detectable surface mag- netic fields (this high frequency was already suggested on the basis of literature reports of fields in nine WDs in the 13 pc vol- ume by Kawka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The fields observed at the surface range in strength, measured by the mean surface field ⟨|B|⟩, from tens of kG to hundreds of MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Such fields can significantly affect WD evolution by altering or suppressing surface convection and internal shear, and by transferring angular momentum between internal layers or during accretion or mass loss (see for example Tremblay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The fields may also introduce additional forces into envelope and atmosphere layers, altering their hy- drostatic structure from that expected when magnetic effects are absent (Landstreet 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' For WDs formed by single-star evolution, which generates most of the large populationsof WDs with masses around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='6M⊙, it has become clear that recently formed WDs (with cooling ages of less than, say, 1 Gyr) are very rarely detectably mag- netic, and when they are magnetic, the fields are usually very weak (Bagnulo & Landstreet 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' As WDs cool, fields begin to appear more frequently and usually become stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' In WDs older than 3 or 4 Gyr, megagauss-scale fields are not uncommon (Bagnulo & Landstreet 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The observed evolution in magnetic field frequency and strength of normal-mass WDs for the first few gigayears of cool- ing may be understood as a slow emergence – as a result of field relaxation to the stellar surface – of the internal fields present in the degenerate cores of the WD precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' An additional contribution to observed surface fields may be due to magnetic fields generated during cooling by a dynamo that acts during the period when the core of the WD is crystallising (Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Beyond the end of crystalli- Article number, page 1 of 6 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 45149corr sation, the only identified evolution mechanisms are continued field relaxation and Ohmic decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Observationally, however, after about 5 Gyr of WD cooling, we have very limited information with which to guide and con- front theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Only small survey samples constrain observed field evolution on cool WDs, such as DQ WDs, where C2 bands show no polarization in strong fields (Berdyugina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Partic- ularly little is known about the magnetic fields of DC WDs, in which no spectral features are seen at all, leading to the ques- tions of whether field strength begins to decay Ohmically and whether the frequency of surface fields continues to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Data that could help us answer these questions are very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' For WDs within 20 pc of the Sun (the 20 pc volume sample), Bagnulo & Landstreet (2021) showed that of 31 DC WDs, only 4 are magnetic white dwarfs (MWDs), and that only 4 of 24 WDs of any spectral class older than ∼ 6 Gyr are magnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' These data are obviously too limited to clearly describe the evolution of fields in these old WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Previous surveys have provided almost no information about magnetic fields in DC WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Fields in such stars cannot be de- tected through the magnetic splitting of spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' They can only be detected via the observation of continuum circular po- larisation (CCP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Kemp 1970), a method of observation hardly employed since the 1970s (Angel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Remarkably, most CCP observations have led to the discovery of magnetic fields in stars that are not featureless but in which the magnetic field is strong enough to shift and broaden spectral lines in a such a way as to make their intensity spectra unrecognisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Only seven featureless DC WDs are presently known to be mag- netic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Five of them were discovered only in the last couple of years (Bagnulo & Landstreet 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Be- fore these results, the only known magnetic DC stars were G195- 19 and G111-49, discovered respectively by Angel & Landstreet (1971) and Putney (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' To improve our knowledge of the magnetic fields in the latest stages of stellar evolution, we have started a volume-limited sur- vey of DC stars in the local 33 pc volume,which is about4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 times larger than the previously explored 20 pc volume and should have a correspondingly larger sample of DCs and DC MWDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' With this sample we expect to find enough DC MWDs to delineate the evolution of their magnetic fields, both in the WDs with He-rich atmospheres that become DCs as soon as their effective tempera- tures reach about 11 000 K (‘young’ DCs), and in DC WDs with Teff below about 5000 K, with cooling ages of around 4 Gyr or more (‘old’ DCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Observations Almost all known MWDs have been discovered via the mag- netic (Zeeman) splitting of spectral lines, observed in stellar flux spectra, or via the Zeeman polarisation of spectral features (Ferrario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Using these methods, fields of a few kG up to 1 GG can be reliably detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, these techniques cannot be used to measure fields in WDs that lack spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' For such stars, it is necessary to rely on continuum polarisation, which Kemp (1970) showed should occur in radiation from a magnetized emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The value of this effect was confirmed by the discovery of a very strong field in the bright WD Grw+70 8247 = WD 1900+705 through the detection of broadband circular po- larisation (BBCP) by Kemp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Broadband circular polarisation is a relatively weak effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Bagnulo & Landstreet(2020) have estimated that a field of ⟨Bz⟩ ∼ 15 MG is required to produce BBCP of order 1 % in optical radiation from a cool WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, with a sensitive polarimeter, especially one with a very stable and well-established zero point, it is possible in principle to detect polarisation of 10−4 or less, corresponding to ∼ 100 kG fields in ‘sufficiently bright’ WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' To detect and measure broadband continuum polarisation, one uses either spectropolarimetry or filter polarimetry with broad, photometry-like filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' It is very difficult to establish the zero point with sufficient accuracy below polarisation levels of the order of 10−3 in spectropolarimetric measurements ofthe con- tinuum (Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Siebenmorgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' therefore, in DC WDs, only megagauss-scale fields can be detected in this way (Bagnulo & Landstreet 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' In contrast, broadband filter polarimeters can be very stable, and instrumental polarisation can be calibrated at the 10−5 level, so detections with such instru- ments of fields of hundreds of kilogauss are in practice limited by the telescope aperture and WD brightness (Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The search for magnetic fields of a fraction of 1 MG or stronger in DC WDs that is reported here was carried out with the DIPol-UF broadband filter polarimeter (Piirola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2020) mounted on the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 m Nordic Optical Telescope (NOT) at the Observatorio del Roque de los Muchachos on the island of La Palma, in the Canaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' This instrument obtains simultaneous circular polarisation (normalized Stokes V/I) measurements in three filter bands isolated by dichroic mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The passbands are centred at about 4450 Å (the B′ band), 5400 Å (the V′ band), and 6400Å (the R′ band) with full widths at half maximum (FWHMs) of 1140, 750, and 960 Å , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' With this instrument on the NOT, we can detect a polarisation degree at the 3σ level of ∼ 10−4 for the Gaia G-band magnitude G ∼ 12, down to a degree of ∼ 10−3 at G ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' This instrument and the filter system are discussed in more detail in our previous paper, which reports the results of the first part of our search for magnetic fields in DC WDs (Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Here we report observations of 23 DC WDs and discovery of 5 new DC MWDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We note that our survey includes nine young DCs of He-rich atmospheres with 11000 <∼ Teff <∼ 5000 K and 14 old WDs with Teff <∼ 5000 K and ages τ >∼ 4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The stars are selected from available classifications and with help from Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Our new observations were obtained between June 27 and July 5, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Instrumental polarisation and alignment of the polarimetric optics During our observing run we obtained seven observations of seven different bright nearby stars, which are believed to have zero circular polarisation, to check for instrumental polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' These observations are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' As in our previous run in July 2021, the high S/N measure- ments of non-polarised stars yield the instrumental polarisation to a precision better than 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' In the B′V′R′ bands, the values of Stokes V/I are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0121 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0109 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0005 %, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0084 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' These are very close to the values obtained in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The instrumental polarisation was sub- tracted from the observed polarisation of all targets, including the measurements of the standard stars reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' In addition, we obtained one measurement of the well-known MWD WD 1900+705, which appears to show a signal of cir- cular polarisation that is nearly constant with time (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Bagnulo & Landstreet2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Our new measurementis compared in Table 1 to one of the same star that we made during the July 2021 run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The agreement is very satisfactory and demonstrates that we can obtain measurements that are precise at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='02 % Article number, page 2 of 6 Andrei V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' : Discovery of magnetic fields in five DC white dwarfs Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Observing log of bright non-polarised standard stars and the highly polarised MWD WD 1900+705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Polarisation values are given assuming as instrumental polarisation the values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0121±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0109±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0005 %, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0084±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004 % in the B′, V′, and R′ filters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' For comparison, we report the polarisation values of WD 1900+705 measured in our 2021 and 2022 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' STAR G DATE UT JD – Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' VI (%) yyyy-mm-dd hh:mm 2400000 (s) B′ V′ R′ HD 107146 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 2022-06-27 21:23 59758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='391 1680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0005±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0006±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0001±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0006 HD 115043 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='7 2022-06-28 21:21 59759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='390 1520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0005±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0001±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0012 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004 HD 122652 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0 2022-06-29 21:18 59760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='387 1520 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0012±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0015±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0019±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0014 HD 122676 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='1 2022-06-30 21:17 59761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='387 1520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0000±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0003±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0018±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0008 HD 124694 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0 2022-07-01 21:16 59762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='386 1520 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0012±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0014±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0002±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0007 HD 135891 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 2022-07-02 21:18 59763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='387 1520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0014±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0006±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0007 HD 117860 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 2022-07-03 21:16 59764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='386 1520 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0004±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0002±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0005±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0006 WD 1900+705 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 2021-07-02 22:22 59398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='432 640 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='756±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='016 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='604±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='016 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='827±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='019 2022-07-02 00:25 59762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='518 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='789±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='016 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='602±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='019 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='838±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='018 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Programme stars and their main physical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Star names in boldface identify WDs in which fields were discovered during the observations reported in this paper (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' STAR G d Teff log g M Age Atmosphere and ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (pc) (K) c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (M⊙) (Gyr) WD 0005+395 LP 240-30 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='4 4680 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='40 DC, H ProbWD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='70 (1,3) WD 0010+543 LSR J0013+5437 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 4123 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='46 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='08 DC, (2, H assumed) WD 0028+035 PB 6002 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 6548 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='40 DC, (2, H assumed) WD 1251+366 LP 267-311 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 4445 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='78 DC, He (1) WD 1315+222 LP 378-956 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='7 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 6235 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='61 DCH, He (1) WD 1346+121 LP 498-66 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 4150 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='58 DCH, He (1) WD 1425+495 CSO 649 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 6895 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='77 DC, (2, H assumed) WD 1427−238 LP 857-45 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='6 4866 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='40 DC, (2, H assumed) WD 1434+437 LP 221-217 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 4685 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='30 DC, H-He (1) WD 1533+469 LP 176-60 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 4310 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='45 DC?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', H (1) WD 1601−073 LP 684-16 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 4920 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='94 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='83 DCH, (2, H assumed) WD 1612+092 LSPM J1614+0906 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 4775 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='57 DC, H (1) WD 1702−016 LP 626-29 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 4700 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='50 DC, (2, H assumed) WD 1737+798 LP 24-66 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 5535 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='72 DC, He (1) WD 1746+450 GD 366 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 9331 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='72 DC, (2, H assumed) WD 1800+508 LP 139-38 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0 4635 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='12 DC, He-H (1) WD 1853+775 LP 25-7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 4850 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='63 DCH, He (1) WD 2058+550 LSR J2059+5517 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='7 4415 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='53 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='15 DC, H-He (1) WD 2109−295 EC 21096-2934 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 9260 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='78 DC, He-H (3) WD 2152−280 LP930-61 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 5220 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='68 DC, He (1) WD 2211+372 LP 287-35 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 6345 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='56 DC?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='H, He (1) WD 2215+368 LP 287-39 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 4485 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='80 DC, H (1) WD 2311−068 G 157-34 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 7360 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='31 DC, He (1) Key to references: 1: Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2: Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 3: Bergeron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Where not found in these references, ages have been interpolated using the tables from Bédard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' level for a G = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 star with about 10 minutes of exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' This shows that the alignment of our polarimetric optics is stable over a few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Results The WDs observed during our 2022 June-July run are listed in Table 2, with their G magnitudes, distances, physical parameters, cooling ages, and some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Physical parameters were obtained from various studies, cited in the table’s notes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' cooling ages are interpolated from the online cooling data provided by the Montreal group (Bédard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The observations are described in the log in Table 3, which gives dates, integration times, and the polarisation data in the three filter bands for each WD observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We list measured BBCP values in boldface if non-zero polarisation is detected at above the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We consider that real polarisation has been detected if a consistent picture of detection is found across the bands, and we highlight star names of WDs in which polarisation is convincingly detected in boldface in Tables2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Of the 23 stars observed, BBCP has been definitely detected in 5 WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The data for these stars are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We observed three of the five WDs in which fields were de- tected in order to fully confirm the weak field detections and to check for possible variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' No variability is detected with con- fidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' There are in addition two further WDs, WD 1434+437 and WD 1533+469, in which marginal polarisation detections have been obtained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' these WDs await further observation to con- Article number, page 3 of 6 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 45149corr Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Observing log of WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Detections are marked in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' STAR DATE UT JD – Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' VI (%) yyyy-mm-dd hh:mm 2400000 (s) B′ V′ R′ WD 0005+395 2022-07-06 04:38 59766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='693 3900 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='017±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='063 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='082±0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='623±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='285±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='040 WD 2215+368 2022-06-30 04:12 59760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='675 4400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='187±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='133±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='101±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='044 WD 2311-068 2022-06-29 04:38 59759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='693 2400 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='011±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='011±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='032±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='032 firm (or not) the fields that may have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, a single pair of measurements does not probe all the possible timescales of variation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' in particular, our measurements require integration of the order of one hour and so cannot probe all the rotation periods that might result from the formation of a MWD from a close binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' With these new discoveries, we almost double the number of DC WDs in which magnetic fields have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' One of the new MWDs discovered, LP 684-16 = WD 1601–073, is quite massive compared to most of the rest of the DC WDs observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Therefore, because of its relatively small radius, it has cooled quite slowly, reaching only Teff = 4920 K, but has a computed cooling time of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='8 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' It is probably the oldest magnetic WD of any spectral type discovered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' For comparison, according to the parameters listed by Bagnulo & Landstreet (2021), the oldest MWD in the 20 pc volume, in which such old MWDs are most likely to be discovered, is WD 1008+290 = LHS 2229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' It is a DQpec star with an age of about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='9 Gyr, almost 2 Gyr younger than LP 684-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We note that no really large polarisation signals, such as that exhibited by WD 1900+705 (see Table 1), are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, the observed level of polarisation in three of the five definite detections reaches the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='6%, so some of the fields detected are probably quite strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Discussion and conclusions We continue to detect MWDs in roughly one-fifth of the DC sample observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Considering that only relatively strong fields can be detected in featureless stars, our results suggest that the frequency of the occurrence of magnetic fields in older WDs may be as high as 25 or even 30 %, consistent with the frequency suggested by Bagnulo & Landstreet (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Polarisation levels in the seven DC MWDs discovered by Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' (2022) and in this paper range from about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='6 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Using the order-of-magnitude estimator of Bagnulo & Landstreet (2020) of a longitudinal field of 15 MG, which leads to BBCP of the order of 1%, inferred fields ⟨Bz⟩ are thus estimated to lie between perhaps 1 and 30 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' From this result, the fields ⟨|B|⟩ that we detect likely lie in the range of roughly 3 to 200 MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Some of the fields produce a polarisation with the same sign in the three filter bands, while in other stars we detect a polar- isation that reverses sign between one filter band and another (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Similar behaviour was found in our earlier survey data (Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022) as well as in other strongly magnetic old WDs (Angel & Landstreet 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Angel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1974, 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Putney 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Article number, page 4 of 6 Andrei V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' : Discovery of magnetic fields in five DC white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Wavelength dependence of circular polarisation detected for five targets (> 3σ confidence level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' A wide variety of polarisation behaviour is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Horizontal bars in the bottom panel show the FWHM of the B′V′R′ filter passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We carried out a second observation for three of our five new discoveries and for one suspected candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' The confirming ob- servations were obtained between one and three days after the discovery observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' For each of these four stars, the repeated observation confirms the presence of the magnetic field first de- tected, except for one R′ observationof WD 1533+469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='In no case do we detect clearly significant variability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' we note, however, that our repeated measurements probe only a very limited range of timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We find that, in practice, a magnetic field can be reliably detected from BBCP at the polarization levels of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='2% with our broadband filter polarimeter on a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 m telescope in a DC WD of G ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We would gain about a factor of 3 in precision, or a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content='5 magnitude increase in limiting magnitude, by going to an 8 m telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' To summarise the current statistical situation, we com- bine the results from the 20 pc volume magnetic field survey (Bagnulo & Landstreet 2021), our exploratory observing run (Berdyugin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022), and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We have collected lit- erature data on and surveyed 30 young DC stars, of which 7 have been found to host magnetic fields, and 43 old DCs, of which 6 are magnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' As more detections of DC MWDs are made, especially in the context of volume-limited surveys such as ours, comparisons with magnetic data for other types of WDs will at first be ham- pered by our current inability to assign precise field strength values to magnetic DC WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, a first comparison can be made between the overall level of polarisation observed in young and old DC MWDs, which may be taken as an indicator of the evolution of the overall field strength with cooling age between these two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' In the currently small sample, it appears that larger polarisation levels (say, above Stokes V/I >∼ 1 %) ap- pear to be at least as common in the old DC MWD population as in the young group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' There is no clear signal of Ohmic field strength decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' However, the available sample is still very small, and as our sample increases we can hope to obtain a statistically more significant constraint and carry out modelling to provide more accurate field strength estimates corresponding to observed polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Such surveys need to be carried on until a clearer statistical view of the magnetism in DC WDs is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Based on observations made with the Nordic Optical Tele- scope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the Uni- versity of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Mucha- chos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' DIPol-UF is a joint effort between University of Turku (Finland) and Leibniz Institute for Solar Physics (Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' We acknowledge support from the Magnus Ehrnrooth foundation and the ERC Advanced Grant HotMol ERC-2011-AdG-291659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' JDL acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 6377-2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Data Availability All raw data and calibrations are available on request from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' References Angel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Borra, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1981, ApJS, 45, 457 Angel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Hintzen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1975, ApJ, 196, L27 Angel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Hintzen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Strittmatter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Martin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1974, ApJ, 190, L71 Angel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1971, ApJ, 164, L15 Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2019, MNRAS, 486, 4655 Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2020, A&A, 643, A134 Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2021, MNRAS, 507, 5902 Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' & Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022, ApJ, 935, L12 Bédard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Bergeron, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Brassard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Fontaine, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2020, ApJ, 901, 93 Berdyugin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Piirola, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Berdyugina, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2022, A&A, 657, A105 Berdyugina, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Berdyugin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Piirola, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2007, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', 99, 091101 Bergeron, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Wesemael, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Fontaine, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2021, AJ, 162, 188 Blouin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Dufour, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Thibeault, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Allard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2019, ApJ, 878, 63 Ferrario, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', de Martino, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Gänsicke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2015, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', 191, 111 Article number, page 5 of 6 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 45149corr Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Mason, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Landi Degl’Innocenti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2007, in As- tronomical Society of the Pacific Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 364, The Future of Photometric, Spectrophotometric and Polarimetric Standardization, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' Sterken, 503 Gentile Fusillo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Tremblay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Cukanovaite, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2021, MNRAS, 508, 3877 Gentile Fusillo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Tremblay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Jordan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2018, MNRAS, 473, 3693 Isern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', García-Berro, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Külebi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Lorén-Aguilar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2017, ApJ, 836, L28 Kawka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Vennes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Schmidt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Wickramasinghe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Koch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2007, ApJ, 654, 499 Kemp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1970, ApJ, 162, 169 Kemp, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Swedlund, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Angel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1970, ApJ, 161, L77 Landstreet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1987, MNRAS, 225, 437 Piirola, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Kosenkov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Berdyugin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Berdyugina, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Poutanen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2020, The Astronomical Journal, 161, 20 Putney, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 1995, ApJ, 451, L67 Siebenmorgen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Voshchinnikov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', & Bagnulo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2014, A&A, 561, A82 Tremblay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Fontaine, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', Freytag, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} +page_content=' 2015, ApJ, 812, 19 Article number, page 6 of 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfigeU/content/2301.03959v1.pdf'} diff --git a/39AzT4oBgHgl3EQfffxn/content/tmp_files/2301.01453v1.pdf.txt b/39AzT4oBgHgl3EQfffxn/content/tmp_files/2301.01453v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..511695241c4b4213b4b426a977e68d511d20cee2 --- /dev/null +++ b/39AzT4oBgHgl3EQfffxn/content/tmp_files/2301.01453v1.pdf.txt @@ -0,0 +1,988 @@ +1 +Information-Theoretic Secure Key Sharing for Wide-Area Mobile +Applications +Guyue Li, Member, IEEE, Hongyi Luo, Jiabao Yu, Aiqun Hu, Senior Member, IEEE and Jiangzhou Wang, Fellow, IEEE +With the rapid growth of handheld devices in the internet +of things (IoT) networks, mobile applications have become +ubiquitous in everyday life. As technology is developed, so do +also the risks and threats associated with it, especially in the +forthcoming quantum era. Existing IoT networks, however, lack a +quantum-resistant secret key sharing scheme to meet confidential +message transmission demands in wide-area mobile applications. +To address this issue, this article proposes a new scheme, channel +reciprocity (CR) based quantum key distribution (QKD) CR- +QKD, which accomplishes the goal of secret key sharing by +combining emerging techniques of QKD and CR-based key +generation (CRKG). Exploiting laws of quantum physics and +properties of wireless channels, the proposed scheme is able +to ensure the secrecy of the key, even against computationally +unbounded adversaries. The basic mechanism is elaborated for +a single-user case and it is extended into a multi-user case by +redesigning a multi-user edge forwarding strategy. In addition, to +make CR-QKD more practical, some enhancement strategies are +studied to reduce the time delay and to improve the secret key +generation rate in a secure manner. A prototype of CR-QKD +is demonstrated in a metropolitan area network, where secret +keys are shared between two remote IoT devices that are roughly +fifteen kilometers apart from each other. The experimental results +have verified that CR-QKD allows a secret key rate of 424 bits +per second with a retransmission rate of 2.1%. +Index Terms—Secret key generation, physical layer security, +quantum key distribution, wide-area mobile applications, Inter- +net of Things +I. INTRODUCTION +Recent years have witnessed a remarkable growth in the +number and variety of mobile devices and applications in +the Internet of Things (IoT) networks. The flourish of IoT, +however, has resulted in the generation of a substantial amount +of private messages exchanged over public channels, which +has grabbed one’s attention. Unfortunately, IoT devices are +susceptible to various threats and security challenges, which +pose hazards for the advancement of IoT in sensitive fields, +such as smart homes, unmanned vehicles, e-health, and mili- +tary networks [1]. In order to avoid being revealed to a third +Guyue Li and Hongyi Luo are with the School of Cyber Science +and Engineering, Southeast University, Nanjing 210096, China (e-mail: +guyuelee@seu.edu.cn; hongyiluo@seu.edu.cn). +Jiabao Yu is with the Purple Mountain Laboratories, Nanjing 210096, China +(e-mail:yujiabao@pmlabs.com.cn). +Aiqun Hu is with National Mobile Communications Research Laboratory, +Southeast University, Nanjing 210096, China (e-mail: aqhu@seu.edu.cn). +Jiangzhou Wang is with the School of Engineering, University of Kent, +Canterbury CT2 7NT, U.K. Email: (e-mail: j.z.wang@kent.ac.uk). +Guyue Li and Aiqun Hu are also with Purple Mountain Laboratories, +Nanjing 210096, China. +party, a message is usually encrypted using a secret key shared +among the communicating devices. Thus, a key prerequisite +of achieving IoT network security is secret key sharing that +avoids eavesdropper interception [2]. +In a classic cryptographic scheme, two legitimate parties, +namely Alice and Bob use the public-key cryptosystem (PKC) +for key distribution. It is extremely difficult for a third party, +namely Eve, to derive the private key or message compu- +tationally, due to the intractability of certain mathematical +problems used in encryption algorithms. However, the emerg- +ing quantum computing technology has the potential to make +some previously-intractable problems tractable [3]. Thus, the +security of computational security-based key distribution will +be rendered insecure by substantial progress in quantum +computing in the coming years, which necessitates the study of +alternative solutions that do not rely on computational security. +In this context, much attention has been paid to emerging +techniques, such as quantum key distribution (QKD) [4] and +channel reciprocity-based key generation (CRKG) [5], which +can provide secret key sharing service with information- +theoretic security, also known as unconditional security or +physical security. +• QKD is a well-known quantum-resistant mechanism, +which distributes secret keys to distant parties by trans- +mitting single photon through a quantum channel [6]. +Employing the laws of quantum physics, QKD can de- +tect eavesdroppers during the key generation process, +in which unauthorized observation of quantum commu- +nication induces a discernible increase of errors. This +sensitivity to eavesdropping makes QKD possible to en- +sure the secrecy of the key, even against computationally +unbounded adversaries. +• CRKG is built on the basis of channel reciprocity, which +means that the channel responses of the forward and +backward communication links are very similar in a time +division duplex (TDD) system. In addition, the dynamic +and complex wireless communication environment makes +the channel responses change over time and hard to +predict. Therefore, legitimate users can share a pair of +common randomness from their radio channel measure- +ments. Since CRKG does not require assistance from a +third party nor expensive infrastructure, it has recently +emerged as a new paradigm that provides a lightweight +and information-theoretic secure key sharing solution for +decentralized or device-to-device sensor applications [7]. +Table I summarizes these typical secret key distribution +arXiv:2301.01453v1 [cs.IT] 4 Jan 2023 + +2 +TABLE I +A SUMMARY AND COMPARATION OF TYPICAL SECRET KEY DISTRIBUTION METHODS +Method +Metric +Security Level +Mobility Support +Distribution Distance +User Cost +PKC +Computational secure +Middle +Long +Middle +QKD +Information theoretically secure +Weak +Long +High +CRKG +Information theoretically secure +Strong +Short +Low +CR-QKD +Information theoretically secure +Strong +Long +Low +methods, and identify their characteristics from perspectives of +security level, mobility support, distribution distance and user +cost. We find that although the separate construction of QKD +and CRKG can be supported in the physical layer, there is no +investigation of a secret key sharing scheme for the security +demands from remote mobile devices. Although point-to-point +connections are suitable to form a backbone quantum core +network to bridge long distances, they are less suitable to +provide the last-mile service needed to give a multitude of +users access to this QKD infrastructure [6]. Similarly, despite +many research efforts in the field of CRKG, its widespread +application is unfortunately hindered by the short distance +between transceivers. With a rapid growth of handheld devices, +wide-area mobile applications, such as remote environmental +and elderly monitoring, have become an inseparable part of +IoT networks. A new architecture needs to be developed +where end-users between two access networks are connected +to a metro network, thus realizing unconditionally secure key +sharing in a more cost-effective and flexible manner. +In this article, we introduce and experimentally demonstrate +the concept of a ‘channel reciprocity-aided quantum key +distribution (CR-QKD)’ based on simple and cost-effective +telecommunication technologies. This scheme can expand the +scope of QKD to IoT networks and therefore vastly broaden +users’ appeal. The contributions of this article are three-fold: +• We introduce a novel secret key sharing architecture, re- +ferred to as CR-QKD, which bridges a backbone quantum +core network and IoT users by exploiting the technique +of CRKG to provide the last-mile service. CR-QKD is +information-theoretically secure and it does not require +IoT users to be equipped with expensive quantum infras- +tructures for exchanging secret keys, thereby significantly +reducing the hardware requirements. +• We propose a multi-user mechanism to realize the con- +cept of CR-QKD with an elaborate design of key align- +ment. We also identify challenges that arise due to the +hybrid architecture of CR-QKD from the perspective +of feasibility and security, respectively. Countermeasures +have been studied to reduce the time delay and to improve +the secret key generation rate in a secure manner. +• We implement a prototype CR-QKD system in a +metropolitan area network, in which secret keys are +shared between two remote IoT devices that are roughly +fifteen kilometers apart from each other. The experimental +results have verified that CR-QKD can provide a secret +key rate of 424 bits per second with a retransmission rate +of 2.1%. +II. AN OVERVIEW OF THE CR-QKD ARCHITECTURE +In this section, we first introduce QKD and CRKG, and then +discuss their combination modes to realize secure communi- +cation in wide-area mobile applications. +A. QKD +QKD protocols exploit a quantum communication channel +and an authenticated classical channel to ensure the exchange +of a cryptographic key between two remote parties with proven +security. Since its inception in [8], QKD protocol design +and analyses have flourished as a field yielding numerous +protocols, security analyses, and practical implementation +methodologies. Although QKD research has made remarkable +progress, these developments have been largely focused on +securing large-scale infrastructures using long distance fiber +transmission and free space transmission between fixed ter- +minals. Some efforts have been made toward handheld free- +space QKD by exploiting a beam-steering module, which +compensates for hand movement of the QKD module at +the transmitter [9, 10]. However, these schemes have limited +transmission range and their QKD receiver is currently difficult +to be miniaturized. In other words, they can not provide a bi- +directional transmission and are thus not applicable to the case +of distributing a quantum key from a core network to an end- +user. In this article, QKD is exploited to form a backbone +quantum core network to bridge long distances. +B. CRKG +CRKG exploits wireless channels between transceivers as +random sources for key generation, and these keys can be +replenished dynamically as wireless channels vary over time. +Eavesdroppers in such situations experience physical channels +independent of those of the legitimate users as long as they +are a few wavelengths away from these legitimate parties, +which is generally the case in wireless networks. So far, +the CRKG field has yielded fruitful results from aspects of +theoretical exploration, modeling, protocol design, and proto- +type implementation in various IoT platforms [11]. However, +these developments have been largely focused on wireless +communication technologies for short-range applications, such +as ZigBee, ultra-wideband, Bluetooth and WiFi. When the +distance is in the order of a few kilometers, the signal-to- +noise ratio is small and the time delay between uplink and +downlink packets becomes large. Therefore, CRKG at a long +distance is challenging to meet the requirement of high cor- +relation between channel parameter measurements for secret +key generation [11]. Due to these reasons, CRKG is more +suitable for secret key sharing between wireless transceivers + +3 +Alice +Bob +K𝑪𝟏 +K𝑪𝟏 +KQ +KQ +K𝐂𝟐 +K𝐂𝟐 +QAP1 +QAP2 +Fig. 1. An illustration of combining QKD and CRKG to realize secure communication between two remote users. +that are within one kilometer apart and thus exploited in this +article to complete the last-mile secret key distribution task +from quantum access points (QAP) to IoT users. +C. The Combination Mode of QKD and CRKG +Neither QKD nor PKG is applicable to long-range IoT +networks, therefore, a critical problem is how to combine their +advantages to apply to the new scenario. Fig. 1 describes the +system model and illustrates one possible combination mode. +Alice and Bob are two distant wireless users, who do not have +direct links with each other. QAP1 and QAP2 are two quantum +nodes that are connected through long-distance optical fibers, +or ground-to-satellite free-space links. QAP1 and QAP2 have +a wireless link to Alice and Bob, respectively. +To complete the secret key distribution between Alice and +Bob, three keys are first shared between Alice and QAP1 (link +1), QAP1 and QAP2 (link 2), and QAP2 and Bob (link 3). +Channel keys are generated from wireless links 1 and 3 by us- +ing the technique of CRKG, while quantum key is distributed +from QAP1 to QAP2, or in the reverse direction, with mature +QKD techniques. Next, the quantum key is securely delivered +to Alice and Bob by encrypting it with channel keys. In other +words, Alice and Bob share a unified key, which is then used to +encrypt and to decrypt the message in the data transmissions. +Therefore, this mode is also abbreviated as unified-key mode. +Notably, Alice and Bob are free to choose wireless and Internet +routes for message transmission. This consideration is due to +the following reasons. First, due to the limited rate of the +quantum link, its message transmission rate is relatively small. +Second, as Alice and Bob are mobile devices, they are more +likely to use communication routes that are different from +those in the key distribution process. Finally, the unified-key +requires less time delay for message transmission as it only +needs one time of message encryption and decryption. The +essential process to obtain unified quantum keys is referred to +as CR-QKD, which is elaborated in the following section. +III. CONCEPTUAL DESIGN OF CR-QKD +In this section, we will first introduce the basic mechanism +of CR-QKD and then study the key aligment, efficiency and +security issues that exist in CR-QKD. +A. Mechanism description +As shown in Fig. 1, Alice and Bob intend to share quantum +key with the help of QAP1 and QAP2, against an adversarial +eavesdropper, Eve, tapping on the quantum channel and lis- +tening to all the exchanges on the classical channels. Similar +to most existing QKD and CRKG protocols, the classical +communication channels are assumed to be authenticated, +in which the identities of the communicating parties have +been verified and the integrity of the transmitted messages +is promised. +The CR-QKD protocol comprises three main phases, i.e, +QKD 1, CRKG and edge forwarding, which will be elaborated +below. +• QKD phase: First, QAP1 prepares and sends to QAP2 +a set of random qubits via a single-photon signal over a +quantum channel. These qubits are selected from a set of +four states with two bases. For every incoming state from +QAP1, QAP2 randomly chooses one of the two bases to +measure and record the results. Once quantum commu- +nication has finished, QAP2 starts base reconciliation by +announcing the position of the detected bits and the basis +used to QAP1 over a classic channel. Then, QAP1 and +QAP2 retain the bits with a coincident basis and discard +the rest. After that, QAP1 publishes a subset of these bits +to QAP2 for eavesdropping detection. If the error rate +between what QAP2 detects and what QAP1 has sent +is high, the eavesdropping is detected and these shared +bits will be invalid. Otherwise, QAP1 and QAP2 perform +information reconciliation and privacy amplification over +the rest of the bits that have not been made public. At +last, QAP1 and QAP2 check whether they obtain the same +result via key verification. If so, they retain the pair of +bits as quantum key KQ, otherwise, they discard both of +them. +• CRKG phase: A CRKG protocol typically contains four +stages, i.e., channel probing, quantization, information +reconciliation, and privacy amplification. Alice and QAP1 +first carry out channel probing, which involves bidirec- +tional measurements within a channel coherence time. +They then convert the analog measurements into digital +1Our study is not bound to specific QKD protocols, and we choose the +BB84 protocol as a representative to introduce the CR-QKD mechanism. + +4 +Pair +𝐴𝑀1-𝐵1 +... +𝐴𝑀1-𝐵1 +... +𝐴𝑀1-𝐵𝑀2 +Number +1 +... +෍ +𝑖=1 +𝑀𝑖 +𝑁𝐴𝑖 +1 +... +𝑁𝐴𝑀1 +𝐾𝑄 +1110…010 +... +0011…010 +... +1011…111 +Pair +𝐴2-𝐵1 +... +𝐴2-𝐵1 +... +𝐴2-𝐵𝑁 +Number +𝑁𝐴1+ +... +𝑁𝐴2,𝐵1 +... +𝑁𝐴2 +𝐾𝑄 +1110…010 +... +0011…010 +... +1011…110 +𝐴1 +𝑲𝑸 +𝑲𝑸 +𝑄𝐴𝑃1 +𝑄𝐴𝑃2 +Quantum +Key Buffer +Quantum +Key Buffer +𝐴2 +𝐴𝑀1 +... +... +𝐵1 +𝐵2 +𝐵𝑀2 +... +... +Request +Request +Pair +𝐴1-𝐵1 +... +𝐴1-𝐵1 +... +𝐴1-𝐵𝑀2 +Number +1 +... +𝑁𝐴1,𝐵1 +... +𝑁𝐴1 +𝐾𝑄 +1010…010 +... +1011…010 +... +1011…011 +Pair +𝐴2-𝐵1 +... +𝐴2-𝐵1 +... +𝐴2-𝐵𝑁 +Number +𝑁𝐴1+ +... +𝑁𝐴2,𝐵1 +... +𝑁𝐴2 +𝐾𝑄 +1110…010 +... +0011…010 +... +1011…110 +Pair +𝐴1-𝐵1 +... +𝐴1-𝐵1 +... +𝐴1-𝐵𝑀2 +Number +1 +... +𝑁𝐴1,𝐵1 +... +𝑁𝐴1 +𝐾𝑄 +1010…010 +... +1011…010 +... +1011…011 +Pair +𝐴1-𝐵𝑀2 +𝐴2-𝐵𝑀2 +... +𝐴𝑀1-𝐵𝑀2 +Number +𝑁𝐴1 +𝑁𝐴1 + 𝑁𝐴2 +... +෍ +𝑖=1 +𝑀1 𝑁𝐴𝑖 +𝐾𝑄 +1011…011 +1111…010 +... +1011…111 +Pair +𝐴𝑀1-𝐵1 +... +𝐴𝑀1-𝐵1 +... +𝐴𝑀1-𝐵𝑀2 +Number +1 +... +෍ +𝑖=1 +𝑀𝑖 +𝑁𝐴𝑖 +1 +... +𝑁𝐴𝑀1 +𝐾𝑄 +1110…010 +... +0011…010 +... +1011…111 +Pair +𝐴2-𝐵1 +... +𝐴2-𝐵1 +... +𝐴2-𝐵𝑁 +Number +𝑁𝐴1+ +... +𝑁𝐴2,𝐵1 +... +𝑁𝐴2 +𝐾𝑄 +1110…010 +... +0011…010 +... +1011…110 +Pair +𝐴2-𝐵1 +... +𝐴2-𝐵1 +... +𝐴2-𝐵𝑁 +Number +𝑁𝐴1+ +... +𝑁𝐴2,𝐵1 +... +𝑁𝐴2 +𝐾𝑄 +1110…010 +... +0011…010 +... +1011…110 +Pair +𝐴1-𝐵1 +... +𝐴1-𝐵1 +... +𝐴1-𝐵𝑀2 +Number +1 +... +𝑁𝐴1,𝐵1 +... +𝑁𝐴1 +𝐾𝑄 +1010…010 +... +1011…010 +... +1011…011 +Fig. 2. An illustration of CR-QKD in a multi-user scenario: edges distribute quantum keys to users according to their needs, where NAi,Bj represents the +number of key groups required by Ai − Bj user pair and NAi represents the total number of key groups required by user Ai. +binaries. There will probably be a mismatch between +these binaries, hence information reconciliation has to be +adopted to correct the mismatch. To avoid information +leakage, privacy amplification is employed to distill the +reconciliated binaries. Finally, after key verification, Alice +and QAP1 retain the pair of bits as channel key KC1 and +Bob and QAP2 retain the pair of bits as channel key KC2. +• Edge forwarding phase: In the last phase, previous +quantum keys shared between QAP1 and QAP2 are +forwarded to Alice and Bob, completing the ultimate task +of secret key sharing. Security is the primary concern +here, as eavesdroppers should not learn any information +about the quantum key through this forwarding process. +With the help of channel keys, it is possible for edges +to encrypt quantum keys with them using the One-Time- +Pad (OTP) encryption algorithm and then forward the +ciphertext to users. So far, the secret key sharing task is +completed. +Although CR-QKD provides a potential solution, it still faces +some challenges to be implemented in practice. We divide +these challenges into three categories and discuss along coun- +termeasures below. +B. Key alignment +OTP is a well-known example of encryption scheme that +provides “perfect secrecy”, however, one challenge here is that +the channel key used for OTP must be at least as long as +the quantum key to be encrypted. As channel keys, KC1 and +KC2, are generated from different wireless channels, their key +generation rates are likely to be different from each other, and +that of the quantum key KQ. As a result, the quantum keys +distributed to Alice and Bob may be disordered. +We address the key alignment issue by segmenting quantum +key and channel key into groups and numbering them before +edge forwarding. Each group has a fixed bit number of LG. +Those quantum key and channel key bits belonging to the +same group are encrypted through a binary XOR operation. +Then, the ciphertext is forwarded, together with the group +number. Alice and Bob eventually obtain the quantum key +by decrypting the ciphertext using their corresponding channel +keys. Here, the trade-off between overhead and real-time must +be taken into account in the selection of the group size. +If the group size is small, the group number will occupy +a field length comparable to that of the ciphertext, and the +communication overhead will become significant. Otherwise, +if the group size is large, the communication overhead is +reduced but it will take a long time to accumulate sufficient +keys for forwarding. +Next, we extend the key alignment issue into a multi- +user scenario, where A = {A1, A2, · · · , AM1} and B = +{B1, B2, · · · , BM2} are two sets of IoT users at the service +range of QAP1 and QAP2, respectively. Users in A desire to +share secret keys with users in B. When CR-QKD is applied +to this case, a new problem arises, i.e., how to distribute +quantum keys from the edge to multiple users, who have +different requirements and channel conditions. In this article, +we introduce a multi-user edge forwarding strategy, which +distributes quantum keys to each user according to its needs. +Fig. 2 illustrates one round of the quantum key distribution +process using this strategy. +To start with, users in A broadcast the name of their +target users for key sharing and the number of required +key groups. After receiving these requests, QAP1 shares the +information with QAP2. Then, QAP2 broadcasts it over the +air and the relevant users in B record them locally. Next, +quantum key sequences are shared between QAP1 and QAP2 +through the above QKD phase. These quantum key sequences +are segmented into groups and numbered, each having LG +bits. QAP1 allocates quantum key groups for each user pair +according to their requests. The mapping relationship of user +pairs and the key group number is transmitted to QAP2. This +allocation information is saved in a quantum key buffer. In this +way, the quantum keys are synchronized at QAP1 and QAP2. +Next, they yield channel keys with these demanding users, +respectively. For each user, the CRKG process is performed +multiple times until it has accumulated sufficient number of +key groups. Finally, QAP1 and QAP2 use these CRKG keys +to encrypt the corresponding quantum keys and broadcast the + +5 +ciphertext together with the user pairs and group number to +end-users. Each end-user obtains quantum keys by decrypting +the related ciphertext with its own CRKG keys. Finally, these +quantum keys are divided into each user pair for message +encryption and this round of quantum key distribution has +come to an end. +C. Efficiency improvement +The basic CR-QKD mechanism is time-consuming as it +interacts heavily to obtain identical keys in both QKD and +CRKG phases. This situation becomes more severe in a multi- +user case. For each round of multi-user key distribution, in a +time division multiple access (TDMA) system, the time delay +is the sum of the time spent on yielding quantum keys and +channel keys plus the time used for key forwarding. The time +spent on quantum keys is calculated by dividing the number +of quantum key bits by the quantum key generation rate. The +time spent on channel keys is equal to the larger one of QAP1 +and QAP2. For each QAP, its time delay is the sum of that +used for yielding channel keys between it and all users. One +approach to reducing the time delay is to make QKD and +CRKG processes work in parallel. However, its reduction ratio +is less than 50% due to the positive forwarding time and the +maximum operation. +Another solution to further reduce the time delay is to +improve the secret key generation rate. In practice, key +generation rates are largely subject to the long time delay +caused by information reconciliation, which exchanges parity +information or syndromes over classic channels to detect and +correct errors in the preliminary key material. According to +OTP with un-identical keys [12], we propose a simplified CR- +QKD mechanism that abolishes the sophisticated information +reconciliation step in the CRKG phase and forwards quantum +keys using non-reconciled channel keys. The challenge is to +decrypt the quantum keys correctly when the non-reconciled +channel keys of two parties are different but highly correlated. +We deem the XOR encryption and decryption modules along +with the physical channel as an equivalent cascade channel. +Then, the tiny differences between keys can be seen as part of +the transmission error, and thus can be corrected by the off- +the-shelf channel coding with a stronger correction capability. +Fig. 3 plots performance improvement ratios of the simplified +CR-QKD mechanism compared with the paralleled CR-QKD +mechanism in terms of time delay and upper bound of secret +key generation rate in a typical WiFi scenario. As shown in +the left panel, the proportion of delay reduction decreases with +the rise of LG, still achieving a reduction ratio above 20% +at LG ≤ 1024. The reduction of HT-Mixed mode is more +remarkable than Non-HT mode, as the former has a larger +time overhead than the latter. The right panel shows that the +growth of the upper bound of the secret key generation rate +is more remarkable when the bit disagreement ratio between +quantized channel measurements gets larger, while it has a +slight fall with the rise of LG. When LG = 1024 and ϵq = 0.1, +the proportion of delay reduction and upper bound of secret +key rate growth are roughly 20% and 10%, respectively. These +simulation results verify the effectiveness of the proposed +simplified CR-QKD mechanism. +32 +64 +128 +256 +512 +1024 +LG/bits +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +The Proportion of Delay Reduction +Non-HT +HT-Mixed +0 +0.05 +0.10 +0.15 +0.20 +q +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +Upper Bound of Secret Key Generation Rate Growth +LG=32bit +LG=256bit +LG=1024bit +Fig. 3. Performance improvements of time delay and secret key generation +rate in a typical WiFi scenario: the transmission distance is set as 150 meters +and the bandwidth is set as 20 MHz. The fixed overhead of a WiFi frame +under the Non-HT (Non-High Throughput) and HT-Mixed mode is 20 us and +40 us, respectively. +D. Security enhancement +Another challenge of CRKG lies in the increased security +risks caused by its hybrid architecture, as security is only +as strong as its weakest link. We assume that the terminal +security of QAP1 and QAP2 is guaranteed by techniques +such as trusted computing. Operations that are relevant to +secret keys are run in a trusted execution environment, thereby +attackers can read neither quantum keys nor channel keys +from the hybrid interface on QAP1 and QAP2. Since the +edge forwarding phase employs the OTP encryption scheme, +its security depends on the key used for OTP. The security +of existing CRKG approaches, however, heavily relies on +the channel variation and thus suffers from vulnerabilities +in slowly varying environments [13]. When users have low +mobility, e.g. in a wireless sensor network, there exist in- +evitable and unknown temporal correlations between adjacent +channel samples, resulting in a large proportion of repeated bit +segments in the quantized bit sequences. Several solutions can +be used to facilitate the practical usage of CRKG in slowly +varying environments. One solution is to introduce helper de- +vices, e.g., relays and reconfigurable intelligent surface (RIS) +to boost the key generation rate and randomness [14]. How- +ever, this solution encounters some practical problems, such +as the unavailability of trust relays and additional hardware +overheads of RIS devices. Another idea is to scramble these +bits segments through some permutation or interleaving tech- +niques. However, the security of the key may be compromised +when the permutation information is public. [15] has proposed +a new physical-layer secret key generation approach with +channel obfuscation, which improved the dynamic property +of channel parameters based on random filtering and random +antenna scheduling, which have mutual remedying parameters +in hiding the obfuscation information. + +6 +YUHUATAI +DISTRICT +JIANGNING +DISTRICT +GULI +RESIDENTIAL +DISTRICT +Jiangjunshan +Tourism +Scenic Area +Fangshan +Scenic Area +XISHANQIAO +RESIDENTIAL +DISTRICT +TIEXINQIAO +RESIDENTIAL +DISTRICT +Qinhuai New River +QAP2 +Total length: 15km +QAP1 +Alice +QAP2 +Bob +Bob +Quantum Key +Eve +Fig. 4. An illustration of the CR-QKD prototype platform in a metropolitan area network at Nanjing, which is the capital of Jiangsu Province, East-central +China. One is located in Yuhuatai District and the other is at Chinese Network Valley in Jiangning District. +IV. CASE STUDY: AN IMPLEMENTATION OF CR-QKD +To realize the concept of CR-QKD, we implement a single- +user confidential transmission prototype system in a metropoli- +tan area network. +A. Experimental Setup +As shown in the left panel of Fig. 4, QAP1 and QAP2 are +two quantum access points at a distance of fifteen kilometers. +Alice and Bob are two remote IoT users in the wireless +service ranges of QAP1 and QAP2, respectively. Without loss +of generality, we zoom in on the wireless access network +at Chinese Network Valley, as depicted in the right panel +of Fig. 4. Here, QAP2 is composed of a QKD terminal +under the series of QKDM-POL40-S for yielding quantum +keys 2, a USRP N210 SDR device embedded with the CBX +daughterboards for providing a wireless connection service, +and a computer under the trusted execution environment for +yielding wireless channel keys and distributing quantum keys. +Both the QKD terminal and USRP N210 are connected to +the computer via the ethernet cable in QAP2. The end-user, +Bob, and a passive eavesdropper, Eve, are realized through two +USRP N210 SDR devices, respectively. We design a TDD +frame for channel sounding, which consists of a sinusoidal +sequence for synchronization and an M-sequence for channel +estimation. The signal operates at 2.605GHz and 20MHz +bandwidth to avoid collisions with ubiquitous 2.4GHz signals +such as WiFi. Once Bob receives the channel sounding signal +from AP2, it will immediately switch to TX mode and send +the same channel sounding signal. By using the same channel +sounding signal for channel estimation, the amplitude part of +the CSI is further preprocessed and quantized to generate the +wireless keys. +B. Performance Results +Considering the comparable experimental scenarios and +results of QAP1 and QAP2, we only take QAP2 as an example +for performance analysis. +2The quantum keys meet strict key randomness, as they conform to the +specification of the GM/T 0005-2012. +Table II summarizes the secret key sharing results from +QAP2 to Bob and Eve in three typical indoor scenarios, +namely office, hall and corridor. First, the measured key +generation rates (KGRs) of the channel keys between QAP2 +and Bob in above scenarios are 315.4, 424.7 and 383.7 bits per +second (bps), respectively. They are sufficient for traditional +symmetric encryption algorithms (such as AES) to update +256-bit keys every second for secure communications. In +the random test, we examined a bit sequence of length 3.4 +million bits that was obtained at the output of the quantization +stage without further processing. The generated channel keys +passed 14 NIST statistical tests, indicating their randomness. +However, while the simplified CR-QKD mechanism leads to +high KGRs and high randomness, removing the complicated +information reconciliation step also results in relatively high +key disagreement rates (KDRs) of 8.1%, 4.7% and 5.8% +between QAP2 and Bob, respectively. The number of person- +nel, the frequency of movement, and the switching time of +USRP affect the reciprocity of uplink and downlink channels, +eventually leading to KDR differences in the noisy office, +occasionally infested corridor, and empty hall. Meanwhile, +along with forwarding quantum keys using non-reconciled +channel keys based on channel error correction coding, the +need arises to retransmit quantum keys when unsuccessfully +decoded. The corresponding retransmission rates (RRs) using +Polar codes from QAP2 to Bob are 11.6%, 2.1%, and 6.7%, +respectively, which are proportional to the KDRs. +To demonstrate the security of our proposed scheme, we +also evaluate the quantum key cracking performance of the +near-end eavesdropper Eve in terms of KDR and cracking rate +(CR). The KDRs between QAP2 and Eve under these three +scenarios are all around 50%, where the line of sight in the +straight corridor contributes to a relatively lower KDR but is +still above 45%. What’s more, the experimental results show +that the CRs of Eve in the three scenarios are all zero, which +means that none of the quantum keys have been cracked. +V. CONCLUSION AND FUTURE DIRECTIONS +Integrating QKD into IoT networks is beneficial for QKD’s +practical deployment and end-user’s security enhancement. + +(0)目7 +TABLE II +THE QUANTUM KEY WIRELESS DISTRIBUTION PERFORMANCE IN THREE +INDOOR SCENARIOS +Scenario +QAP2 - Bob +QAP2 - Eve +Metrics +KGR/bps +NIST +KDR +RR +KDR +CR +Office +315.4 +14 +8.1% +11.6% +48.1% +0% +Hall +424.7 +14 +4.7% +2.1% +49.2% +0% +Corridor +383.7 +14 +5.8% +6.7% +45.3% +0% +This article proposed a framework of CR-QKD over IoT +networks. QKD and CRKG assembly were adopted for se- +cret key sharing over backbone core networks and the last- +mile wireless access networks in CR-QKD, respectively. The +demonstration of CR-QKD prototype represented a major step +towards real-world information theoretically security for wide- +area mobile applications, such as confidential VoLTE and +confidential VoWiFi. +Some open issues in future work are given below. +• Device Authentication: Considering the hybrid archi- +tecture of CR-QKD, it is more vulnerable to spoofing +attacks from either user’s side or QAP’s side. However, +neither QKD nor CRKG provides a means to authenticate +the transmission source. Therefore, source authentication +in CR-QKD should be further studied by using asym- +metric cryptography techniques or emerging physical- +layer techniques, such as radio frequency fingerprinting +identification and physical unclonable function [1]. +• Untrusted QAPs: The proposed CR-QKD scheme relies +on the trust of the intermediate QAPs. In this paper, we +use techniques of trust computing to ensure that the the +information stored in QAP is protected from external +software attacks. When a trusted platform is not available, +designing a scheme that relaxes this assumption could +also be a very good future research direction. +• Performance Optimization: In this article, we presented +a multi-user edge forwarding strategy, in which quantum +keys were allocated as needed. Unfortunately, its perfor- +mance metrics, e.g., delay, secret key generation rate, and +energy efficiency, are limited by those user pairs with +weak channel reciprocity. How to optimize these perfor- +mance metrics by allocating power or spectrum resources +among different user pairs becomes an interesting topic +and needs to be investigated. +• System Integration and Compatibility: Our prototype +was built on the USRP platform, which was different +from commercial-off-the-shelf (COTS) devices. It is un- +known whether these performances are still achievable on +existing communication standards and whether CR-QKD +will affect the network efficiency. More studies should be +done on its system integration and compatibility issues, +including frame format design, key management scheme +and efficiency evaluation in practical communication sys- +tems. +VI. ACKNOWLEDGMENT +We thank our colleagues Prof. Linning Peng, Mr. Yanjun +Ding, Dr. Dong Wang, Mr. Siyun Wu and Dr. Xuyang Wang +from the Purple Mountain Laboratories, for their help with +the experimental platform. This work was supported in part +by the National Key Research and Development Program of +China under Grant 2020YFE0200600 and 2022YFB2902202, +in part by the National Natural Science Foundation of China +under Grant 62171121, in part by the Natural Science Foun- +dation of Jiangsu Province under Grant BK20211160 and in +part by Jiangsu Provincial Key Laboratory of Network and +Information Security under Grant BM2003201. +REFERENCES +[1] K. Sood, S. Yu, D. D. N. Nguyen, Y. Xiang, B. Feng, and X. Zhang, +“A tutorial on next generation heterogeneous IoT networks and node +authentication,” IEEE Internet of Things Magazine, vol. 4, no. 4, pp. +120–126, 2021. +[2] M. S. Hossain, G. Muhammad, S. M. M. Rahman, W. Abdul, A. Ale- +laiwi, and A. Alamri, “Toward end-to-end biomet rics-based security for +IoT infrastructure,” IEEE Wireless Communications, vol. 23, no. 5, pp. +44–51, 2016. +[3] S. Imre, “Quantum communications: explained for communication en- +gineers,” IEEE Communications Magazine, vol. 51, no. 8, pp. 28–35, +2013. +[4] C. Wang and A. Rahman, “Quantum-enabled 6G wireless networks: +Opportunities and challenges,” IEEE Wireless Communications, vol. 29, +no. 1, pp. 58–69, 2022. +[5] G. Li, C. Sun, J. Zhang, E. Jorswieck, B. Xiao, and A. Hu, “Physical +layer key generation in 5G and beyond wireless communications: +Challenges and opportunities,” Entropy, vol. 21, 2019. +[6] B. Fr¨ohlich, J. F. Dynes, M. Lucamarini, A. W. Sharpe, Z. Yuan, and +A. J. Shields, “A quantum access network,” Nature, vol. 501, pp. 69–72, +2013. +[7] U. M. Maurer, “Secret key agreement by public discussion from common +information,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 733–742, May +1993. +[8] C. H. Bennett and G. Brassard, “Quantum cryptography: Public key +distribution and coin tossing,” Proc. IEEE Int. Conf. Comput., Syst. +Signal Process., vol. 175, p. 175–179, 1984. +[9] H. Chun, I. Choi, G. Faulkner, L. Clarke, B. Barber, G. George, +C. Capon, A. Niskanen, J. Wabnig, D. O’Brien, and D. Bitauld, +“Handheld free space quantum key distribution with dynamic motion +compensation,” Opt. Express, vol. 25, no. 6, pp. 6784–6795, Mar 2017. +[10] Elmabrok, Osama, Razavi, and Mohsen, “Wireless quantum key distribu- +tion in indoor environments,” Journal of the Optical Society of America +B Optical Physics, vol. 35, no. 2, pp. 197–207, 2018. +[11] J. Zhang, G. Li, A. Marshall, A. Hu, and L. Hanzo, “A new frontier for +IoT security emerging from three decades of key generation relying on +wireless channels,” IEEE Access, vol. 8, pp. 138 406–138 446, 2020. +[12] G. Li, Z. Zhang, J. Zhang, and A. Hu, “Encrypting wireless communica- +tions on the fly using one-time pad and key generation,” IEEE Internet +of Things Journal, vol. 8, no. 1, pp. 357–369, 2021. +[13] N. Aldaghri and H. Mahdavifar, “Physical layer secret key generation +in static environments,” IEEE Trans. Inf. Forensics Security, vol. 15, pp. +2692–2705, Feb. 2020. +[14] G. Li, L. Hu, P. Staat, H. Elders-Boll, C. Zenger, C. Paar, and A. Hu, +“Reconfigurable intelligent surface for physical layer key generation: +Constructive or destructive?” IEEE Wireless Communications, pp. 1–12, +2022. +[15] G. Li, H. Yang, J. Zhang, H. Liu, and A. Hu, “Fast and secure key +generation with channel obfuscation in slowly varying environments,” +in Proc. IEEE INFOCOM, Virtual Conference, May 2022, pp. 1–10. + diff --git a/39AzT4oBgHgl3EQfffxn/content/tmp_files/load_file.txt b/39AzT4oBgHgl3EQfffxn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf7f81d8de3acd4e046c7722f9204cd2b260027d --- /dev/null +++ b/39AzT4oBgHgl3EQfffxn/content/tmp_files/load_file.txt @@ -0,0 +1,587 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf,len=586 +page_content='1 Information-Theoretic Secure Key Sharing for Wide-Area Mobile Applications Guyue Li, Member, IEEE, Hongyi Luo, Jiabao Yu, Aiqun Hu, Senior Member, IEEE and Jiangzhou Wang, Fellow, IEEE With the rapid growth of handheld devices in the internet of things (IoT) networks, mobile applications have become ubiquitous in everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' As technology is developed, so do also the risks and threats associated with it, especially in the forthcoming quantum era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Existing IoT networks, however, lack a quantum-resistant secret key sharing scheme to meet confidential message transmission demands in wide-area mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' To address this issue, this article proposes a new scheme, channel reciprocity (CR) based quantum key distribution (QKD) CR- QKD, which accomplishes the goal of secret key sharing by combining emerging techniques of QKD and CR-based key generation (CRKG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Exploiting laws of quantum physics and properties of wireless channels, the proposed scheme is able to ensure the secrecy of the key, even against computationally unbounded adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The basic mechanism is elaborated for a single-user case and it is extended into a multi-user case by redesigning a multi-user edge forwarding strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In addition, to make CR-QKD more practical, some enhancement strategies are studied to reduce the time delay and to improve the secret key generation rate in a secure manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' A prototype of CR-QKD is demonstrated in a metropolitan area network, where secret keys are shared between two remote IoT devices that are roughly fifteen kilometers apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The experimental results have verified that CR-QKD allows a secret key rate of 424 bits per second with a retransmission rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Index Terms—Secret key generation, physical layer security, quantum key distribution, wide-area mobile applications, Inter- net of Things I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' INTRODUCTION Recent years have witnessed a remarkable growth in the number and variety of mobile devices and applications in the Internet of Things (IoT) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The flourish of IoT, however, has resulted in the generation of a substantial amount of private messages exchanged over public channels, which has grabbed one’s attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Unfortunately, IoT devices are susceptible to various threats and security challenges, which pose hazards for the advancement of IoT in sensitive fields, such as smart homes, unmanned vehicles, e-health, and mili- tary networks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In order to avoid being revealed to a third Guyue Li and Hongyi Luo are with the School of Cyber Science and Engineering, Southeast University, Nanjing 210096, China (e-mail: guyuelee@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' hongyiluo@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Jiabao Yu is with the Purple Mountain Laboratories, Nanjing 210096, China (e-mail:yujiabao@pmlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Aiqun Hu is with National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: aqhu@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Jiangzhou Wang is with the School of Engineering, University of Kent, Canterbury CT2 7NT, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Email: (e-mail: j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='wang@kent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Guyue Li and Aiqun Hu are also with Purple Mountain Laboratories, Nanjing 210096, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' party, a message is usually encrypted using a secret key shared among the communicating devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Thus, a key prerequisite of achieving IoT network security is secret key sharing that avoids eavesdropper interception [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In a classic cryptographic scheme, two legitimate parties, namely Alice and Bob use the public-key cryptosystem (PKC) for key distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' It is extremely difficult for a third party, namely Eve, to derive the private key or message compu- tationally, due to the intractability of certain mathematical problems used in encryption algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, the emerg- ing quantum computing technology has the potential to make some previously-intractable problems tractable [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Thus, the security of computational security-based key distribution will be rendered insecure by substantial progress in quantum computing in the coming years, which necessitates the study of alternative solutions that do not rely on computational security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In this context, much attention has been paid to emerging techniques, such as quantum key distribution (QKD) [4] and channel reciprocity-based key generation (CRKG) [5], which can provide secret key sharing service with information- theoretic security, also known as unconditional security or physical security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QKD is a well-known quantum-resistant mechanism, which distributes secret keys to distant parties by trans- mitting single photon through a quantum channel [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Employing the laws of quantum physics, QKD can de- tect eavesdroppers during the key generation process, in which unauthorized observation of quantum commu- nication induces a discernible increase of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' This sensitivity to eavesdropping makes QKD possible to en- sure the secrecy of the key, even against computationally unbounded adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CRKG is built on the basis of channel reciprocity, which means that the channel responses of the forward and backward communication links are very similar in a time division duplex (TDD) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In addition, the dynamic and complex wireless communication environment makes the channel responses change over time and hard to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Therefore, legitimate users can share a pair of common randomness from their radio channel measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Since CRKG does not require assistance from a third party nor expensive infrastructure, it has recently emerged as a new paradigm that provides a lightweight and information-theoretic secure key sharing solution for decentralized or device-to-device sensor applications [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Table I summarizes these typical secret key distribution arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='01453v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='IT] 4 Jan 2023 2 TABLE I A SUMMARY AND COMPARATION OF TYPICAL SECRET KEY DISTRIBUTION METHODS Method Metric Security Level Mobility Support Distribution Distance User Cost PKC Computational secure Middle Long Middle QKD Information theoretically secure Weak Long High CRKG Information theoretically secure Strong Short Low CR-QKD Information theoretically secure Strong Long Low methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' and identify their characteristics from perspectives of security level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' mobility support,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' distribution distance and user cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We find that although the separate construction of QKD and CRKG can be supported in the physical layer, there is no investigation of a secret key sharing scheme for the security demands from remote mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Although point-to-point connections are suitable to form a backbone quantum core network to bridge long distances, they are less suitable to provide the last-mile service needed to give a multitude of users access to this QKD infrastructure [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Similarly, despite many research efforts in the field of CRKG, its widespread application is unfortunately hindered by the short distance between transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' With a rapid growth of handheld devices, wide-area mobile applications, such as remote environmental and elderly monitoring, have become an inseparable part of IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' A new architecture needs to be developed where end-users between two access networks are connected to a metro network, thus realizing unconditionally secure key sharing in a more cost-effective and flexible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In this article, we introduce and experimentally demonstrate the concept of a ‘channel reciprocity-aided quantum key distribution (CR-QKD)’ based on simple and cost-effective telecommunication technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' This scheme can expand the scope of QKD to IoT networks and therefore vastly broaden users’ appeal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The contributions of this article are three-fold: We introduce a novel secret key sharing architecture, re- ferred to as CR-QKD, which bridges a backbone quantum core network and IoT users by exploiting the technique of CRKG to provide the last-mile service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CR-QKD is information-theoretically secure and it does not require IoT users to be equipped with expensive quantum infras- tructures for exchanging secret keys, thereby significantly reducing the hardware requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We propose a multi-user mechanism to realize the con- cept of CR-QKD with an elaborate design of key align- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We also identify challenges that arise due to the hybrid architecture of CR-QKD from the perspective of feasibility and security, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Countermeasures have been studied to reduce the time delay and to improve the secret key generation rate in a secure manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We implement a prototype CR-QKD system in a metropolitan area network, in which secret keys are shared between two remote IoT devices that are roughly fifteen kilometers apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The experimental results have verified that CR-QKD can provide a secret key rate of 424 bits per second with a retransmission rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' AN OVERVIEW OF THE CR-QKD ARCHITECTURE In this section, we first introduce QKD and CRKG, and then discuss their combination modes to realize secure communi- cation in wide-area mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QKD QKD protocols exploit a quantum communication channel and an authenticated classical channel to ensure the exchange of a cryptographic key between two remote parties with proven security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Since its inception in [8], QKD protocol design and analyses have flourished as a field yielding numerous protocols, security analyses, and practical implementation methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Although QKD research has made remarkable progress, these developments have been largely focused on securing large-scale infrastructures using long distance fiber transmission and free space transmission between fixed ter- minals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Some efforts have been made toward handheld free- space QKD by exploiting a beam-steering module, which compensates for hand movement of the QKD module at the transmitter [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, these schemes have limited transmission range and their QKD receiver is currently difficult to be miniaturized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In other words, they can not provide a bi- directional transmission and are thus not applicable to the case of distributing a quantum key from a core network to an end- user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In this article, QKD is exploited to form a backbone quantum core network to bridge long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CRKG CRKG exploits wireless channels between transceivers as random sources for key generation, and these keys can be replenished dynamically as wireless channels vary over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Eavesdroppers in such situations experience physical channels independent of those of the legitimate users as long as they are a few wavelengths away from these legitimate parties, which is generally the case in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' So far, the CRKG field has yielded fruitful results from aspects of theoretical exploration, modeling, protocol design, and proto- type implementation in various IoT platforms [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, these developments have been largely focused on wireless communication technologies for short-range applications, such as ZigBee, ultra-wideband, Bluetooth and WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' When the distance is in the order of a few kilometers, the signal-to- noise ratio is small and the time delay between uplink and downlink packets becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Therefore, CRKG at a long distance is challenging to meet the requirement of high cor- relation between channel parameter measurements for secret key generation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Due to these reasons, CRKG is more suitable for secret key sharing between wireless transceivers 3 Alice Bob K𝑪𝟏 K𝑪𝟏 KQ KQ K𝐂𝟐 K𝐂𝟐 QAP1 QAP2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' An illustration of combining QKD and CRKG to realize secure communication between two remote users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' that are within one kilometer apart and thus exploited in this article to complete the last-mile secret key distribution task from quantum access points (QAP) to IoT users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The Combination Mode of QKD and CRKG Neither QKD nor PKG is applicable to long-range IoT networks, therefore, a critical problem is how to combine their advantages to apply to the new scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1 describes the system model and illustrates one possible combination mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Alice and Bob are two distant wireless users, who do not have direct links with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QAP1 and QAP2 are two quantum nodes that are connected through long-distance optical fibers, or ground-to-satellite free-space links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QAP1 and QAP2 have a wireless link to Alice and Bob, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' To complete the secret key distribution between Alice and Bob, three keys are first shared between Alice and QAP1 (link 1), QAP1 and QAP2 (link 2), and QAP2 and Bob (link 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Channel keys are generated from wireless links 1 and 3 by us- ing the technique of CRKG, while quantum key is distributed from QAP1 to QAP2, or in the reverse direction, with mature QKD techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Next, the quantum key is securely delivered to Alice and Bob by encrypting it with channel keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In other words, Alice and Bob share a unified key, which is then used to encrypt and to decrypt the message in the data transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Therefore, this mode is also abbreviated as unified-key mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Notably, Alice and Bob are free to choose wireless and Internet routes for message transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' This consideration is due to the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' First, due to the limited rate of the quantum link, its message transmission rate is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Second, as Alice and Bob are mobile devices, they are more likely to use communication routes that are different from those in the key distribution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Finally, the unified-key requires less time delay for message transmission as it only needs one time of message encryption and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The essential process to obtain unified quantum keys is referred to as CR-QKD, which is elaborated in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CONCEPTUAL DESIGN OF CR-QKD In this section, we will first introduce the basic mechanism of CR-QKD and then study the key aligment, efficiency and security issues that exist in CR-QKD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Mechanism description As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1, Alice and Bob intend to share quantum key with the help of QAP1 and QAP2, against an adversarial eavesdropper, Eve, tapping on the quantum channel and lis- tening to all the exchanges on the classical channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Similar to most existing QKD and CRKG protocols, the classical communication channels are assumed to be authenticated, in which the identities of the communicating parties have been verified and the integrity of the transmitted messages is promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The CR-QKD protocol comprises three main phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='e, QKD 1, CRKG and edge forwarding, which will be elaborated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QKD phase: First, QAP1 prepares and sends to QAP2 a set of random qubits via a single-photon signal over a quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' These qubits are selected from a set of four states with two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' For every incoming state from QAP1, QAP2 randomly chooses one of the two bases to measure and record the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Once quantum commu- nication has finished, QAP2 starts base reconciliation by announcing the position of the detected bits and the basis used to QAP1 over a classic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Then, QAP1 and QAP2 retain the bits with a coincident basis and discard the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' After that, QAP1 publishes a subset of these bits to QAP2 for eavesdropping detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' If the error rate between what QAP2 detects and what QAP1 has sent is high, the eavesdropping is detected and these shared bits will be invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Otherwise, QAP1 and QAP2 perform information reconciliation and privacy amplification over the rest of the bits that have not been made public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' At last, QAP1 and QAP2 check whether they obtain the same result via key verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' If so, they retain the pair of bits as quantum key KQ, otherwise, they discard both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CRKG phase: A CRKG protocol typically contains four stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=', channel probing, quantization, information reconciliation, and privacy amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Alice and QAP1 first carry out channel probing, which involves bidirec- tional measurements within a channel coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' They then convert the analog measurements into digital 1Our study is not bound to specific QKD protocols, and we choose the BB84 protocol as a representative to introduce the CR-QKD mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 4 Pair 𝐴𝑀1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴𝑀1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴𝑀1-𝐵𝑀2 Number 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' \u0dcd 𝑖=1 𝑀𝑖 𝑁𝐴𝑖 +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴𝑀1 𝐾𝑄 1110…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 0011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…111 Pair 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵𝑁 Number 𝑁𝐴1+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2 𝐾𝑄 1110…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 0011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…110 𝐴1 𝑲𝑸 𝑲𝑸 𝑄𝐴𝑃1 𝑄𝐴𝑃2 Quantum Key Buffer Quantum Key Buffer 𝐴2 𝐴𝑀1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐵1 𝐵2 𝐵𝑀2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Request Request Pair 𝐴1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴1-𝐵𝑀2 Number 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴1,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴1 𝐾𝑄 1010…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…011 Pair 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵𝑁 Number 𝑁𝐴1+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2 𝐾𝑄 1110…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 0011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…110 Pair 𝐴1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴1-𝐵𝑀2 Number 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴1,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴1 𝐾𝑄 1010…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…011 Pair 𝐴1-𝐵𝑀2 𝐴2-𝐵𝑀2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴𝑀1-𝐵𝑀2 Number 𝑁𝐴1 𝑁𝐴1 + 𝑁𝐴2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' \u0dcd 𝑖=1 𝑀1 𝑁𝐴𝑖 𝐾𝑄 1011…011 1111…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…111 Pair 𝐴𝑀1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴𝑀1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴𝑀1-𝐵𝑀2 Number 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' \u0dcd 𝑖=1 𝑀𝑖 𝑁𝐴𝑖 +1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴𝑀1 𝐾𝑄 1110…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 0011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…111 Pair 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵𝑁 Number 𝑁𝐴1+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2 𝐾𝑄 1110…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 0011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…110 Pair 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴2-𝐵𝑁 Number 𝑁𝐴1+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴2 𝐾𝑄 1110…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 0011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…110 Pair 𝐴1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴1-𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝐴1-𝐵𝑀2 Number 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴1,𝐵1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 𝑁𝐴1 𝐾𝑄 1010…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1011…011 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' An illustration of CR-QKD in a multi-user scenario: edges distribute quantum keys to users according to their needs, where NAi,Bj represents the number of key groups required by Ai − Bj user pair and NAi represents the total number of key groups required by user Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' There will probably be a mismatch between these binaries, hence information reconciliation has to be adopted to correct the mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' To avoid information leakage, privacy amplification is employed to distill the reconciliated binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Finally, after key verification, Alice and QAP1 retain the pair of bits as channel key KC1 and Bob and QAP2 retain the pair of bits as channel key KC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Edge forwarding phase: In the last phase, previous quantum keys shared between QAP1 and QAP2 are forwarded to Alice and Bob, completing the ultimate task of secret key sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Security is the primary concern here, as eavesdroppers should not learn any information about the quantum key through this forwarding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' With the help of channel keys, it is possible for edges to encrypt quantum keys with them using the One-Time- Pad (OTP) encryption algorithm and then forward the ciphertext to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' So far, the secret key sharing task is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Although CR-QKD provides a potential solution, it still faces some challenges to be implemented in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We divide these challenges into three categories and discuss along coun- termeasures below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Key alignment OTP is a well-known example of encryption scheme that provides “perfect secrecy”, however, one challenge here is that the channel key used for OTP must be at least as long as the quantum key to be encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' As channel keys, KC1 and KC2, are generated from different wireless channels, their key generation rates are likely to be different from each other, and that of the quantum key KQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' As a result, the quantum keys distributed to Alice and Bob may be disordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We address the key alignment issue by segmenting quantum key and channel key into groups and numbering them before edge forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Each group has a fixed bit number of LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Those quantum key and channel key bits belonging to the same group are encrypted through a binary XOR operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Then, the ciphertext is forwarded, together with the group number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Alice and Bob eventually obtain the quantum key by decrypting the ciphertext using their corresponding channel keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Here, the trade-off between overhead and real-time must be taken into account in the selection of the group size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' If the group size is small, the group number will occupy a field length comparable to that of the ciphertext, and the communication overhead will become significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Otherwise, if the group size is large, the communication overhead is reduced but it will take a long time to accumulate sufficient keys for forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Next, we extend the key alignment issue into a multi- user scenario, where A = {A1, A2, · · · , AM1} and B = {B1, B2, · · · , BM2} are two sets of IoT users at the service range of QAP1 and QAP2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Users in A desire to share secret keys with users in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' When CR-QKD is applied to this case, a new problem arises, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=', how to distribute quantum keys from the edge to multiple users, who have different requirements and channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In this article, we introduce a multi-user edge forwarding strategy, which distributes quantum keys to each user according to its needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 2 illustrates one round of the quantum key distribution process using this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' To start with, users in A broadcast the name of their target users for key sharing and the number of required key groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' After receiving these requests, QAP1 shares the information with QAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Then, QAP2 broadcasts it over the air and the relevant users in B record them locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Next, quantum key sequences are shared between QAP1 and QAP2 through the above QKD phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' These quantum key sequences are segmented into groups and numbered, each having LG bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QAP1 allocates quantum key groups for each user pair according to their requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The mapping relationship of user pairs and the key group number is transmitted to QAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' This allocation information is saved in a quantum key buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In this way, the quantum keys are synchronized at QAP1 and QAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Next, they yield channel keys with these demanding users, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' For each user, the CRKG process is performed multiple times until it has accumulated sufficient number of key groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Finally, QAP1 and QAP2 use these CRKG keys to encrypt the corresponding quantum keys and broadcast the 5 ciphertext together with the user pairs and group number to end-users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Each end-user obtains quantum keys by decrypting the related ciphertext with its own CRKG keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Finally, these quantum keys are divided into each user pair for message encryption and this round of quantum key distribution has come to an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Efficiency improvement The basic CR-QKD mechanism is time-consuming as it interacts heavily to obtain identical keys in both QKD and CRKG phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' This situation becomes more severe in a multi- user case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' For each round of multi-user key distribution, in a time division multiple access (TDMA) system, the time delay is the sum of the time spent on yielding quantum keys and channel keys plus the time used for key forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The time spent on quantum keys is calculated by dividing the number of quantum key bits by the quantum key generation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The time spent on channel keys is equal to the larger one of QAP1 and QAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' For each QAP, its time delay is the sum of that used for yielding channel keys between it and all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' One approach to reducing the time delay is to make QKD and CRKG processes work in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, its reduction ratio is less than 50% due to the positive forwarding time and the maximum operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Another solution to further reduce the time delay is to improve the secret key generation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In practice, key generation rates are largely subject to the long time delay caused by information reconciliation, which exchanges parity information or syndromes over classic channels to detect and correct errors in the preliminary key material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' According to OTP with un-identical keys [12], we propose a simplified CR- QKD mechanism that abolishes the sophisticated information reconciliation step in the CRKG phase and forwards quantum keys using non-reconciled channel keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The challenge is to decrypt the quantum keys correctly when the non-reconciled channel keys of two parties are different but highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We deem the XOR encryption and decryption modules along with the physical channel as an equivalent cascade channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Then, the tiny differences between keys can be seen as part of the transmission error, and thus can be corrected by the off- the-shelf channel coding with a stronger correction capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 3 plots performance improvement ratios of the simplified CR-QKD mechanism compared with the paralleled CR-QKD mechanism in terms of time delay and upper bound of secret key generation rate in a typical WiFi scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' As shown in the left panel, the proportion of delay reduction decreases with the rise of LG, still achieving a reduction ratio above 20% at LG ≤ 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The reduction of HT-Mixed mode is more remarkable than Non-HT mode, as the former has a larger time overhead than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The right panel shows that the growth of the upper bound of the secret key generation rate is more remarkable when the bit disagreement ratio between quantized channel measurements gets larger, while it has a slight fall with the rise of LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' When LG = 1024 and ϵq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1, the proportion of delay reduction and upper bound of secret key rate growth are roughly 20% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' These simulation results verify the effectiveness of the proposed simplified CR-QKD mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 32 64 128 256 512 1024 LG/bits 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7 The Proportion of Delay Reduction Non-HT HT-Mixed 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='20 q 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='35 Upper Bound of Secret Key Generation Rate Growth LG=32bit LG=256bit LG=1024bit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Performance improvements of time delay and secret key generation rate in a typical WiFi scenario: the transmission distance is set as 150 meters and the bandwidth is set as 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The fixed overhead of a WiFi frame under the Non-HT (Non-High Throughput) and HT-Mixed mode is 20 us and 40 us, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Security enhancement Another challenge of CRKG lies in the increased security risks caused by its hybrid architecture, as security is only as strong as its weakest link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We assume that the terminal security of QAP1 and QAP2 is guaranteed by techniques such as trusted computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Operations that are relevant to secret keys are run in a trusted execution environment, thereby attackers can read neither quantum keys nor channel keys from the hybrid interface on QAP1 and QAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Since the edge forwarding phase employs the OTP encryption scheme, its security depends on the key used for OTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The security of existing CRKG approaches, however, heavily relies on the channel variation and thus suffers from vulnerabilities in slowly varying environments [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' When users have low mobility, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' in a wireless sensor network, there exist in- evitable and unknown temporal correlations between adjacent channel samples, resulting in a large proportion of repeated bit segments in the quantized bit sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Several solutions can be used to facilitate the practical usage of CRKG in slowly varying environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' One solution is to introduce helper de- vices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=', relays and reconfigurable intelligent surface (RIS) to boost the key generation rate and randomness [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' How- ever, this solution encounters some practical problems, such as the unavailability of trust relays and additional hardware overheads of RIS devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Another idea is to scramble these bits segments through some permutation or interleaving tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, the security of the key may be compromised when the permutation information is public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [15] has proposed a new physical-layer secret key generation approach with channel obfuscation, which improved the dynamic property of channel parameters based on random filtering and random antenna scheduling, which have mutual remedying parameters in hiding the obfuscation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 6 YUHUATAI DISTRICT JIANGNING DISTRICT GULI RESIDENTIAL DISTRICT Jiangjunshan Tourism Scenic Area Fangshan Scenic Area XISHANQIAO RESIDENTIAL DISTRICT TIEXINQIAO RESIDENTIAL DISTRICT Qinhuai New River QAP2 Total length: 15km QAP1 Alice QAP2 Bob Bob Quantum Key Eve Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' An illustration of the CR-QKD prototype platform in a metropolitan area network at Nanjing, which is the capital of Jiangsu Province, East-central China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' One is located in Yuhuatai District and the other is at Chinese Network Valley in Jiangning District.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CASE STUDY: AN IMPLEMENTATION OF CR-QKD To realize the concept of CR-QKD, we implement a single- user confidential transmission prototype system in a metropoli- tan area network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Experimental Setup As shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 4, QAP1 and QAP2 are two quantum access points at a distance of fifteen kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Alice and Bob are two remote IoT users in the wireless service ranges of QAP1 and QAP2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Without loss of generality, we zoom in on the wireless access network at Chinese Network Valley, as depicted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Here, QAP2 is composed of a QKD terminal under the series of QKDM-POL40-S for yielding quantum keys 2, a USRP N210 SDR device embedded with the CBX daughterboards for providing a wireless connection service, and a computer under the trusted execution environment for yielding wireless channel keys and distributing quantum keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Both the QKD terminal and USRP N210 are connected to the computer via the ethernet cable in QAP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The end-user, Bob, and a passive eavesdropper, Eve, are realized through two USRP N210 SDR devices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' We design a TDD frame for channel sounding, which consists of a sinusoidal sequence for synchronization and an M-sequence for channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The signal operates at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='605GHz and 20MHz bandwidth to avoid collisions with ubiquitous 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='4GHz signals such as WiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Once Bob receives the channel sounding signal from AP2, it will immediately switch to TX mode and send the same channel sounding signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' By using the same channel sounding signal for channel estimation, the amplitude part of the CSI is further preprocessed and quantized to generate the wireless keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Performance Results Considering the comparable experimental scenarios and results of QAP1 and QAP2, we only take QAP2 as an example for performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 2The quantum keys meet strict key randomness, as they conform to the specification of the GM/T 0005-2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Table II summarizes the secret key sharing results from QAP2 to Bob and Eve in three typical indoor scenarios, namely office, hall and corridor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' First, the measured key generation rates (KGRs) of the channel keys between QAP2 and Bob in above scenarios are 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='4, 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7 and 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7 bits per second (bps), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' They are sufficient for traditional symmetric encryption algorithms (such as AES) to update 256-bit keys every second for secure communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In the random test, we examined a bit sequence of length 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='4 million bits that was obtained at the output of the quantization stage without further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The generated channel keys passed 14 NIST statistical tests, indicating their randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, while the simplified CR-QKD mechanism leads to high KGRs and high randomness, removing the complicated information reconciliation step also results in relatively high key disagreement rates (KDRs) of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='8% between QAP2 and Bob, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The number of person- nel, the frequency of movement, and the switching time of USRP affect the reciprocity of uplink and downlink channels, eventually leading to KDR differences in the noisy office, occasionally infested corridor, and empty hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Meanwhile, along with forwarding quantum keys using non-reconciled channel keys based on channel error correction coding, the need arises to retransmit quantum keys when unsuccessfully decoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The corresponding retransmission rates (RRs) using Polar codes from QAP2 to Bob are 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='6%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1%, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7%, respectively, which are proportional to the KDRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' To demonstrate the security of our proposed scheme, we also evaluate the quantum key cracking performance of the near-end eavesdropper Eve in terms of KDR and cracking rate (CR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The KDRs between QAP2 and Eve under these three scenarios are all around 50%, where the line of sight in the straight corridor contributes to a relatively lower KDR but is still above 45%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' What’s more, the experimental results show that the CRs of Eve in the three scenarios are all zero, which means that none of the quantum keys have been cracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' CONCLUSION AND FUTURE DIRECTIONS Integrating QKD into IoT networks is beneficial for QKD’s practical deployment and end-user’s security enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' (0)目7 TABLE II THE QUANTUM KEY WIRELESS DISTRIBUTION PERFORMANCE IN THREE INDOOR SCENARIOS Scenario QAP2 - Bob QAP2 - Eve Metrics KGR/bps NIST KDR RR KDR CR Office 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='4 14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='6% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1% 0% Hall 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='1% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='2% 0% Corridor 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='8% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='7% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='3% 0% This article proposed a framework of CR-QKD over IoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' QKD and CRKG assembly were adopted for se- cret key sharing over backbone core networks and the last- mile wireless access networks in CR-QKD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' The demonstration of CR-QKD prototype represented a major step towards real-world information theoretically security for wide- area mobile applications, such as confidential VoLTE and confidential VoWiFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Some open issues in future work are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Device Authentication: Considering the hybrid archi- tecture of CR-QKD, it is more vulnerable to spoofing attacks from either user’s side or QAP’s side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' However, neither QKD nor CRKG provides a means to authenticate the transmission source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Therefore, source authentication in CR-QKD should be further studied by using asym- metric cryptography techniques or emerging physical- layer techniques, such as radio frequency fingerprinting identification and physical unclonable function [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Untrusted QAPs: The proposed CR-QKD scheme relies on the trust of the intermediate QAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' In this paper, we use techniques of trust computing to ensure that the the information stored in QAP is protected from external software attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' When a trusted platform is not available, designing a scheme that relaxes this assumption could also be a very good future research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Performance Optimization: In this article, we presented a multi-user edge forwarding strategy, in which quantum keys were allocated as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Unfortunately, its perfor- mance metrics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=', delay, secret key generation rate, and energy efficiency, are limited by those user pairs with weak channel reciprocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' How to optimize these perfor- mance metrics by allocating power or spectrum resources among different user pairs becomes an interesting topic and needs to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' System Integration and Compatibility: Our prototype was built on the USRP platform, which was different from commercial-off-the-shelf (COTS) devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' It is un- known whether these performances are still achievable on existing communication standards and whether CR-QKD will affect the network efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' More studies should be done on its system integration and compatibility issues, including frame format design, key management scheme and efficiency evaluation in practical communication sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank our colleagues Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Linning Peng, Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Yanjun Ding, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Dong Wang, Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Siyun Wu and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Xuyang Wang from the Purple Mountain Laboratories, for their help with the experimental platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' This work was supported in part by the National Key Research and Development Program of China under Grant 2020YFE0200600 and 2022YFB2902202, in part by the National Natural Science Foundation of China under Grant 62171121, in part by the Natural Science Foun- dation of Jiangsu Province under Grant BK20211160 and in part by Jiangsu Provincial Key Laboratory of Network and Information Security under Grant BM2003201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Sood, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Yu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Nguyen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Xiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Feng, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zhang, “A tutorial on next generation heterogeneous IoT networks and node authentication,” IEEE Internet of Things Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 120–126, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hossain, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Muhammad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Rahman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Abdul, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Ale- laiwi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Alamri, “Toward end-to-end biomet rics-based security for IoT infrastructure,” IEEE Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 44–51, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Imre, “Quantum communications: explained for communication en- gineers,” IEEE Communications Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 28–35, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Wang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Rahman, “Quantum-enabled 6G wireless networks: Opportunities and challenges,” IEEE Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 58–69, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zhang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Jorswieck, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Xiao, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hu, “Physical layer key generation in 5G and beyond wireless communications: Challenges and opportunities,” Entropy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 21, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Fr¨ohlich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Dynes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Lucamarini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Sharpe, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Yuan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Shields, “A quantum access network,” Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 501, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 69–72, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [7] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 733–742, May 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Bennett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Brassard, “Quantum cryptography: Public key distribution and coin tossing,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=', Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 175, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 175–179, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Chun, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Choi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Faulkner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Clarke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Barber, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' George, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Capon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Niskanen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Wabnig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' O’Brien, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Bitauld, “Handheld free space quantum key distribution with dynamic motion compensation,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Express, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 6784–6795, Mar 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [10] Elmabrok, Osama, Razavi, and Mohsen, “Wireless quantum key distribu- tion in indoor environments,” Journal of the Optical Society of America B Optical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 197–207, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Marshall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hanzo, “A new frontier for IoT security emerging from three decades of key generation relying on wireless channels,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 138 406–138 446, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zhang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hu, “Encrypting wireless communica- tions on the fly using one-time pad and key generation,” IEEE Internet of Things Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 357–369, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Aldaghri and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Mahdavifar, “Physical layer secret key generation in static environments,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Forensics Security, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 2692–2705, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Staat, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Elders-Boll, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zenger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Paar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hu, “Reconfigurable intelligent surface for physical layer key generation: Constructive or destructive?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' IEEE Wireless Communications, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1–12, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' [15] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Liu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' Hu, “Fast and secure key generation with channel obfuscation in slowly varying environments,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' IEEE INFOCOM, Virtual Conference, May 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} +page_content=' 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AzT4oBgHgl3EQfffxn/content/2301.01453v1.pdf'} diff --git a/3dAyT4oBgHgl3EQfb_e6/content/tmp_files/2301.00275v1.pdf.txt b/3dAyT4oBgHgl3EQfb_e6/content/tmp_files/2301.00275v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c7731aaf220d3b20469044204a895f36d074223 --- /dev/null +++ b/3dAyT4oBgHgl3EQfb_e6/content/tmp_files/2301.00275v1.pdf.txt @@ -0,0 +1,1228 @@ +Normal and anomalous diffusion in a bouncing ball over an irregular surface +Ana Laura Boscoloa,∗, Valdir Barbosa da Silva Juniora, Luiz Antonio Barreiroa +aSão Paulo State University (Unesp), Institute of Geosciences and Exact Sciences, +Physics Department, CEP 13506-900, Rio Claro, São Paulo, Brazil +Abstract +The problem of a bouncing ball on a non-planar surface is investigated. We discovered that surface undulation adds +a horizontal component to the impact force, which acquires a random character. Some aspects of Brownian motion +are found in the horizontal distribution of the particle. On the x-axis, normal and super diffusion are observed. For +the probability density’s functional form, a scaling hypothesis is presented. +Keywords: +Scaling Hypothesis, Anomalous Diffusion, Brownian Motion +1. Introduction +Diffusion is a common natural phenomenon and generally occurs when a system moves toward the equilibrium +state [1]. Many domains employ the notion of diffusion, including physics (particle diffusion), chemistry, biology, +sociology, economics, and finance [2, 3, 4]. They all represent the fundamental concept of diffusion, which asserts +that a substance or collection expands away from a point or location where that material or collection is more +concentrated. In a diffusion process in a set of moving elements - energy, linear momentum, atoms, molecules, cells, +animals, etc - each element performs a random trajectory. As a result of this highly irregular individual movement, +the ensemble diffuses. Many non-linear systems also present a diffusive behavior in your phase space. Modeling +such a dynamic system has become one of the most challenging subjects among scientists. The modeling helps to +understand in many cases how the system evolves in time [5, 6, 7]. +On a macroscopic level, the average collective behavior, in contrast to the microscopic individual movement, +shows great regularity and follows well-defined dynamic laws. The non-linear dynamic formulation of these transport +phenomena, as well as the diffusion equation, are two ways to describe the diffusion phenomena. The form of the +temporal dependence of the mean squared distance (MSD), +� +x2� +∝ t2µ, or, equivalently, of the variance, allows +classifying the type of diffusion. For µ = 1/2 we have the usual or normal diffusion, which can be described by +Fick’s laws. Otherwise, we have an anomalous diffusion (or non-Fickian diffusion). When µ > 1/2 the case is +classified as superdiffusive [8, 9] and for µ < 1/2 we have the subdiffusive case [10, 11]. Indeed, a wide diversity of +systems presents a non-linear growth of the mean squared displacement. +In this work, we explore the diffusive behavior that occurs in a free-falling particle colliding with a non-planar +surface. Compared to a flat surface, on which the falling particles maintain their velocity in the horizontal direction, +a non-planar surface introduces changes in the horizontal component of velocity after each collision. This creates +a spread in the absolute value of the horizontal component of velocity as well as in its signal. Thus, in section +2 we study the dynamics of the model, in which the equations of motion are established, and how the iterative +process takes place. Some special points are explored in 2.3, for which no diffusion is observed. In section 3, the +randomness of the horizontal component of the collision force is studied. Also, the diffusion in the signal of the +horizontal component of velocity and its relation to the random walk problem is explored. Section 4devoted to +discussing the behavior of the mean square displacement in relation to the initial collision point and the Probability +Distribution Function (PDF) numerically and analitically. In section 5, the conclusions and final considerations +about the problem addressed are presented. +2. The Model +We now discuss how to construct the equations of the mapping that describe the dynamics of the particles. +The model under study consists of an ensemble of non-interacting classical particles of mass m travelling in the +∗Corresponding author +Email address: ana.boscolo@unesp.br (Ana Laura Boscolo) +Preprint submitted to Elsevier +January 3, 2023 +arXiv:2301.00275v1 [nlin.CD] 31 Dec 2022 + +2.1 +The Map +2 +presence of a constant gravitational field g and colliding with a non-flat ground via elastic collisions. The ground +is parametrically described by: +x(p) = α p +y(p) = β [1 + cos (p)] , +(1) +The figure 1 shows an example of a ground. +Figure 1: Graph obtained from equations (1) using the parameters α = 0.01 and β = 0.001. +Here it is worth noting that if the β parameter is null then the floor becomes flat, recovering the traditional +Bouncer model [12] with a static floor. However, different from the traditional Bouncer model, if β ̸= 0, the particles +gain an extra degree of freedom, with movement in the x-direction too. Also, as in the Bouncer model, the action of +the constant gravitational field g is responsable for the return mechanism of the particle for the next collision with +the floor. The conservation of energy during the collision is controlled by a parameter which is called the coefficient +of restitution and it is denoted by γ. Indeed γ plays a key role in our model. For γ = 1 the conservative dynamics +is observed. However, if 0 < γ < 1 we found a dissipative behavior. +2.1. The Map +We now move on to the study of the temporal evolution of particles, obtaining the position coordinates of +the collision points and their respective velocities. The dynamic evolution of the particle can be described by the +Newton’s equation of motion +mdv +dt = F grav + F col, +(2) +where F grav = mg is the gravitational force acting on the particle and F col represents the instantaneous force of +collision with the ground. We will assume that the collision force only changes the velocity component orthogonal +to the surface. It is also an acceptable assumption that during the collision process the force F col has an extremely +rapid variation. +A typical path taken by the particles is shown in the figure 2. After the nth collision at the point defined by the +parameter pn, the particle travels in the gravitational field until it collides at the point pn+1. This journey takes a +δtn,n+1 time and continues incessantly if no dissipation is taken into account. +Figure 2: Schematic drawing of the trajectory of a particle, with its collision points and the respective normal vectors. +The normal vectors at each collision point are also shown. The unit normal and tangent vectors at the point pn +can be written in terms of the Cartesian vectors as +ˆnn = (−λn i + j) +� +1 + λ2n +and ˆtn = (i + λn j) +� +1 + λ2n +(3) + +0.015 +0.010 +0.005 +0.000 +0.10 +0.05 +0.00 +0.05 +0.10 +3.ina +fin+1 +Stn,n+1 +Pn +Pn+12.1 +The Map +3 +where λn is the local inclination of the ground, which for the functions in (1), is given by +λn = (dy/dp)pn +(dx/dp)pn += −β +αsin (pn) . +(4) +Since motion in the gravitational field is a well-known problem, the fundamental question in determining the +dynamic evolution of the particle will be to find the points of collision with the ground. To proceed with this +determination, we define the following two functions +GX(p, t) = x (p) − +� +x (pn) + v(r) +xn t +� +GY (p, t) = y (p) − +� +y (pn) + v(r) +yn t − g +2t2� +, +(5) +where +� +v(r) +xn , v(r) +yn +� +is the velocity of the particle after it collides at point pn. The next point pn+1 and the travel time +δtn,n+1 = (tn+1−tn) spent by the particle between pn and pn+1 are obtained by solving the system of transcendental +equations +� +GX(pn+1, δtn,n+1) = 0 +GY (pn+1, δtn,n+1) = 0. +(6) +In such a way, if the particles make a trip with N collisions, the total time spent will be +tN = +N +� +n=1 +δtn−1,n with t0 = 0. +(7) +In our model, we assume that only the component of the velocity normal to the surface at the collision point is +altered (inverted) [13]. Then, at the instant of collision, the law of reflection relating the incident velocity vector +v(i) +n to the reflected velocity vector v(r) +n +is, +v(r) +n += +� +v(i) +n · ˆtn +� +ˆtn − γn +� +v(i) +n · ˆnn +� +ˆnn. +(8) +Obviously, the velocity vector, incident at a point pn+1, is related to the velocity vector reflected at the previous point pn as +v(i) +n+1 = v(r) +xn i + +� +v(r) +yn − g δtn,n+1 +� +j. +Now we can define the following dimensionless variables ¯x(p) = x(p)/gt2 +N, ¯y(p) = y(p)/gt2 +N, ¯v(r) +n += v(r) +n /gtN and φn = tn/tN +, where tN is defined in (7). Therefore, the dimensionless velocity vector components in (8) take the form +¯v(r) +xn+1 = +� +1 − γn+1λ2 +n+1 +� +¯v(r) +xn + λn+1 (1 + γn+1) +� +¯v(r) +yn − δφn,n+1 +� +1 + λ2 +n+1 +¯v(r) +yn+1 = +λn+1 (1 + γn+1) ¯v(r) +xn + +� +λ2 +n+1 − γn+1 +� � +¯v(r) +yn − δφn,n+1 +� +1 + λ2 +n+1 +. +(9) +and the system (6) becomes +� +� +� +� +� +� +� +� +� +pn+1 = pn + ¯v(r) +xn +¯α δφn,n+1 +cos (pn+1) = cos (pn) + ¯v(r) +yn +¯β δφn,n+1 − 1 +2¯β δφ2 +n,n+1 +(10) +where ¯α = α/gt2 +N and ¯β = β/gt2 +N. Given the values of pn, ¯v(r) +xn and ¯v(r) +yn of the nth iteraction, the set of equations +(10) produce the values of pn+1 and the travel time δφn,n+1 which allows us to find ¯v(r) +xn+1 and ¯v(r) +yn+1through (9). +After that, the iteractive process restart. +The last ingredient is the dimensionless energy +¯En = 1 +2 +�� +¯v(r) +xn +�2 ++ +� +¯v(r) +yn +�2� ++ ¯yn +(11) +which is used to establish the initial conditions to be chosen so that all particles in the ensemble have the same +initial energy. + +2.2 +Conservative case +4 +2.2. Conservative case +We shall only consider the conservative scenario, when γn=γn+1 = 1. Since we choose ¯β ≪ 1, it is appropriate +to consider that the point of collision with the ground has a height ¯y(pn) ≃ ¯y(pn+1) ≃ 0 , but with local slope not +necessarily zero. This approach avoids transcendental equations and simplifies the calculation. As a consequence, +the second of the equations in (10) yields δφn,n+1 = φn,n+1 − φn = 2¯v(r) +yn . Finally, a simplified form of the map +equations used to explain motion is expressed as +¯v(r) +xn+1 =F1 +� +¯v(r) +xn , ¯v(r) +yn , pn +� +¯v(r) +yn+1 = +���F2 +� +¯v(r) +xn , ¯v(r) +yn , pn +���� +pn+1 =F3 +� +¯v(r) +xn , ¯v(r) +yn , pn +� +(12) +where +F1 +� +¯v(r) +xn , ¯v(r) +yn , pn +� += +� +1 − ¯λ2 +n +� +¯v(r) +xn − 2¯λn¯v(r) +yn +1 + ¯λ2n +F2 +� +¯v(r) +xn , ¯v(r) +yn , pn +� +=2¯λn¯v(r) +xn + +� +1 − ¯λ2 +n +� +¯v(r) +yn +1 + ¯λ2n +F3 +� +¯v(r) +xn , ¯v(r) +yn , pn +� +=pn + 2 +α ¯v(r) +xn ¯v(r) +yn +(13) +and were defined +¯λn = λn+1 = +∂ ¯YS/∂p +∂ ¯ +XS/∂p +���� +pn+ 2 +α ¯v(r) +xn ¯v(r) +yn += − +¯β +¯αsin +� +pn + 2 +α ¯v(r) +xn ¯v(r) +yn +� +. +(14) +The ground was assumed to be flat, as a consequence there is a small possibility of the particle presenting a negative +value for ¯v(r) +y . This non-physical situation is bypassed by introducing the modulus in the second equation of (12). +This means that if such a case happens, the particle is reinjected back to the dynamics with the same velocity but +with a positive direction. +2.2.1. Jacobian Matrix +The Jacobian matrix for this dynamical system may be simply calculated using equations (12-14), +J = +� +� +� +∂F1 +∂vx +∂F1 +∂vy +∂F1 +∂p +∂F2 +∂vx +∂F2 +∂vy +∂F2 +∂p +∂F3 +∂vx +∂F3 +∂vy +∂F3 +∂p +� +� +� +leading to1 +∂F1 +∂vx += +¯α4 + 4 ¯βvy +2 cos +� +p + +2vxvy +¯ +α +� � +¯α2 − ¯β2 sin2 � +p + +2vxvy +¯ +α +�� +− ¯β4 sin4 � +p + +2vxvy +¯ +α +� +− 4¯α ¯β2vxvy sin +� +2p + +4vxvy +¯ +α +� +� +¯α2 + ¯β2 sin2 � +p + +2vxvy +¯ +α +��2 +∂F1 +∂vy += +2 ¯β +� +¯α +� +−2 ¯βvx +2 sin +� +2p + +4vxvy +¯ +α +� ++ ¯α2 sin +� +p + +2vxvy +¯ +α +� ++ ¯β2 sin3 � +p + +2vxvy +¯ +α +�� ++ 2vxvy cos +� +p + +2vxvy +¯ +α +� � +¯α2 − ¯β2 sin2 � +p + +2vxvy +¯ +α +��� +� +¯α2 + ¯β2 sin2 � +p + +2vxvy +¯ +α +��2 +∂F1 +∂p += +2¯α ¯β cos +� +p + +2vxvy +¯ +α +� � +¯α2vy − ¯β sin +� +p + +2vxvy +¯ +α +� � +¯βvy sin +� +p + +2vxvy +¯ +α +� ++ 2¯αvx +�� +� +¯α2 + ¯β2 sin2 � +p + +2vxvy +¯ +α +��2 +1In Jacobian expressions, we utilize (vx, vy, p) rather than (¯v(r) +xn , ¯v(r) +yn , pn) to simplify notation. + +2.3 +Periodic points +5 +∂F2 +∂vx += +− +2 ¯β +� +¯α +� +2 ¯βvy +2 sin +� +2p + +4vxvy +¯ +α +� ++ ¯α2 sin +� +p + +2vxvy +¯ +α +� ++ ¯β2 sin3 � +p + +2vxvy +¯ +α +�� ++ 2vxvy cos +� +p + +2vxvy +¯ +α +� � +¯α2 − ¯β2 sin2 � +p + +2vxvy +¯ +α +��� +� +¯α2 + ¯β2 sin2 � +p + +2vxvy +¯ +α +��2 +∂F2 +∂vy += +¯α4 − ¯β +� +4vx +2 cos +� +p + +2vxvy +¯ +α +� � +¯α2 − ¯β2 sin2 � +p + +2vxvy +¯ +α +�� ++ ¯β3 sin4 � +p + +2vxvy +¯ +α +� ++ 4¯α ¯βvxvy sin +� +2p + +4vxvy +¯ +α +�� +� +¯α2 + ¯β2 sin2 � +p + +2vxvy +¯ +α +��2 +∂F2 +∂p += +− +2¯α ¯β cos +� +p + +2vxvy +¯ +α +� � +¯β sin +� +p + +2vxvy +¯ +α +� � +2¯αvy − ¯βvx sin +� +p + +2vxvy +¯ +α +�� ++ ¯α2vx +� +� +¯α2 + ¯β2 sin2 � +p + +2vxvy +¯ +α +��2 +∂F3 +∂vx += +2vy +¯α +∂F3 +∂vy += +2vx +¯α +∂F3 +∂p += +1 +This Jacobian matrix’s determinant is equal to one, confirming that the system is indeed conservative. +2.3. Periodic points +We can anticipate the occurrence of some exceptional points using the physics of the problem. These are known +as fixed points, to which the dynamical system returns after one iteration (period-one fixed point), two iterations +(period-two fixed point), or n iterations (period-n fixed point). The figure 3 illustrates two fixed points: (a) Fixed +points for period one and (b) Fixed points for period two. +Figure 3: Examples of fixed points: (a) Fixed point of period one. The dynamical system returns to the point in phase space at each +iteration and (b) the system returns to the point after 2 iterations. +2.3.1. Period-one Point +It is evident that period-one fixed points, as shown in portion (a) of the figure, have a zero local slope. So, as long +as the x component of the initial velocity is zero, the system will not experience any diffusion in the horizontal axis. +A period-one point is obtained by solving the following equations: ¯v(r) +xn+1 = ¯v(r) +xn = 0, ¯v(r) +yn+1 = ¯v(r) +yn and pn+1 = pn +with ¯λn = 0 (zero slope). We can verify the fact considering first eq (14) +¯λn = 0 ⇒ sin +� +pn + 2 +α ¯v(r) +xn ¯v(r) +yn +� += 0 +⇒ +¯v(r) +xn =0 +pn = mπ, +where m is a integer. These points indicate the locations of the peaks and valleys in Figure 1 - part (a). Thus +¯v(r) +xn+1 += +F1 +� +0, ¯v(r) +yn , mπ +� += 0 +¯v(r) +yn+1 += +F2 +� +0, ¯v(r) +yn , mπ +� += ¯v(r) +yn +(15) +pn+1 += +F3 +� +0, ¯v(r) +yn , mπ +� += mπ +We have the following physical situation: If a particle is chosen whose horizontal component of velocity is zero, in +a zero slope point, clearly the x-coordinate of the particle will never change and the particle does not scatter in the +x-direction. + +P +Pn = Pn+1 +pn +Pn+1 +x +T元 +X +(a) +(b)2.3 +Periodic points +6 +2.3.2. Period-two Points +We now consider points with non-zero slope. In general, the particle gains a non-zero horizontal component +to the velocity and then diffuses along the horizontal axis. Nevertheless, depending on the initial conditions, it is +possible for the particle to strike the surface at point pn with velocity ⃗vn, reflect there, then it reaches point pn+1 +with velocity ⃗vn+1, where it will then reflect again and go back to point pn with velocity ⃗vn. Part (b) of Fig. 3 +depicts an illustration of this kind.. Inspired by the figure, consider points connected by ¯v(r) +xn+2 = −¯v(r) +xn+1 = ¯v(r) +xn , +¯v(r) +yn+2 = ¯v(r) +yn+1 = ¯v(r) +yn , pn+2 = pn, ¯y(pn+1) = ¯y(pn) and opposite local slopes λn+1 = −λn. +Taking into account the figure 3 portion (b) , the points pn and pn+1 must be connected by +� +pn = −π − χ +pn+1 = π + χ +with 0 < χ < π +where we are solely concerned with the most straightforward solution. Then, with the help of equations (10), we +can write +¯v(r) +xn ¯v(r) +yn = ¯α (π + χ) . +(16) +In addition, the first of the equations (13) yields +¯v(r) +yn = +¯α +¯βsin (χ) ¯v(r) +xn . +(17) +These results allow us to determine both ¯v(r) +xn and ¯v(r) +yn as functions of χ. So the period two fixed point is written as +¯v(r) +xn += +± +� +¯β (π + χ) sin (χ) +¯v(r) +yn += +¯α (π + χ) +�¯β (π + χ) sin (χ) +, +pn += +∓ (π + χ) +Figure 4 illustrates these fixed points. The middle points in black in this picture indicate the period-1 fixed +points. The graphic also illustrates the effect of the β−parameter on the formation of period-2 fixed points. The +points are calculated by altering the value of χ from 0 to π, and each color indicates a β parameter value: red +(β = 0.00001), green (β = 0.00002),..., purple (β = 0.0001). α = 0.001 is used for all points. +Figure 4: Period one fixed points are represented by the black dots in the center of the line. The other points are the period 2 fixed +points.The gray dots at the end of the curves are the points obtained with the value χ = π/2. + +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +8.03 +-0.02 +-0.01 +0.00 +0.01 +0.02 +0.03 +Vx7 +The choice χ = π/2 is used to calculate the gray dots in the figure 4. Each curve is divided into two branches by +these points. The points that make up the branches we name external have χ > π/2, whereas the points that make +up the branches we term internal have χ < π/2 . Consider the eigenvalues of the Jacobian matrix to categorize +the stability of these points. The external points (χ ≥ π/2) can be classified as node-type stable points since the +modules of their Jacobian matrix eigenvalues are all equal to 1. On the other hand, because all of the eigenvalues +are real with one positive and the others negatives, the internal points (χ < π/2) are categorized as unstable points +of the saddle type. Therefore, the gray dots in the phase space represent saddle-node bifurcations [14]. +Many additional sorts of fixed points may exist, but the purpose of this paper is to study the diffusion process +along the x-axis. +3. Diffusion Proccess +3.1. The stochastic character of force +Clearly, unless we are in some special initial point, the particles must diffuse in the x-direction. This diffusion is +caused by the collision force with the ground. Due to the irregular nature of the ground, the collision force F col has +components in both horizontal and vertical directions. It is intuitive to notice that the horizontal component presents +different magnitudes and directions at each collision. To understand the behavior of this horizontal component of +the collision force, we can describe it as +¯Fcolx(φn) = ∆¯v +¯τ +���� +pn += ¯v(r) +xn − ¯v(i) +xn +¯τ += ¯v(r) +xn − ¯v(r) +xn−1 +¯τ +where ¯τ is the dimensionless collision time, which is extremely small. We will also assume that the collision force +is approximately constant during the collision time and a typical example of what this force looks like is shown in +figure 5. +Figure 5: Typical behavior of the horizontal component of the collision force. Here we have used ¯α = 0.01 and ¯β = 0.005. The graph +has two regions with different scales. On the left we have the region magnified between φ = 0.000 and φ = 0.020 and on the right, after +a cut in the graph, the normal scale from φ = 0.5 to φ = 1.0 is shown. +The width of each rectangle represents the collision time and despite the dynamics being well known and +the irregularities in the ground having a periodicity, the numerical results presented show that the effects of the +horizontal component of this force has a behavior comparable to a stochastic force. It is actually extremely difficult +to tell whether a sequence is random or chaotic, but there are some proposed procedures to distinguish between +these two behaviors. In this work we will make use of the permutation entropy (PE) method [15, 16] to establish the +randomness of the time series produced by the collision force. Denoting the time series as {St}t=1,...,T the method +consists in defining subsets of order O, forming the set S = {{S1, S2, . . . , SO}, {S2, S3, . . . , SO+1}, . . . , {ST −O+1, . . . , +ST −1, ST }}. We then compare consecutive values from each subset to establish the associated permutation. For +example, {S1 < S2 < . . . < SO} represents the permutation {1, 2, ..., O}, while {S2 < S1 < . . . < SO} represents + +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +0.0 +0.005 +0.010 +0.015 +0.020 +0.6 +0.8 +1.08 +the permutation 2, 1, ..., O and so on, yielding the set of all permutations associated with the sequence S, named +Π(S). Then, the set of all O! possible permutations πi of the numbers {1, 2, ..., O} are constructed. The relative +frequency of each permutation πi can be calculated by counting the number of times the permutation πi is found +in the set Π(S) divided by the total number of sequences, +Pi = Number of times that πi appears in Π(S) +T − O + 1 +. +(18) +and the normalized permutation entropy function is written as, +PEO = − +1 +log2(O!) +O! +� +i=1 +Pi log2(Pi). +(19) +Formulas (18) and (19) were applied to the temporal sequences of collision forces for three different initial +conditions and also different orders O. The table 1 shows the results obtained. +floor parameters +initial condiction +O = 3 +O = 4 +O = 5 +O = 6 +α = 0.01 +β = 0.05 +p0 = −0.033 +0.998569 +0.995189 +0.981222 +0.92671 +p0 = 0.032 +0.999633 +0.995120 +0.982245 +0.925946 +α = 0.01 +β = 0.0005 +p0 = −0.033 +0.998874 +0.994082 +0.986440 +0.935262 +p0 = 0.032 +0.999501 +0.996295 +0.984878 +0.934281 +Table 1: The initial conditions are chosen in order to vary the initial point (x(p0), y(p0)) and keeping the energy ¯E = 4 constant. +The smaller the PEO is, the more regular and more deterministic the time series is. Contrarily, the closer to 1 +the value of PEO is, the more noisy and random the time series is. The results allow us to assume that the force +is random. +4. Probability Distribution Function (PDF) +This section’s major purpose is to establish the probability distribution function (PDF) Ψ(x, t), which provides +us the probability of the particle being on the coordinate x at time t, and what it has to do with normal and +superdiffusive processes. Among the various diffusive processes, Brownian motion is the prototype for the description +of non-equilibrium dynamical systems. Due to the stochastic behavior of the collision force, the jumps performed +by the particles also reproduce characteristics of random walk. We can comprehend this by calculating the chance +of each particle going to the right. After each impact, we obtain the x−component of the velocity. Then, by +examining the sign of these velocities and associating +1 for vx > 0 and 0 for vx < 0, we can count the number of +jumps to the right and derive the evolution of this probability as the number of jumps increases. It is appropriate +at this point to introduce an index that specifies the initial condition (ν), which is used to compute the Probability +Density Function (PDF) for the complete ensemble. So, starting with an initial state labeled by ν, the probability +of jumping to the right after n jumps is calculated as follows: +Pr−jump(n, ν) = 1 +n +n +� +i=1 +SgnPlus(v(ν) +x,i ) where SgnPlus(v(ν) +x,i ) = +� +1 +if vx > 0 +0 +if vx < 0 +Figure 6, on the left, shows examples of the time progression of individual particle jumps for four distinct initial +conditions and two ground parameter adjustments, as well as the corresponding PDFs Ψ(x, t). With time evolution, +the left/right jump probabilities for a ground with α = 0.01 and β = 0.005 tend to be 0.5 very quickly as we can +see into upper graphic on the left. However, if the beta parameter is set to β = 0.0005 the graph indicates an initial +oscillation, but the probability ultimately tends to reach 0.5. +The coordinates of the collision points and the travel time between one point and the next are obtained from +the mapping given in equations (9) and (10). It is obvious that the travel time varies between jumps. However, for +our analysis, it is critical to obtain the particle’s position as a function of time with equal time intervals. This is +simple because the particle moves in a gravitational field g, and we can easily calculate its position as a function of +time. The time is then normalized so that the maximum time equals one. So, to get the probability distribution, +for all iterative processes, we begin by subtracting the starting position of the particles. As a result, all of the +particles in the ensemble start from the same position. In our scenario, we have 2000 particles performing 4000 + +9 +Figure 6: The first graphic of each column contains time evolution examples for the likelihood of a single particle jumping to the right. +The difference is in the β parameter value, which is lowered to one-tenth and one-hundredth of its initial value in the columns on the +left. The evolution of the 4-particle leaps (4 initial conditions) is explored in the graphs. The different initial conditions for the particles +are obtained by changing the initial parameter p in the functions x(p) and y(p) in Eq (1) and keeping the energy ¯E = 4 constant. The +selected p-parameters are shown in the figures. The respective contour plots for the probability distributions are shown on the right. +leaps, totaling 8 million collision points, but it is clear that the number of points as a function of time depends on +the choice of interval dt and can be much higher. To demonstrate the procedure, the simulation is configured so +that each particle in the ensemble has an energy of E = 4. The outcomes for two different types of grounds are +shown in Figure 6 on the right. +The first PDF graph was obtained with the parameters α = 0.01 and β = 0.005, and shows a probability density +region following a format very similar to a Gaussian distribution. The second pdf, obtained with the parameters +α = 0.01 and β = 0.0005, has an extremely anomalous diffusion in the early part of its time evolution, however +when the time evolution takes place, the PDF apparently starts to show a Gaussian behavior. In order to have a +better understanding of this behavior, we studied the moments associated with each distribution. Inspired by the +Gaussian form of normal diffusion, with an anomalous diffusion we make a scaling hypothesis [17] so that we can +express the anomalous distribution as +Ψµ(x, t) = +� a +π +1 +tµ exp +� +−a +� x +tµ +�2� +. +(20) +The associated moments are easily obtained as +⟨|x(t)|m⟩ = +∞ +ˆ +−∞ +xmΨµ(x, t) dx = +1 +√ +amπ Γ +�m + 1 +2 +� +tmµ. +(21) +The result shows a behavior of MSD as +� +x2� +∝ t2µ, therefore normal distribution has a scale parameter µ = 1/2. +If µ < 1/2 we have a subdifussive process and for µ > 1/2 we found a superdiffusive behavior. Figure 7 shows +the results of the moments calculations for two different grounds. We can observe that at left we obtain the scale +µ = 0.5 and at right we obtain µ = 0.65. So, we have two distinct behaviors: at left a normal diffusion and at right +we have a superdiffusive behavior. + +0.8 +500 +0.00200 +.0.6 +x-coordinate +0.00175 +0.00150 +α = 0.01 +0.00125 +pue +0.4 +0.00100 +β = 0.005 +p=2.82156 +0.00075 +0.2 +p=-2.50841 +0.00050 +p=2.19525 +500 +0.00025 +0.0 +p=-1.88209 +0 +1000 +2000 +3000 +4000 +number of iteractions +0.2 +0.4 +0.6 +0.8 +1.0 +time +1.0F +4000 +0.8 +2000 +0.00030 +X-coordinate +0.00025 +α= 0.01 +0.00020 +and +0.4 +p=-2.82156 +0.00015 +β=0.0005 +p=-2.50841 +2000 +0.00010 +0.2 +p=-2.19525 +0.00005 +0.0E +p=1.88209 +4000 +0 +1000 +2000 +3000 +4000 +number of iteractions +0.2 +0.4 +0.8 +0.8 +1.0 +time10 +Figure 8: PDF’s rescaled by the factor ξ = x2/ +� +x(t)2� +for four distinct times. The theoretical prediction given in Eq. (22) with an +a = 3.75 × 10−8 is shown by the black dot-dashed line. +Figure 7: The red lines represent equations of lines with powers mµ. +At left we have a normal diffusion and at right we have a +superdiffusive diffusion. We can see that almost half of the time evolution has passed before the superdiffusive behavior with scale +µ = 0.65 manifests. +The scaling hypothesis is carried forward using equation (21) to obtain t2µ = a√π +� +x(t)2� +/Γ (3/2), which +enables us to specify the subsequent function +F(ξ) = tµΨµ(x, t) = +� a +π exp +� +−Γ (3/2) +√π +ξ +� +(22) +where ξ = x2/ +� +x(t)2� +. Using the PDF data for the superdiffusive process (µ = 0.65) we obtain F(ξ) numerically +and the results for t = 0.76, t = 0.765, t = 0.89 and t = 0.995 are presented in the figure 8 . The only parameter +that can be adjusted in the theoretical forecast stated in Eq (22) is the value of a. We get a remarkable agreement +with the simulation findings when we choose a = 3.75 × 10−8. The black dot-dashed line on the graph denotes the +theoretical result obtained in equation (22). We observe that the theoretical modeling and the simulation outcome +start to diverge for periods of time less than 76.5% of the overall duration of the iterative procedure. Rescaling +the data, all simulation points for times more than this amount lie exactly on the same curve. This was already a +foregone conclusion if we look at the second PDF in the figure 6, which shows quite anomalous behavior for times +less than 0.8. Before this time has elapsed the particles display a strongly anomalous diffusion with a scale that +must rely on the moment being estimated, ⟨|x|m⟩ ∝ tmµ(m), [18]. +5. Conclusions and Outlook +In this work, we have studied a falling particle in the gravitational field colliding with a non-plane surface. We +could observe that the horizontal component of the collision force presented a stochastic behavior. This was verified + +α=0.01andβ=0.005 +α = 0.01and β +3=0.0005 +1016 +1016 +<1 x(t) /*)~ ++x0.65 +<1 x(t) 1*)~t+x0.5 +1012 +1012 +<1 x() 3)~ 3-0.65 + 0. The parameter space +is also a function space. The model M is often described implicitly, by an ordinary or partial +differential equation where one or more coefficients are determined by the parameter θ. +Assumption 2.1. There exists a unique best model M† ∈ M and a unique best parameter +θ† ∈ Θ such that among all model-unknown pairs (M′, θ′) ∈ M × Θ, the corresponding ‘best +state’ u† := M†(θ†) describes the phenomenon of interest most accurately. +The assumption of a unique θ† is commonly made in the context of frequentist statistics, +where θ† is often referred to as the ‘true parameter’. However, in the context of experimental +design for statistical inverse problems where observations are assumed to have the form (1.1), +the question of whether a parameter is the true parameter makes sense only when a parameter- +to-state map or model has been specified. Hence, for θ† to have the interpretation of the ‘true +parameter’, we must also fix a unique ‘true model’ M†. +Remark 2.2. In this paper, we shall consider the setting where the best model M† in Assump- +tion 2.1 is unknown. However, one may also consider the best model M† to be a model that is +known but is too expensive to use ‘frequently’, where the meaning of ‘frequently’ depends on the +5 + +context or the target application. The model M can be considered as an emulator, a surrogate, +or a reduced order model; M ideally has the property that it approximates M† reasonably well +and is cheaper to evaluate than M†. +Assumption 2.3. There is a fixed collection of measurement functions (ℓi)i∈I indexed by a +countable set I ⊂ N, where each ℓi is a continuous linear mapping from U to Rd for some d ∈ N. +In addition, every admissible observation operator O has the form +O : U → RNd, +u �→ (ℓi1(u), . . . , ℓiN (u)), +ij ∈ I, +(2.1) +where the (ℓij)N +j=1 are distinct. Associated to the collection (ℓij)N +j=1 of measurement functions in +(2.1) is a collection of random variables (εij)N +j=1 that are independent and identically N(0, Σ0)- +distributed, where Σ0 ∈ Rd×d is positive definite. +The random variables (εij)N +j=1 represent +additive measurement noise. +If U = C(D, ∥ · ∥L∞), then pointwise evaluation functionals of the form ℓij(u) := u(xij) for +some xij ∈ D give an example of the measurement functions in (2.1). +An important consequence of the assumptions on the measurement noise (εij)N +j=1 in Assump- +tion 2.3 is that the RNd-valued random variable satisfies +ε := (εi1, . . . , εiN ) ∼ N(0, Σε), +Σε = diag(Σ0, . . . Σ0) ∈ RNd×Nd. +The last equation above means that Σε is a block-diagonal matrix with identical blocks that +do not depend on O. In particular, Σε depends on the choice of O only via the number N +of observations. One could achieve greater generality by allowing the (εij)N +j=1 in (2.1) to be +statistically correlated or to have different distributions. This generality would allow Σε to +depend not only on N but on O itself. +In the remainder of this section, we will describe the posterior measures that we shall analyse +in this paper. For a measurable misfit Φ : Θ → R, we will write +Z(Φ) := +� +Θ +exp(−Φ(θ′))dµθ(θ′) +to denote the normalisation constant that makes θ′ �→ exp(−Φ(θ′))Z(Φ)−1 a probability density +function with respect to µθ, whenever the normalisation constant belongs to (0, ∞). +Approximate posterior +Given O as in (2.1) and some model M ∈ M , we assume that an +observation y is a noisy observation of state, i.e. a realisation of the Y := RNd-valued random +variable +Y := O ◦ M(θ†) + ε. +(2.2) +Given Assumption 2.3, the observation model (2.2), a prior µθ on θ, the data y and Bayes’ law, +we obtain the approximate misfit Φy,A and the approximate posterior +Φy,A(θ′) := 1 +2∥y − O ◦ M(θ′)∥2 +Σ−1 +ε , +(2.3a) +dµy,A +θ +(θ′) := exp(−Φy,A(θ′)) +Z(Φy,A) +dµθ(θ′). +(2.3b) +6 + +Best posterior +Recall from Assumption 2.1 that u† := M†(θ†) best describes the phenomenon +of interest. For an arbitrary O ∈ O, the ‘best model’ is given by replacing M with M† in (2.2): +Y = O ◦ M†(θ†) + ε. +The corresponding best misfit Φy,† and best posterior µy,† +θ +are defined by +Φy,†(θ′) := 1 +2∥y − O ◦ M†(θ′)∥2 +Σ−1 +ε , +(2.4a) +dµy,† +θ (θ′) := exp(−Φy,†(θ′)) +Z(Φy,†) +dµθ(θ′). +(2.4b) +Given M ∈ M , the corresponding ‘model error’ is given by the difference +δ† := M† − M ∈ D, +D := M† − M . +(2.5) +We refer to D as the ‘model error space’. If the model space M is a vector space, then the +model error space D and M coincide. If M† is not known or too expensive to evaluate, then +so is δ†. For the unique, fixed θ† in Assumption 2.1, define the corresponding ‘state error’ +δ†(θ†) = M†(θ†) − M(θ†) ∈ U. +Rewriting (2.5) as M† = M + δ†, substituting the latter equation into the best observation +model, and using the linearity of O, we obtain +Y = O ◦ M(θ†) + O ◦ δ†(θ†) + ε. +The observation model (2.2) thus corresponds to the assumption of zero observed state error +O ◦ δ†(θ†). +Enhanced noise posterior +One approach that aims to account for the observed state error +is to group the observed state error O ◦ δ†(θ†) with the noise ε to obtain O ◦ δ†(θ†) + ε, and +to model this random variable with an ‘enhanced noise’ random variable [12, 13]. This is also +known as the ‘pre-marginalisation’ approach, e.g. [15]. We approximate the unknown state +error δ†(θ†) with a random variable u, and make the following assumption. +Assumption 2.4. The random variable u that approximates the unknown state error δ†(θ†) is +Gaussian with mean mu and covariance Σu, and is independent of θ ∼ µθ and ε ∼ N(0, Σε). +Given the distributional assumptions in Assumption 2.4 and the linearity assumption on O in +Assumption 2.3, it follows from the properties of Gaussian random variables that the enhanced +noise random variable Ou + ε has the law N(Omu, Σε + OΣuO∗). This yields the enhanced +noise observation model +Y = O ◦ M(θ†) + Ou + ε, +which yields the enhanced noise misfit and enhanced noise posterior +Φy,E(θ′) := 1 +2∥y − O ◦ M(θ′) − Omu∥2 +(Σε+OΣuO∗)−1, +(2.6a) +dµy,E +θ +(θ′) := exp(−Φy,E(θ′)) +Z(Φy,E) +dµθ(θ′). +(2.6b) +7 + +Joint parameter-error posterior +In the enhanced noise approach presented in (2.6), we account +for the uncertainty due to the state error δ†(θ†) by approximating it using a random variable u. +The only unknown that we aim to infer is θ†. In the joint parameter-error inference approach, +one aims to infer (θ†, δ†) jointly, by using a random variable (θ, δ) with prior µθ,δ and using +Bayes’ formula. +Assumption 2.5. The joint prior on the random variable (θ, δ) is a product measure of the +form µθ ⊗ µδ, for µθ ∈ M1(Θ) and µδ ∈ M1(D). +The assumption that the prior µθ,δ on (θ, δ) has product structure is equivalent to the as- +sumption that θ and δ are independent random variables. +Under the observation model +Y = O ◦ M(θ) + O ◦ δ(θ) + ε +and under the distributional assumptions on ε in Assumption 2.3, we have the joint misfit and +joint posterior +Φy,J(θ′, δ′) := 1 +2∥y − O ◦ M(θ′) − O ◦ δ′(θ′)∥2 +Σ−1 +ε , +(2.7a) +dµy,J +θ,δ(θ′, δ′) := exp(−Φy,J(θ′, δ′)) +Z(Φy,J) +dµθ ⊗ µδ(θ′, δ′). +(2.7b) +One important disadvantage of jointly inferring the parameter and model error is that the di- +mension of the space on which one performs inference increases; this tends to make the inference +task more computationally expensive. It is also known that the problem of identifiability may +arise, but we shall not consider the problem of identifiability here. On the other hand, jointly +inferring the parameter and model error is consistent with the Bayesian approach of treating +all unknowns as random variables and updating these distributions using the data. In addition, +the joint inference approach offers the possibility to improve a possibly incorrect model M ∈ M +by posterior estimates of δ†, and thus also the possibility of obtaining better estimates of both +the parameter θ† as well as the state u† = M†(θ†). +Marginal posterior +The marginal approach involves first using the joint inference approach to +obtain the joint posterior µy,J +θ,δ on (θ, δ), and then integrating over all δ′ ∈ D: +µy,M +θ +(S) = +� +S×D +dµy,J +θ,δ(θ′, δ′), +S ∈ B(Θ). +(2.8) +The marginal posterior µy,M +θ +in (2.8) can be approximated by using Monte Carlo integration +of the joint posterior µy,J +θ,δ over δ′ ∈ D. The marginal approach inherits the problem of high +computational cost from the joint inference approach. On the other hand, it has an advantage +over the enhanced noise approach, namely that it involves a Bayesian update of the distribution +of δ. +3. Error bounds on misfits and posteriors +In this section we compare the approaches presented above, by using some local Lipschitz +stability bounds with respect to the Kullback–Leibler divergence. Recall that the Kullback– +Leibler divergence between two probability measures µ and ν on a metric space (E, dE) is given +8 + +by +dKL(µ∥ν) := +�� +E log dµ +dν dµ +µ ≪ ν ++∞ +otherwise. +(3.1) +Given µ ∈ M1(E) and Φ ∈ L1 +µ(E; R), define µΦ ∈ M1(E) by +dµΦ +dµ (x′) = exp(−Φ(x′)) +Z(Φ) +, +Z(Φ) := +� +E +exp(−Φ(x′))dµ(x′). +For a measure µ on some measurable space (E, E) and a measurable function f : E → R, +ess infµf denotes the essential infimum of f with respect to µ. +The following local Lipschitz stability result is due to [23]. +Theorem 3.1. Let µ ∈ M1(E), Φ(1) ∈ L1 +µ(E; R≥0), and Φ(2) ∈ L1 +µ(E; R). +Assume that +ess infµΦ(1) = 0. Then +dKL(µΦ(1)∥µΦ(2)) +≤ 2 exp +� +− min{ess infµΦ(2), 0} + ∥Φ(1)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ +� +∥Φ(1) − Φ(2)∥L1µ, +and thus µΦ(1) is absolutely continuous with respect to µΦ(2). In particular, +max{dKL(µΦ(1)∥µΦ(2)), dKL(µΦ(2)∥µΦ(1))} +≤ 2 exp +� +− min{ess infµΦ(2), 0} + 2∥Φ(1)∥L1µ + 2∥Φ(2)∥L1µ +� +∥Φ(1) − Φ(2)∥L1µ, +and thus µΦ(1) is mutually equivalent to µΦ(2). +Proof. The first statement follows by combining [23, Theorem 11] and [23, Proposition 6]. By +the triangle inequality, +max{∥Φ(1)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ, ∥Φ(2)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ} ≤ 2∥Φ(1)∥L1µ + 2∥Φ(2)∥L1µ, +and thus the second statement follows from the first. +Below, we will use Theorem 3.1 to bound the Kullback–Leibler error between pairs of poste- +riors, for the posteriors defined in Section 2. +Recall that the Hellinger metric between µ, ν ∈ M1(E) is defined by +d2 +H(µ, ν) := +� +E +�� +dµ +dλ − +� +dν +dλ +�2 +dλ, +where λ is any measure such that both µ ≪ λ and ν ≪ λ. The definition of dH(µ, ν) does not +depend on the choice of ν. The Hellinger metric and Kullback–Leibler divergence satisfy +d2 +H(µ, ν) ≤ dKL(µ∥ν), +see e.g. [25, Lemma 2.4]. Hence, the bounds on the Kullback–Leibler error that we present +below imply bounds with respect to the Hellinger metric. +9 + +3.1. Kullback–Leibler error of posteriors on parameter space +3.1.1. Error with respect to the best posterior +Error of approximate posterior with respect to best posterior +In Lemma 3.2 below, we bound +the L1 +µθ error between the approximate misfit Φy,A and the best misfit Φy,† defined in (2.3a) +and (2.4a) respectively. We express the bound in terms of the average observed model error +O ◦ δ†. +Lemma 3.2. Suppose Φy,† ∈ L1 +µθ. If Φy,A ∈ L1 +µθ, then +∥Φy,† − Φy,A∥L1µθ ≤ 2−1/2∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +. +(3.2) +where +∥∥O ◦ δ†∥2 +Σ−1 +ε ∥L1µθ ≤ 21/2� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +. +(3.3) +See Appendix A.1.1 for the proof of Lemma 3.2. +Remark 3.3. Combining (3.2) and (3.3) yields +∥Φy,† − Φy,A∥L1µθ ≤ +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +�2. +Since +∥Φy,†∥L1µθ + ∥Φy,A∥L1µθ ≤ +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +�2, +it follows that (3.2) and (3.3) together are not optimal: they yield a worse bound on ∥Φy,† − +Φy,A∥L1µθ than the bound we could obtain using the triangle inequality. However, the bound +(3.2) is useful, because it bounds ∥Φy,† − Φy,A∥L1µθ in terms of the average observed model error +∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ and quantities that are assumed to be finite. +Proposition 3.4. If Φy,A, Φy,† ∈ L1 +µθ(Θ, R), then +max{dKL(µy,A +θ +∥µy,† +θ ), dKL(µy,† +θ ∥µy,A +θ +)} ≤C∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ +for +C = C(∥Φy,A∥L1µθ , ∥Φy,†∥L1µθ ) := 21/2 exp(2∥Φy,†∥L1µθ + 2∥Φy,A∥L1µθ ) +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +. +The constant C in Proposition 3.4 is not optimal. The main value in defining C is to show +that, given the hypotheses, the constant C is finite. +Proof of Proposition 3.4. By the definition (2.3a) of Φy,A, it follows that ess infµΦy,A ≥ 0, so +min{ess infµθΦy,A, 0} = 0. Given that Φy,A, Φy,† ∈ L1 +µθ, we may apply Lemma 3.2, and also the +second statement of Theorem 3.1 with Φ(1) ← Φy,†, Φ(2) ← Φy,A, and µ ← µθ, which yields +max{dKL(µy,A +θ +∥µy,† +θ ), dKL(µy,† +θ ∥µy,A +θ +)} +≤2 exp +� +2∥Φy,A∥L1µθ + 2∥Φy,†∥L1µθ +� +∥Φy,† − Φy,A∥L1µθ +≤21/2 exp +� +2∥Φy,A∥L1µθ + 2∥Φy,†∥L1µθ +�� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ . +This completes the proof of Proposition 3.4. +10 + +If M† is not completely known, then neither are Φy,† nor δ†. Furthermore, it may be difficult +to compute ∥Φy,A∥L1µθ exactly. Thus, it will in general be difficult to compute the constant C +in Proposition 3.4. +Proposition 3.4 shows that the Kullback–Leibler divergences of the approximate posterior +µy,A +θ +with respect to the best posterior µy,† +θ +and vice versa are controlled by the average ob- +served model error ∥∥O ◦δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ . In order for dKL(µy,A +θ +∥µy,† +θ ) or dKL(µy,† +θ ∥µy,A +θ +) to be small, +Proposition 3.4 suggests that one could choose the observation operator O such that δ† takes +values in or near ker(O) with high µθ-probability. In particular, if the observation operator O +satisfies +P(O ◦ δ†(θ) = 0) = 1 +(3.4) +then the corresponding approximate posterior and the best posterior coincide. +This is not +surprising, since if (3.4) holds then Φy,† = Φy,A coincide µθ-almost everywhere, by Lemma 3.2, +and hence µy,† +θ +and µy,A +θ +coincide. +The condition (3.4) can be useful for guiding the choice of observation operator O even if δ† is +not fully known. For example, if the state space U is a Hilbert space, and if one can determine +a priori that δ† takes values in some proper subspace V of U without knowing δ† exactly, then +any choice of observation operator O such that V ⊆ ker(O) will yield an approximate posterior +µy,A +θ +that agrees with the best posterior µy,† +θ . The problem then consists in choosing O so that +ker(O) is as small as possible, while satisfying the constraint that the model error takes values +in ker(O) µθ-almost surely. The payoff in choosing O in this way is that Bayesian inference with +µy,A +θ +will be as good as Bayesian inference with µy,† +θ . Thus, the key idea in choosing observation +operators to mitigate the effect of model error on Bayesian inference is to exploit all available +knowledge about the model error. +Remark 3.5. Recall from Remark 2.2 that we may also interpret M as a reduced-order model +or surrogate for a more accurate but costly model M†. The preceding discussion then implies +that, for suitably chosen observation operators, Bayesian inference with µy,A +θ +will be as good as +µy,† +θ , and have smaller computational cost. +Error of enhanced noise posterior with respect to best posterior +Lemma 3.6 below bounds +the L1 +µθ error between the misfits Φy,† and Φy,E from (2.4a) and (2.6a) respectively. The bound +indicates the importance of the shifted observed model error term O ◦ (δ† − mu) and difference +Σ−1 +ε +− (Σε + OΣuO∗)−1 of covariance matrices. +Define the scalar +CE := ∥Σ−1/2 +ε +(Σε + OΣuO∗)1/2∥. +(3.5) +By Lemma A.1, CE satisfies +∥z∥Σ−1 +ε +≤ CE∥z∥(Σε+OΣuO∗)−1, +z ∈ Rd. +Lemma 3.6. Suppose Φy,† ∈ L1 +µθ. If Φy,E ∈ L1 +µθ, then for CE as in (3.5), +∥Φy,† − Φy,E∥L1µθ ≤2−1/2∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ +� +∥Φy,†∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +(3.6) ++ 2−1∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ . +Furthermore, +∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ ≤21/2� +∥Φy,†∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +, +∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ ≤(CE + 1)∥2Φy,E∥L1µθ . +11 + +See Appendix A.1.1 for the proof of Lemma 3.6. +Proposition 3.7. If Φy,E, Φy,† ∈ L1 +µθ(Θ, R), then +max{dKL(µy,† +θ ∥µy,E +θ +), dKL(µy,E +θ +∥µy,† +θ )} +≤C +� +∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ + ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ +� +for C = C(∥Φy,E∥L1µθ , ∥Φy,†∥L1µθ , CE), where +C := exp +� +2∥Φy,†∥L1µθ + 2∥Φy,E∥L1µθ +� +max +�√ +2 +� +∥Φy,†∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +, 1 +� +. +As with Proposition 3.4, the importance of the constant C above is that C is finite under the +hypotheses of Proposition 3.7. +Proof of Proposition 3.7. By the definition (2.6a) of Φy,E, it follows that ess infµΦy,E ≥ 0, so +min{ess infµθΦy,E, 0} = 0. Given that Φy,E, Φy,† ∈ L1 +µθ, we may apply Theorem 3.1 with Φ(1) ← +Φy,†, Φ(2) ← Φy,E, and µ ← µθ, to obtain +max{dKL(µy,† +θ ∥µy,E +θ +), dKL(µy,E +θ +∥µy,† +θ )} +≤2 exp +� +2∥Φy,E∥L1µθ + 2∥Φy,†∥L1µθ +� +∥Φy,† − Φy,E∥L1µθ +≤ exp +� +2∥Φy,E∥L1µθ + 2∥Φy,†∥L1µθ +� +max{ +√ +2 +� +∥Φy,†∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +, 1} +× +� +∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ + ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ +� +where the second inequality follows from the bound (3.6) of Lemma 3.6. +The significance of Proposition 3.7 is similar to that of Proposition 3.4. The main differences +follow from the fact the L1 +µθ error between the enhanced noise misfit Φy,E and the best misfit +Φy,† — and hence also the Kullback–Leibler error between µy,E +θ +and µy,† +θ +— is now controlled +by the sum +∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ + ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ . +(3.7) +Recall from Section 2 that the enhanced noise approach consists in modelling the unknown +state correction term δ†(θ†) ∈ U by a Gaussian random variable u ∼ N(mu, Σu). Since Σ−1 +ε +is +invertible by Assumption 2.3, the first term in (3.7) vanishes if and only if δ† − mu ∈ ker(O) +µθ-almost surely. This condition differs from the sufficient condition for µy,† +θ += µy,A +θ +that was +implied by Lemma 3.2, namely, that δ† ∈ ker(O) µθ-almost surely. The difference consists in +the mu term. +By recalling (1.3), the second term in (3.7) vanishes if and only if +P +� +y − O ◦ M(θ) − Omu ∈ ker +� +Σ−1 +ε +− (Σε + OΣuO∗)−1� += 1. +(3.8) +By a rearrangement of the Woodbury formula that we obtained from [7, Eq. (3)], we have +Σ−1 +ε +− (Σε + OΣuO∗)−1 = Σ−1 +ε OΣuO∗Σ−1 +ε (Σε + OΣuO)−1Σ−1 +ε . +The equation above holds for non-invertible OΣuO∗. If OΣuO∗ = 0, then both sides of the +equation above vanish, and thus the condition (3.8) follows immediately. Since Σu describes the +12 + +covariance of the U-valued random model u of the state error δ†(θ†), the condition OΣuO∗ = 0 +has the equivalent formulation that the RNd-valued random variable Ou is constant P-almost +surely. Since Ou = O(u − mu) + Omu, the latter condition is equivalent to u − mu ∈ ker(O) +P-almost surely. More generally, if OΣuO∗ is nonzero but has non-trivial kernel, then Σ−1 +ε +− +(Σε + OΣuO∗)−1 also has a non-trivial kernel, and it may be possible for (3.8) to be satisfied. +If one knows a priori that the image of Θ under δ† is contained in some affine subspace x + V +of U for a linear subspace V of U, then one can exploit this information. For example, given +the enhanced noise model N(mu, Σu), one should choose the observation operator O so that +V ⊆ mu + ker(O) and u takes values in mu + ker(O) P-almost surely. In this case, the enhanced +noise posterior µy,E +θ +and the best posterior µy,† +θ +will coincide. +3.1.2. Error of enhanced noise posterior with respect to approximate posterior +Lemma 3.8. Suppose Φy,A ∈ L1 +µθ. If Φy,E ∈ L1 +µθ, then for CE as in (3.5), +∥Φy,A − Φy,E∥L1µθ ≤2−1/2∥Omu∥Σ−1 +ε +� +∥Φy,A∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +(3.9) ++ 2−1∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ +where ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ satisfies the bound in Lemma 3.6. +For the proof of Lemma 3.8, see Appendix A.1.2. +Proposition 3.9. If Φy,A, Φy,E ∈ L1 +µθ(Θ, R), then +max{dKL(µy,A +θ +∥µy,E +θ +), dKL(µy,E +θ +∥µy,A +θ +)} +≤C +� +∥Omu∥Σ−1 +ε ++ ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ +� +for C = C(∥Φy,E∥L1µθ , ∥Φy,A∥L1µθ , CE), where +C := exp +� +2∥Φy,A∥L1µθ + 2∥Φy,E∥L1µθ +� +max +�√ +2 +� +∥Φy,A∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +, 1 +� +. +Proof of Proposition 3.9. By Theorem 3.1 and Lemma 3.8, +max{dKL(µy,A +θ +∥µy,E +θ +), dKL(µy,E +θ +∥µy,A +θ +)} +≤2 exp +� +2∥Φy,E∥L1µθ + 2∥Φy,A∥L1µθ +� +∥Φy,A − Φy,E∥L1µθ +≤ exp +� +2∥Φy,E∥L1µθ + 2∥Φy,A∥L1µθ +� +max{ +√ +2 +� +∥Φy,A∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� +, 1} +× +� +∥Omu∥Σ−1 +ε ++ ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ +� +. +This completes the proof of Proposition 3.9. +Proposition 3.9 implies that if, given an enhanced noise model with mean mu and covariance +Σu, one chooses the observation operator O so that +∥Omu∥Σ−1 +ε += ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ = 0, +then the enhanced noise posterior µy,E +θ +and the approximate posterior µy,A +θ +coincide. Equiva- +lently, Omu = 0 and (3.8) together imply that µy,E +θ += µy,A +θ +. Thus, such a choice of observation +13 + +operator yields an enhanced noise posterior µy,E +θ +that does not account for model error, in which +case it would be simpler to use the approximate posterior µy,A +θ +instead. By the contrapositive +statement, if µy,E +θ +and µy,A +θ +differ, then either Omu does not vanish, or (3.8) does not hold. If +OΣuO∗ = 0, then (3.8) does not hold if and only if +P(y − O ◦ M(θ) − Omu ̸= 0) > 0. +We expect that in many cases, the latter condition will hold — and thus that µy,A +θ +and µy,E +θ +will +differ — for a large collection of observation operators. +3.2. Kullback–Leibler error of joint parameter-error posterior +Recall from (2.7) that the joint misfit and the joint posterior are defined by +Φy,J(θ′, δ′) := 1 +2∥y − O ◦ M(θ′) − O ◦ δ′(θ′)∥2 +Σ−1 +ε , +dµy,J +θ,δ(θ′, δ′) := exp(−Φy,J(θ′, δ′)) +Z(Φy,J) +dµθ ⊗ µδ(θ′, δ′). +In this section, we shall compare µy,J +θ +∈ M1(Θ×D) with the other posterior measures considered +so far. To do this, we need to redefine these posteriors as measures on Θ × D. +For • ∈ {A, †, E}, define the lifted misfit and lifted posterior by +Φy,•(θ′, δ′) := Φy,•(θ′), +(3.11a) +dµy,• +θ,δ(θ′, δ′) := exp(−Φy,•(θ′, δ′)) +Z(Φy,•) +dµθ ⊗ µδ, +(3.11b) +where the definition (3.11b) follows from Assumption 2.5, namely that the joint prior is a +product measure. +Note that in (3.11a), we use the notation Φy,• to refer to two functions defined on different +domains. The following lemma shows that this abuse of notation is not problematic. +Lemma 3.10. Let µθ, µδ, Φy,• and µy,• +θ,δ be as in (3.11). Then for • ∈ {A, †, E}, +dµy,• +θ,δ(θ′, δ′) = dµy,• +θ (θ′) ⊗ µδ(δ′). +(3.12) +If Φy,• : Θ → R belongs to L1 +µθ, then its lifted version Φy,• defined in (3.11a) belongs to L1 +µθ⊗µδ, +and for every q > 0, +∥Φy,•∥Lq +µθ⊗µδ = ∥Φy,•∥Lq +µθ . +(3.13) +Proof. The second statement follows immediately from the definition (3.11a). +For the first +statement, observe that +� +Θ×D +exp(−Φy,•(θ′, δ′))dµθ ⊗ µδ(θ′, δ′) = +� +Θ +exp(−Φy,•(θ′))dµθ(θ′). +Thus, the normalisation constant Z(Φy,•) for the lifted posterior µy,• +θ,δ in (3.11b) agrees with the +corresponding normalisation constant for the posterior µy,• +θ . This implies (3.12). +14 + +Notation: +Recall from Section 1.3 that P denotes the probability measure in the probability +space on which we define all random variables. In this subsection, we will sometimes write the +random variables θ and δ explicitly, and take Lp-norms with respect to P instead of µθ ⊗ µδ. +For example, +∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥L1 +P = +� +Θ×D +∥O ◦ (δ† − δ′)(θ′)∥2 +Σ−1 +ε dµθ ⊗ µδ(θ′, δ′). +See Lemma 3.11 below for an example where we use this notation. +Error with respect to lifted best posterior +Lemma 3.11. Let Φy,† be defined as in (3.11a) with • = †. If Φy,† ∈ L1 +µθ and Φy,J ∈ L1 +µθ⊗µδ, +then +∥Φy,† − Φy,J∥L1 +µθ⊗µδ ≤ 2−1/2∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1µθ +� +. +(3.14) +Furthermore, +∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P ≤ 21/2� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1µθ +� +. +Proposition 3.12. Let Φy,† and Φy,J be as in Lemma 3.11 and let µy,† +θ,δ be defined as in (3.11b) +with • = †. Then +max{dKL(µy,† +θ,δ∥µy,J +θ,δ), dKL(µy,J +θ,δ∥µy,† +θ,δ)} ≤ C∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P , +where +C = 21/2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,†∥L1µθ +�� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1µθ +� +. +See Appendix A.2 for the proofs of Lemma 3.11 and Proposition 3.12. +By Proposition 3.12, a sufficient condition for µy,J +θ,δ to coincide with µy,† +θ,δ is +P +� +(δ† − δ)(θ) ∈ ker(O) +� += 1. +(3.15) +For example, suppose that one knows a priori that there exist a vector x ∈ U and a linear +subspace V of U such that +{δ†(θ′) : θ′ ∈ Θ} ⊆ x + V. +Suppose that (D, ⟨·, ·⟩D) is a Hilbert space and δ ∼ µδ = N(mδ, Σδ). Then mδ + Σ1/2 +δ +D is the +Cameron–Martin space of µδ, and the support supp(µδ) of µδ is the closure of mδ +Σ1/2 +δ +D with +respect to ∥ · ∥D. Assume there exists y ∈ U and a closed linear subspace W ⊆ U such that +{δ′(θ′) : θ′ ∈ Θ, δ′ ∈ supp(µδ)} ⊆ y + W. +If for every v ∈ V and w ∈ W it holds that (x+v)−(y +w) ∈ ker(O), then the condition (3.15) +holds. +The preceding example suggests that, in general, it may be difficult to choose µδ and O so +that µy,† +θ,δ and µy,J +θ,δ coincide. This is to be expected, since the lifted best posterior measure µy,† +θ,δ +is a product measure with δ-marginal equal to µδ, whereas for many reasonable choices of µδ +the joint posterior measure µy,J +θ,δ will not have δ-marginal equal to µδ. +15 + +Error with respect to lifted approximate posterior +Lemma 3.13. Let Φy,A be defined as in (3.11a) with • = A. If Φy,A ∈ L1 +µθ and Φy,J ∈ L1 +µθ⊗µδ, +then +∥Φy,A − Φy,J∥L1 +µθ⊗µδ ≤ 2−1/2∥∥O ◦ δ(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,A∥1/2 +L1µθ +� +. +(3.16) +Furthermore, +∥∥O ◦ δ(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P ≤ 21/2� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,A∥1/2 +L1µθ +� +. +Proposition 3.14. Let Φy,A and Φy,J be as in Lemma 3.13 and let µy,A +θ,δ be defined as in (3.11b) +with • = A. Then +max{dKL(µy,A +θ,δ ∥µy,J +θ,δ), dKL(µy,J +θ,δ∥µy,A +θ,δ )} ≤ C∥∥O ◦ δ(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P , +where +C = 21/2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,A∥L1µθ +�� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,A∥1/2 +L1µθ +� +. +See Appendix A.2 for the proofs of Lemma 3.13 and Proposition 3.14. +By Proposition 3.14, a sufficient condition for µy,A +θ,δ and µy,J +θ,δ to coincide is that δ(θ) ∈ ker(O), +P-almost surely. By the same reasoning that we used when considering µy,† +θ,δ and µy,J +θ,δ, we do not +expect µy,A +θ,δ and µy,J +θ,δ to coincide, since the former is a product measure with δ-marginal equal +to µδ, and the latter will not have this property in general. +Connection with parametrised background data-weak approach +Recall the definition (2.1) of +the observation operator as O := (ℓi1, . . . , ℓiN ). Suppose the state space U is a Hilbert space and +d = 1, i.e. suppose that the measurement functions (ℓij)N +j=1 are continuous linear functionals +on U with Riesz representatives denoted by (Rℓij)N +j=1 ⊂ U. Then the condition δ(θ) ∈ ker(O) is +equivalent to δ(θ) ∈ (span(Rℓij, j = 1, . . . , N))⊥. We may reformulate the necessary condition +for µy,A +θ,δ and µy,J +θ,δ to differ, namely that δ(θ) /∈ ker(O) with positive P-probability, as δ(θ) ∈ +span(Rℓij, j = 1, . . . , N) with positive P-probability. +Given the interpretation of δ(θ) as a +state correction term, the condition that δ(θ) ∈ span(Rℓij, j = 1, . . . , N) closely resembles the +‘variational update’ from the parametrised background data-weak approach for state inference; +see e.g. [17, Section 2.3]. +It is possible to state and prove the analogues of the preceding bounds for the lifted enhanced +noise posterior µy,E +θ,δ . These bounds are not relevant for the main goal of this paper. However, +for the sake of completeness, we state them in Appendix A.2. +3.3. Kullback–Leibler error of marginal posterior +Recall from (2.8) that the marginal posterior is defined by +µy,M +θ +(S) = +� +S×D +dµy,J +θ,δ(θ′, δ′), +S ∈ B(Θ). +In Section 3.2, we bounded the Kullback–Leibler error of the joint posterior µy,J +θ,δ with respect +to the lifted posteriors µy,• +θ,δ for • ∈ {A, †, E} that were defined in (3.11b), and we observed in +Lemma 3.10 that the θ-marginal of the lifted posterior µy,• +θ,δ is exactly µy,• +θ . +16 + +In this section, we shall bound the Kullback–Leibler error of the marginal posterior µy,M with +respect to the θ-marginals of the lifted posteriors that we considered in Section 3.2, i.e. with +respect to µy,† +θ , µy,A +θ +, and µy,E +θ +. +Below, µy,• +δ|θ denotes the regular version of the posterior distribution of δ conditioned on θ, for +• ∈ {A, †, E, J}. We assume that such regular conditional distributions exist and are unique up +to sets of measure zero. +Proposition 3.15. Let Φy,J ∈ L1 +µθ⊗µδ and • ∈ {A, †, E}. Suppose Φy,• : Θ → R≥0 belongs to +L1 +µθ. Then +dKL(µy,M +θ +∥µy,• +θ ) =dKL(µy,J +θ,δ∥µy,• +θ,δ) − +� +Θ +dKL(µy,J +δ|θ∥µy,• +δ|θ)dµθ, +dKL(µy,• +θ ∥µy,M +θ +) =dKL(µy,• +θ,δ∥µy,J +θ,δ) − +� +Θ +dKL(µy,• +δ|θ∥µy,J +δ|θ)dµθ. +In particular, +max{dKL(µy,M +θ +∥µy,• +θ ), dKL(µy,• +θ ∥µy,M +θ +)} +≤ max{dKL(µy,J +θ,δ∥µy,• +θ,δ), dKL(µy,• +θ,δ∥µy,J +θ,δ)} − min +� � +Θ +dKL(µy,J +δ|θ∥µy,• +δ|θ)dµθ, +� +Θ +dKL(µy,• +δ|θ∥µy,J +δ|θ)dµθ +� +. +Proof. By nonnegativity of the Kullback–Leibler divergence, the first statement implies the +second statement. +The first statement follows by recalling the chain rule for the Kullback–Leibler divergence, +see e.g. [26, Exercise 3.2]: +dKL(µy,J +θ,δ∥µy,• +θ,δ) =dKL(µy,M +θ +∥µy,• +θ ) + +� +Θ +dKL(µy,J +δ|θ∥µy,• +δ|θ)dµy,M +θ +, +(3.17) +dKL(µy,• +θ,δ∥µy,J +θ,δ) =dKL(µy,• +θ ∥µy,M +θ +) + +� +Θ +dKL(µy,• +δ|θ∥µy,J +δ|θ)dµy,• +θ . +(3.18) +Above, we used the definition (2.8) of the marginal posterior µy,M +θ +, and the fact expressed in +(3.12), namely that the θ-marginal of µy,• +θ,δ is µy,• +θ . +Proposition 3.15 is important for the following reasons. First, it implies that the Kullback– +Leibler error of the marginal posterior µy,M +θ +with respect to any of the above-mentioned poste- +riors on Θ cannot be larger than the Kullback–Leibler error of the joint posterior µy,J +θ,δ and the +corresponding lifted version of the posterior on Θ. In other words, marginalisation can only +reduce the Kullback–Leibler error. As a result, for • ∈ {A, †, E}, the Kullback–Leibler error +of the marginal posterior µy,M +θ +with respect to µy,• +θ +satisfies the same bounds as the Kullback– +Leibler error of the joint posterior µy,J +θ,δ with respect to the lifted posterior µy,• +θ,δ. Thus, the same +statements regarding sufficient conditions for the coincidence of the posteriors on Θ × D that +were made after Proposition 3.12, Proposition 3.14, and Proposition A.4, also apply to the +marginalised versions of the posteriors in the above-mentioned propositions. +4. Conclusion +We considered Bayesian inverse problems in the presence of model error in the following set- +ting: the data is finite-dimensional; the noise is additive, Gaussian, and independent; and the +17 + +parameter-to-observable map is the composition of a possibly nonlinear model with a linear +observation operator. We assumed that there exists a unique best model and best parameter, +such that the resulting best state most accurately describes the phenomenon of interest. The +‘model error’ is then the difference between the model that one uses and the best model. +We described some existing deterministic approaches for accounting for model error and +used the local Lipschitz stability property of posteriors with respect to perturbations in the +likelihood to bound the symmetrised Kullback–Leibler error between pairs of posteriors. These +bounds have two important properties: first, they control the Kullback–Leibler error in terms of +quantities that depend on the observation operator and the objects used to account for model +error; and second, the other terms in the bounds are finite under mild hypotheses, namely +L1-integrability of the misfits with respect to the prior. +The bounds yield sufficient conditions on the observation operator and the model error-aware +approach to yield a posterior that performs almost as well as the best posterior that uses the +best model. They also yield necessary conditions for a model error-aware approach to yield +a posterior that differs from the posterior yielded by the model error-agnostic approach. A +recurring theme in the sufficient conditions is the importance of the kernel of the observation +operator. We expect that the design of observation operators that approximately or exactly +satisfy the sufficient conditions will be challenging and highly problem-specific. It would be +of interest to study the setting of nonlinear observation operators and other approaches for +accounting for model error. +Acknowledgements +The research of HCL has been partially funded by the DFG — Project-ID 318763901 — +SFB1294. +A. Technical lemmas +Let d ∈ N and Mi ∈ Rd×d, i = 1, 2 be symmetric and positive definite. +For convenience, we state the following lemma. +Lemma A.1. Suppose that M1, M2 ∈ Rd×d are symmetric, that M1 is positive definite, and that +M2 is nonnegative definite. Then M1+M2 is symmetric positive definite and M−1 +1 −(M1+M2)−1 +is symmetric nonnegative definite. In addition, +∥z∥(M1+M2)−1 +∥(M1 + M2)−1/2M1/2 +1 +∥ +≤ ∥z∥M−1 +1 +≤ ∥M−1/2 +1 +(M1 + M2)1/2∥∥z∥(M1+M2)−1, +z ∈ Rd. +(A.1) +Proof. Let λmin(A) denote the smallest eigenvalue of a matrix A. By the assumptions on M1 +and M2, M1 + M2 is positive definite. +As the difference of two symmetric matrices, M−1 +1 +− (M1 + M2)−1 is symmetric. To show +that it is nonnegative, we use the following rearrangement of the Woodbury formula, which we +take from [7, Equation (3)]: +(M1 + M2)−1 =M−1 +1 +− M−1 +1 M2(I + M−1 +1 M2)−1M−1 +1 +⇐⇒ M−1 +1 +− (M1 + M2)−1 =M−1 +1 M2(I + M−1 +1 M2)−1M−1 +1 . +18 + +Invertibility of I+M−1 +1 M2 = M−1 +1 (M1+M2) follows from the invertibility of M−1 +1 +and M1+M2. +From the second equation above, M−1 +1 +− (M1 + M2)−1 inherits the nonnegative definiteness of +M2. +Recall the notation (1.3) for a matrix-weighted norm. The first inequality of (A.1) follows by +∥z∥(M1+M2)−1 = ∥(M1 + M2)−1/2z∥ = ∥(M1 + M2)−1/2M1/2 +1 +M−1/2 +1 +z∥ +≤ ∥(M1 + M2)−1/2M1/2 +1 +∥∥M−1/2 +1 +z∥ += ∥(M1 + M2)−1/2M1/2 +1 +∥∥z∥M−1 +1 . +The second inequality of (A.1) follows by +∥z∥M−1 +1 += ∥M−1/2 +1 +z∥ = ∥M−1/2 +1 +(M1 + M2)1/2(M1 + M2)−1/2z∥ +≤ ∥M−1/2 +1 +(M1 + M2)1/2∥∥(M1 + M2)−1/2z∥ += ∥M−1/2 +1 +(M1 + M2)1/2∥∥z∥(M1+M2)−1. +This completes the proof of Lemma A.1. +We will use the following bound in the proofs below. +Lemma A.2. Let (E, dE) be a metric space, µ ∈ M1(E), and f and g be Rd-valued measurable +functions on E. Let M1 ∈ Rd×d be symmetric and positive definite and M2 ∈ Rd×d be symmetric +nonnegative definite. Then +∥∥f∥2 +M−1 +1 +− ∥f + g∥2 +M−1 +1 ∥L1µ ≤ ∥∥g∥2 +M−1 +1 ∥1/2 +L1µ +� +∥∥f + g∥2 +M−1 +1 ∥1/2 +L1µ + ∥∥f∥2 +M−1 +1 ∥1/2 +L1µ +� +. +(A.2) +More generally, +∥∥f∥2 +M−1 +1 +− ∥f + g∥2 +(M1+M2)−1∥L1µ +≤∥∥g∥2 +M−1 +1 ∥1/2 +L1µ +� +∥M−1/2 +1 +(M1 + M2)1/2∥ · ∥∥f + g∥2 +(M1+M2)−1∥1/2 +L1µ + ∥∥f∥2 +M−1 +1 ∥1/2 +L1µ +� +(A.3) ++ ∥∥f + g∥2 +M−1 +1 +−(M1+M2)−1∥L1µ, +where ∥f + g∥2 +M−1 +1 +−(M1+M2)−1 := ∥f + g∥2 +M−1 +1 +− ∥f + g∥2 +(M1+M2)−1. In addition, +∥∥g∥2 +M−1 +1 ∥1/2 +L1µ ≤ ∥∥f∥2 +M−1 +1 ∥1/2 +L1µ + ∥M−1/2 +1 +(M1 + M2)1/2∥∥∥f + g∥2 +(M1+M2)−1∥1/2 +L1µ +(A.4) +∥∥f + g∥2 +M−1 +1 +−(M1+M2)−1∥L1µ ≤ +� +1 + ∥M−1/2 +1 +(M1 + M2)1/2∥ +� +∥∥f + g∥2 +(M1+M2)−1∥L1µ. +(A.5) +Proof. We claim that for arbitrary a, b ∈ Rd, symmetric positive definite M1 ∈ Rd×d, and +symmetric nonnegative definite M2 ∈ Rd×d, +∥a∥2 +M−1 +1 +− ∥a + b∥2 +(M1+M2)−1 = −⟨b, 2a + b⟩M−1 +1 ++ ∥a + b∥2 +M−1 +1 +−(M1+M2)−1. +(A.6) +Recall that (M1 + M2)−1 exists and is positive definite, by Lemma A.1. +19 + +Using the matrix-weighted inner product and norm notation from (1.3), +∥a∥2 +M−1 +1 +− ∥a + b∥2 +M−1 +1 +=a⊤M−1 +1 a − (a + b)⊤M−1 +1 (a + b) +=a⊤M−1 +1 a − (a⊤M−1 +1 a + 2a⊤M−1 +1 b + b⊤M−1 +1 b) += − b⊤M−1 +1 (2a + b) += − ⟨b, 2a + b⟩M−1 +1 . +This implies (A.6), since +∥a∥2 +M−1 +1 +− ∥a + b∥2 +M−1 +1 ++ ∥a + b∥2 +M−1 +1 +− ∥a + b∥2 +(M1+M2)−1 += − ⟨b, 2a + b⟩M−1 +1 ++ ∥a + b∥2 +M−1 +1 +− ∥a + b∥2 +(M1+M2)−1. +Now let f, g, and µ be as in the statement of the lemma. Then by (A.6) and the triangle +inequality, +∥∥f∥2 +M−1 +1 +− ∥f + g∥2 +(M1+M2)−1∥L1µ =∥ − ⟨g, 2f + g⟩M−1 +1 ++ ∥f + g∥2 +M−1 +1 +−(M1+M2)−1∥L1µ +≤∥⟨g, 2f + g⟩M−1 +1 ∥L1µ + ∥∥f + g∥2 +M−1 +1 +−(M1+M2)−1∥L1µ. +(A.7) +Next, +∥⟨g, 2f + g⟩M−1 +1 ∥L1µ ≤∥∥g∥M−1 +1 ∥2f + g∥M−1 +1 ∥L1µ +≤∥∥g∥M−1 +1 ∥L2µ∥∥2f + g∥M−1 +1 ∥L2µ +≤∥∥g∥M−1 +1 ∥L2µ +� +∥∥f + g∥M−1 +1 ∥L2µ + ∥∥f∥M−1 +1 ∥L2µ +� +=∥∥g∥2 +M−1 +1 ∥1/2 +L1µ +� +∥∥f + g∥2 +M−1 +1 ∥1/2 +L1µ + ∥∥f∥2 +M−1 +1 ∥1/2 +L1µ +� +. +(A.8) +The first and second inequalities follow by applying the Cauchy–Schwarz inequality with respect +to ⟨·, ·⟩M−1 +1 +and ∥·∥L1µ respectively. The third inequality and the equation follow from the ∥·∥L2µ- +triangle inequality and the definition of the Lp +µ norm for p = 1, 2. By (A.8), we bound the first +term on the right-hand side of (A.7). By using M2 ← 0, the second term on the right-hand side +of (A.7) vanishes. Thus (A.2) follows from (A.7). +Next, we bound the first term inside the parentheses on the right-hand side of (A.8). By +(A.1), +∥∥f + g∥2 +M−1 +1 ∥1/2 +L1µ ≤ ∥M−1/2 +1 +(M1 + M2)1/2∥∥∥f + g∥2 +(M1+M2)−1∥1/2 +L1µ . +Using the above bound yields (A.3). +To prove (A.4), +∥∥g∥2 +M−1 +1 ∥1/2 +L1µ =∥∥ − f + (f + g)∥M−1 +1 ∥L2µ +≤∥∥f∥M−1 +1 ∥L2µθ + ∥∥f + g∥M−1 +1 ∥L2µ +≤∥∥f∥2 +M−1 +1 ∥1/2 +L1µθ + ∥M−1/2 +1 +(M1 + M2)1/2∥∥∥f + g∥2 +(M1+M2)−1∥1/2 +L1µ +20 + +where the last inequality uses (A.1). To prove (A.5), observe that +∥∥f + g∥2 +M−1 +1 +−(M1+M2)−1∥L1µ =∥∥f + g∥2 +M−1 +1 +− ∥f + g∥2 +(M1+M2)−1∥L1µ +≤∥∥f + g∥2 +M−1 +1 ∥L1µ + ∥∥f + g∥2 +(M1+M2)−1∥L1µ +≤(∥M−1/2 +1 +(M1 + M2)1/2∥ + 1)∥∥f + g∥2 +(M1+M2)−1∥L1µ +where the first and second inequality follow from the triangle inequality and (A.1) in Lemma A.1. +This completes the proof of Lemma A.2. +A.1. Proof of lemmas in Section 3.1 +A.1.1. Proofs for Section 3.1.1 +Proof of error of approximate posterior with respect to best posterior +Lemma 3.2 bounds +∥Φy,†−Φy,A∥Lq +µθ in terms of the observed model error O◦δ†, under the hypothesis that Φy,† ∈ L1 +µθ +and Φy,A ∈ L1 +µθ. The bound is given in (3.2): +∥Φy,† − Φy,A∥L1µθ ≤ 2−1/2∥∥O ◦ δ†∥Σ−1 +ε ∥L2µθ +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +. +Proof of Lemma 3.2. Recall from (2.4a) and (2.3a) that Φy,†(θ′) = 1 +2∥y − O ◦ M†(θ′)∥2 +Σ−1 +ε +and +Φy,A(θ′) = 1 +2∥y − O ◦ M(θ′)∥2 +Σ−1 +ε +respectively. By these definitions, +∥2 · 1 +2∥y − O ◦ M†∥2 +Σ−1 +ε ∥1/2 +L1µθ = ∥2Φy,†∥1/2 +L1µθ = +√ +2∥Φy,†∥1/2 +L1µθ +(A.9) +and similarly +∥∥y − O ◦ M∥2 +Σ−1 +ε ∥1/2 +L1µθ = +√ +2∥Φy,A∥1/2 +L1µθ . +(A.10) +Now set f ← y − O ◦ M†, g ← O ◦ δ†, µ ← µθ, M1 ← Σε, and M2 ← 0. By (2.5), we have +f + g = y − O ◦ (M† − δ†) = y − O ◦ M. Hence ∥f∥2 +M−1 +1 += 2Φy,† and ∥f + g∥2 +M−1 +1 += 2Φy,A. +Applying (A.2) from Lemma A.2 with these choices yields +2∥Φy,† − Φy,A∥L1µθ ≤∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ +� +∥∥y − O ◦ M†∥2 +Σ−1 +ε ∥1/2 +L1µθ + ∥∥y − O ◦ M∥2 +Σ−1 +ε ∥1/2 +L1µθ +� +=∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ +√ +2 +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +, +where we used (A.9) and (A.10) for the equation. This proves (3.2). The bound on ∥∥O ◦ +δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ in the statement of Lemma 3.2 follows from (A.4) in Lemma A.2 with the choices +above: +∥∥O ◦ δ†∥2 +Σ−1 +ε ∥1/2 +L1µθ ≤ +√ +2 +� +∥Φy,†∥1/2 +L1µθ + ∥Φy,A∥1/2 +L1µθ +� +. +This completes the proof of Lemma 3.2. +Proof of error of enhanced noise posterior with respect to best posterior +Lemma 3.6 bounds +∥Φy,† −Φy,E∥L1µθ in terms of the observed model error O◦δ† and the Gaussian model N(mu, Σδ) +21 + +of δ†(θ†). +In particular, under the hypotheses that Φy,† ∈ L1 +µθ and Φy,E ∈ L1 +µθ, then for +CE := ∥Σ−1/2 +ε +(Σε + OΣuO∗)1/2∥ as in (3.5), the bound (3.6) +∥Φy,† − Φy,E∥L1µθ ≤2−1/2∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ +� +∥Φy,†∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� ++ 2−1∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ , +holds, and all terms on the right-hand side are finite. +Proof of Lemma 3.6. In the same way that we proved (A.9), we can use the definition (2.6a) of +Φy,E to prove +∥∥y − O ◦ M − Omu∥2 +(Σε+OΣuO∗)−1∥1/2 +L1µθ = +√ +2∥Φy,E∥1/2 +L1µθ . +(A.11) +Let f ← y − O ◦ M†, g ← O ◦ (δ† − mu), µ ← µθ, M1 ← Σε, and M2 ← OΣuO∗. Then +f + g = y − O ◦ (M† − δ†) − Omu = y − O ◦ M − Omu, +∥f∥2 +M−1 +1 += 2Φy,†, and ∥f + g∥2 +(M1+M2)−1 = 2Φy,E. Applying (A.3) from Lemma A.2 yields +2∥Φy,† − Φy,E∥L1µθ ≤∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ +� +CE∥2Φy,E∥1/2 +L1µθ + ∥2Φy,†∥1/2 +L1µθ +� +(A.12) ++ ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ . +This proves (3.6). By (A.4) and CE as in (3.5), +∥∥O ◦ (δ† − mu)∥2 +Σ−1 +ε ∥1/2 +L1µθ ≤ ∥2Φy,†∥1/2 +L1µθ + CE∥2Φy,E∥1/2 +L1µθ . +By (A.5) from Lemma A.2, +∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ ≤ (CE + 1)∥2Φy,E∥L1µθ . +This completes the proof of Lemma 3.6. +A.1.2. Proofs for Section 3.1.2 +Lemma 3.8 asserts that, under the hypotheses that Φy,A ∈ L1 +µθ and Φy,E ∈ L1 +µθ, then for +CE := ∥Σ−1/2 +ε +(Σε + OΣuO∗)1/2∥ as in (3.5), the bound (3.9) +∥Φy,A − Φy,E∥L1µθ ≤2−1/2∥Omu∥Σ−1 +ε +� +∥Φy,A∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� ++ 2−1∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ , +holds, and all terms on the right-hand side are finite. +Proof of Lemma 3.8. Let f ← y −O ◦M, g ← −Omu, M1 ← Σ−1 +ε , M2 ← OΣuO∗, and µ ← µθ. +Then ∥f∥2 +M−1 +1 += 2Φy,A and ∥f + g∥2 +(M1+M2)−1 = 2Φy,E. Applying (A.3) from Lemma A.2 yields +2∥Φy,A − Φy,E∥L1µθ +≤∥Omu∥Σ−1 +ε +√ +2 +� +∥Φy,A∥1/2 +L1µθ + CE∥Φy,E∥1/2 +L1µθ +� ++ ∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ , +for CE := ∥Σ−1/2 +ε +(Σε + OΣuO∗)1/2∥ as in (3.5). This proves (3.9). Next, (A.5) in Lemma A.2 +yields +∥∥y − O ◦ M − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1µθ ≤ 2 +� +CE + 1)∥Φy,E∥L1µθ . +This completes the proof of Lemma 3.8. +22 + +A.2. Proofs for Kullback–Leibler error of joint parameter-error posterior +Proof of error of joint parameter-error posterior with respect to lifted best posterior +In +Lemma 3.11, one assumes that Φy,† as defined in (2.4a) belongs to L1 +µθ, and also that Φy,J ∈ +L1 +µθ⊗µδ. The resulting bound (3.14) is +∥Φy,† − Φy,J∥L1 +µθ⊗µδ ≤ 2−1/2∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1µθ +� +. +Proof of Lemma 3.11. Let f ← y − O ◦ M†(θ), g ← O ◦ (δ† − δ)(θ), M1 ← Σε, M2 ← 0, and +µ ← P. Then ∥f∥2 +M−1 +1 += 2Φy,†(θ, δ) and ∥f + g∥2 +M−1 +1 += 2Φy,J(θ, δ). Using the same argument +that we used to prove (A.9), it follows from (2.7a) that +∥∥y − O ◦ M(θ) − O ◦ δ(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P = +√ +2∥Φy,J∥1/2 +L1 +µθ⊗µδ +. +(A.13) +By (A.9) and (3.13) from Lemma 3.10, +∥∥y − O ◦ M†(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P = +√ +2∥Φy,†∥1/2 +L1µθ = +√ +2∥Φy,†∥1/2 +L1 +µθ⊗µδ +. +(A.14) +Applying (A.2) Lemma A.2 with these choices yields (3.14): +2∥Φy,A − Φy,J∥L1 +µθ⊗µδ ≤∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P +√ +2 +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1µθ +� +. +Next, +∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P ≤ +√ +2 +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1µθ +� +, +follows from (A.13), (A.14), and (A.4) of Lemma A.2 with the choices stated above. +Proof of Proposition 3.12. Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,† are nonnegative, by (2.7a) +and (2.4a). Applying Theorem 3.1 and Lemma 3.11 yields +max{dKL(µy,† +θ,δ∥µy,J +θ,δ), dKL(µy,J +θ,δ∥µy,† +θ,δ)} +≤2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,†∥L1 +µθ⊗µδ +� +∥Φy,† − Φy,J∥L1 +µθ⊗µδ +≤21/2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,†∥L1 +µθ⊗µδ +�� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,†∥1/2 +L1 +µθ⊗µδ +� +× ∥∥O ◦ (δ† − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P . +Using the definition of C given in the statement of Proposition 3.12 and (A.14) completes the +proof. +Proof of error of joint parameter-error posterior with respect to lifted approximate posterior +In Lemma 3.13, one assumes that Φy,A as defined in (2.3a) belongs to L1 +µθ, and also that +Φy,J ∈ L1 +µθ⊗µδ. The resulting bound (3.16) is +∥Φy,A − Φy,J∥L1 +µθ⊗µδ ≤ 2−1/2∥∥O ◦ δ(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,A∥1/2 +L1µθ +� +. +The proof of Lemma 3.13 is very similar to the proof of Lemma 3.11 above. +23 + +Proof of Lemma 3.13. Let f ← y −O ◦M(θ), g ← O ◦(−δ)(θ), M1 ← Σε, M2 ← 0, and µ ← P. +Then ∥f + g∥2 +M−1 +1 += 2Φy,J(θ, δ) and ∥f∥2 +M−1 +1 += 2Φy,A(θ, δ). Analogously to (A.14), we have by +(2.3a) and (3.13) of Lemma 3.10 that +∥∥y − O ◦ M(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P = +√ +2∥Φy,A∥1/2 +L1µθ = +√ +2∥Φy,A∥1/2 +L1 +µθ⊗µδ +. +(A.15) +Applying (A.2) of Lemma A.2 with the choices above yields (3.16): +2∥Φy,A − Φy,J∥L1 +µθ⊗µδ ≤∥∥O ◦ (−δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P +√ +2 +� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,A∥1/2 +L1µθ +� +. +Next, +∥∥O ◦ (−δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P ≤ +√ +2 +� +∥Φy,A∥1/2 +L1µθ + ∥Φy,J∥1/2 +L1 +µθ⊗µδ +� +follows from (A.13), (A.15), and (A.4) of Lemma A.2. +Proof of Proposition 3.14. Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,A are nonnegative, by (2.7a) +and (2.3a). Applying Theorem 3.1 and Lemma 3.13 yields +max{dKL(µy,A +θ,δ ∥µy,J +θ,δ), dKL(µy,J +θ,δ∥µy,A +θ,δ )} +≤2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,A∥L1 +µθ⊗µδ +� +∥Φy,A − Φy,J∥L1 +µθ⊗µδ +≤21/2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,A∥L1 +µθ⊗µδ +�� +∥Φy,J∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,A∥1/2 +L1 +µθ⊗µδ +� +× ∥∥O ◦ (−δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P . +Using the definition of C given in the statement of Proposition 3.14 and (A.15) completes the +proof. +Proof of error of joint parameter-error posterior with respect to lifted enhanced noise +posterior +For the sake of completeness, we compare the joint posterior with the lifted enhanced +noise posterior. +Lemma A.3. Let Φy,E be defined as in (3.11a) with • = E. If Φy,E ∈ L1 +µθ and Φy,J ∈ L1 +µθ⊗µδ, +then +∥Φy,E − Φy,J∥L1 +µθ⊗µδ ≤2−1/2∥∥O ◦ (δ(θ) − mu)∥2 +Σ−1 +ε ∥1/2 +L1 +P +� +CE∥Φy,E∥1/2 +L1µθ + ∥Φy,J∥1/2 +L1 +µθ⊗µδ +� +(A.16) ++ 2−1∥∥y − O ◦ M(θ) − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1 +P. +Furthermore, +∥∥O ◦ (δ(θ) − mu)∥2 +Σ−1 +ε ∥1/2 +L1 +P ≤ +√ +2 +� +CE∥Φy,E∥1/2 +L1µθ + ∥Φy,J∥1/2 +L1 +µθ⊗µδ +� +∥∥y − O ◦ M(θ) − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1 +P ≤ 2(CE + 1)∥Φy,E∥L1µθ . +Proof of Lemma A.3. Let f ← y − O ◦ (M(θ) + δ(θ)), g ← O ◦ (δ(θ) − mu), M1 ← Σε, +M2 ← OΣuO∗, and µ ← P. Then ∥f∥2 +M−1 +1 += 2Φy,J(θ, δ) and ∥f + g∥2 +(M1+M2)−1 = 2Φy,E(θ, δ). +Analogously to (A.14), we have by (2.6a) and (3.13) of Lemma 3.10 that +∥∥y − O ◦ M(θ) − Omu∥2 +(Σε+OΣuO∗)−1∥1/2 +L1 +P = +√ +2∥Φy,E∥1/2 +L1µθ = +√ +2∥Φy,E∥1/2 +L1 +µθ⊗µδ +. +(A.17) +24 + +Applying (A.3) of Lemma A.2 with the choices above yields (A.16): +2∥Φy,E − Φy,J∥L1 +µθ⊗µδ ≤∥∥O ◦ (δ(θ) − mu)∥2 +Σ−1 +ε ∥1/2 +L1 +P +� +CE∥2Φy,E∥1/2 +L1µθ + ∥2Φy,J∥1/2 +L1 +µθ⊗µδ +� ++ ∥∥y − O ◦ M(θ) − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1 +P. +Next, we apply (A.4) and (A.5) from Lemma A.2, and use (A.17): +∥∥O ◦ (δ(θ) − mu)∥2 +Σ−1 +ε ∥1/2 +L1 +P ≤ +√ +2 +� +CE∥Φy,E∥1/2 +L1µθ + ∥Φy,J∥1/2 +L1 +µθ⊗µδ +� +∥∥y − O ◦ M(θ) − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1 +P ≤ 2(CE + 1)∥Φy,E∥L1µθ . +This completes the proof of Lemma A.3. +Proposition A.4. Let Φy,E and Φy,J be as in Lemma A.3, and let µy,E +θ,δ be as in (3.11b) with +• = E. Then +max{dKL(µy,E +θ,δ ∥µy,J +θ,δ), dKL(µy,J +θ,δ∥µy,E +θ,δ )} ≤ C∥∥O ◦ (mu − δ)(θ)∥2 +Σ−1 +ε ∥1/2 +L1 +P , +where +C = exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,E∥L1µθ +� +max{21/2� +CE∥Φy,E∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,J∥1/2 +L1 +µθ⊗µδ +� +, 1}. +Proof of Proposition A.4. Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,E are nonnegative, by (2.7a) +and (2.6a). Applying Theorem 3.1 and Lemma A.3 yields +max{dKL(µy,E +θ,δ ∥µy,J +θ,δ), dKL(µy,J +θ,δ∥µy,E +θ,δ )} +≤2 exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,E∥L1 +µθ⊗µδ +� +∥Φy,E − Φy,J∥L1 +µθ⊗µδ +≤ exp +� +2∥Φy,J∥L1 +µθ⊗µδ + 2∥Φy,E∥L1 +µθ⊗µδ +� +max{21/2� +CE∥Φy,E∥1/2 +L1 +µθ⊗µδ ++ ∥Φy,J∥1/2 +L1 +µθ⊗µδ +� +, 1} +× +� +∥∥O ◦ (mu − δ)(θ)∥2 +Σ−1 +ε ∥L1 +P + ∥∥y − O ◦ M(θ) − Omu∥2 +Σ−1 +ε +−(Σε+OΣuO∗)−1∥L1 +P +� +. +Using the definition of C given in the statement of Proposition A.4 and (3.13) from Lemma 3.10 +completes the proof. +References +[1] Assyr Abdulle and Giacomo Garegnani, Random time step probabilistic methods for un- +certainty quantification in chaotic and geometric numerical integration, Stat. Comput. 30 +(2020), no. 4, 907–932. +[2] Alen Alexanderian, Ruanui Nicholson, and No´emi Petra, Optimal design of large-scale +nonlinear Bayesian inverse problems under model uncertainty, 2022, arXiv:2211.03952. +[3] Alen Alexanderian, No´emi Petra, Georg Stadler, and Isaac Sunseri, Optimal design of +large-scale Bayesian linear inverse problems under reducible model uncertainty: Good to +know what you don’t know, SIAM/ASA J. Uncertainty Quantif. 9 (2021), no. 1, 163–184. +[4] Jenn´y Brynjarsd´ottir and Anthony O’Hagan, Learning about physical parameters: the im- +portance of model discrepancy, Inverse Probl. 30 (2014), no. 11, 114007. +25 + +[5] Daniela Calvetti, Matthew Dunlop, Erkki Somersalo, and Andrew Stuart, Iterative updat- +ing of model error for Bayesian inversion, Inverse Probl. 34 (2018), no. 2, 025008. +[6] Lianghao Cao, Thomas O’Leary-Roseberry, Prashant K. Jha, J. Tinsley Oden, and +Omar Ghattas, Residual-based error correction for neural operator accelerated infinite- +dimensional Bayesian inverse problems, 2022, arXiv:2210.03008. +[7] Feng Cheng Chang, Inversion of a perturbed matrix, Appl. Math. Lett. 19 (2006), no. 2, +169–173. +[8] Duc-Lam Duong, Tapio Helin, and Jose Rodrigo Rojo-Garcia, Stability estimates for the +expected utility in Bayesian optimal experimental design, 2022, arXiv:2211.04399. +[9] Subhashis Ghosal and Aad van der Vaart, Fundamentals of Nonparametric Bayesian In- +ference, vol. 44, Cambridge: Cambridge University Press, 2017. +[10] Konstantinos Gourgoulias, Markos A. Katsoulakis, Luc Rey-Bellet, and Jie Wang, How +biased is your model? Concentration inequalities, information and model bias, IEEE Trans. +Inform. Theory 66 (2020), no. 5, 3079–3097. +[11] Eric Joseph Hall and Markos A. Katsoulakis, Robust information divergences for model- +form uncertainty arising from sparse data in random PDE, SIAM/ASA J. Uncertain. Quan- +tif. 6 (2018), no. 4, 1364–1394. +[12] Jari Kaipio and Erkki Somersalo, Statistical and computational inverse problems, vol. 160, +Springer Science & Business Media, 2005. +[13] +, Statistical inverse problems: discretization, model reduction and inverse crimes, +J. Comput. Appl. Math. 198 (2007), no. 2, 493–504. +[14] Marc C. Kennedy and Anthony O’Hagan, Bayesian calibration of computer models, J. R. +Stat. Soc., Ser. B, Stat. Methodol. 63 (2001), no. 3, 425–464. +[15] Ville Kolehmainen, Tanja Tarvainen, Simon R. Arridge, and Jari P. Kaipio, Marginalization +of uninteresting distributed parameters in inverse problems – application to diffuse optical +tomography, Int. J. Uncertain. Quantif. 1 (2011), no. 1, 1–17. +[16] Karina Koval, Alen Alexanderian, and Georg Stadler, Optimal experimental design under +irreducible uncertainty for linear inverse problems governed by PDEs, Inverse Probl. 36 +(2020), no. 7, 075007. +[17] Yvon Maday and Tommaso Taddei, Adaptive PBDW approach to state estimation: noisy +observations; user-defined update spaces, SIAM J. Sci. Comput. 41 (2019), no. 4, b669– +b693. +[18] Ruanui Nicholson, No´emi Petra, and Jari P Kaipio, Estimation of the Robin coefficient field +in a Poisson problem with uncertain conductivity field, Inverse Probl. 34 (2018), no. 11, +115005. +[19] A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P. Kaipio, Compensation of errors +due to discretization, domain truncation and unknown contact impedances in electrical +impedance tomography, Meas. Sci. Technol. 20 (2009), no. 10, 105504. +26 + +[20] K. Sargsyan, H. N. Najm, and R. Ghanem, On the statistical calibration of physical models, +Int. J. Chem. Kinet. 47 (2015), no. 4, 246–276. +[21] Khachik Sargsyan, Xun Huan, and Habib N. Najm, Embedded model error representation +for Bayesian model calibration, Int. J. Uncertain. Quantif. 9 (2019), no. 4, 365–394. +[22] Andrea Scarinci, Michael Fehler, and Youssef Marzouk, Bayesian inference under model +misspecification using transport-Lagrangian distances: an application to seismic inversion, +2021, arXiv:2105.07027. +[23] Bj¨orn Sprungk, On the local Lipschitz stability of Bayesian inverse problems, Inverse Probl. +36 (2020), 055015. +[24] Andrew M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica 19 (2010), +451–559. +[25] Alexandre B. Tsybakov, Introduction to nonparametric estimation, Springer series in statis- +tics, vol. 160, Springer Science & Business Media, 2009. +[26] Martin J. Wainwright, High-dimensional statistics, Cambridge Series in Statistical and +Probabilistic Mathematics, vol. 48, Cambridge University Press, Cambridge, 2019. +27 + diff --git a/8dE4T4oBgHgl3EQfCwtu/content/tmp_files/load_file.txt b/8dE4T4oBgHgl3EQfCwtu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1431208453530e949a1dd27044870e4caaa4b87 --- /dev/null +++ b/8dE4T4oBgHgl3EQfCwtu/content/tmp_files/load_file.txt @@ -0,0 +1,1034 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf,len=1033 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='04863v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='ST] 12 Jan 2023 Choosing observation operators to mitigate model error in Bayesian inverse problems Nada Cvetkovi´c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Han Cheng Lie2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Harshit Bansal1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' and Karen Veroy–Grepl1 1Centre for Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Scientific computing and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Eindhoven University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Groene Loper 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 5612 AE Eindhoven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' the Netherlands 2 Institut f¨ur Mathematik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Universit¨at Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Campus Golm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Haus 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Karl-Liebknecht-Straße 24–25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Potsdam OT Golm 14476,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Germany Abstract In Bayesian inverse problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' ‘model error’ refers to the discrepancy between the parameter- to-observable map that generates the data and the parameter-to-observable map that is used for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Model error is important because it can lead to misspecified likelihoods, and thus to incorrect inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We consider some deterministic approaches for accounting for model error in inverse problems with additive Gaussian observation noise, where the parameter-to-observable map is the composition of a possibly nonlinear parameter-to-state map or ‘model’ and a linear state-to-observable map or ‘observation operator’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using local Lipschitz stability estimates of posteriors with respect to likelihood perturbations, we bound the symmetrised Kullback–Leibler divergence of the posterior generated by each approach with respect to the posterior associated to the true model and the posterior associated to the wrong model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Our bounds lead to criteria for choosing observation operators that mitigate the effect of model error on the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Keywords: Model error, Bayesian inverse problems, experimental design, misspecified like- lihood, posterior error bounds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Introduction In many applications, one considers an inverse problem where the data is a noisy observation of the true state of some phenomenon of interest, where the true state is the output of a parameter- to-state mapping or ‘model’ corresponding to an unknown parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' That is, for the unknown true parameter θ† and the true model M†, the true state is u† = M†(θ†), and the data y is a realisation of the random variable Y := O ◦ M†(θ†) + ε (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) for an observation operator O and additive noise ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The inverse problem consists in inferring the data-generating parameter θ† from the data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We consider Bayesian inverse problems with finite-dimensional data, centred Gaussian obser- vation noise, and linear observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In this setting, ε has the normal distribution 1 N(mε, Σε) for mε = 0 ∈ Rn and positive definite covariance Σε ∈ Rn×n, and the observation operator O is a linear mapping from the ‘state space’ U of candidate states to the ‘data space’ Y = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' One fixes a possibly infinite-dimensional parameter space Θ consisting of candidate values of θ†, and describes the unknown true parameter θ† using a random variable θ with prior law µθ supported on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Under certain assumptions on the prior µθ, the observation operator O, and the model M†, the posterior measure µy,† θ of θ|y that corresponds to O ◦M† is well-defined and admits the following likelihood with respect to the prior µθ: Θ ∋ θ′ �→ dµy,† θ dµθ (θ′) = exp(− 1 2∥y − O ◦ M†(θ′)∥2 Σ−1 ε ) � Θ exp(− 1 2∥y − O ◦ M†(ˆθ)∥2 Σ−1 ε )dµθ(ˆθ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [24, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1] for sufficient conditions for well-definedness in the case of a Gaussian prior µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In practice, the true model M† : Θ → U is not available, so an approximate model M : Θ → U is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Alternatively, M† may be available but impractical or costly to evaluate: in the context of multifidelity or reduced order models, M† may be the element of a collection M of models that yields the most accurate predictions of state, and M may be a reduced-order model, an emulator, or a surrogate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' a model that yields less accurate predictions but can be evaluated more cheaply and quickly than M†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We shall refer to the difference δ† := M† − M as the ‘model error of the appproximate model’, or simply the ‘model error’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Model error is also known as ‘model inadequacy’, ‘model discrepancy’, or ‘structural uncertainty’, for example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' see [14, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We do not use the term ‘model misspecification’, since this term is used in the statistics literature to refer to the distinct problem where the parameter space Θ does not contain the true parameter θ†;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [9, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In the context of inverse problems, model error is important because it may lead to a wrong or ‘misspecified’ likelihood, which in turn may lead to incorrect inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The negative effects may persist even after applying or approximating common limits from statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For example, numerical experiments in [4, Section 3] show how ignoring the model error results in posterior distributions that do not converge to the true data-generating parameter as the number of observations grows larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' An analytical example in [1, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1] considers the problem of inferring the initial condition of an initial value problem on the time interval [0, T] from a noisy observation of the state at time T, and shows that the posterior density contracts around the wrong initial condition in the limit of small observation noise and T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This raises the question of how to mitigate the effect of model error in Bayesian inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We approach this question from the point of view of selecting an appropriate observation operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using that O is a linear mapping and substituting M† with M + δ† in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) yields Θ ∋ θ′ �→ exp(− 1 2∥y − O ◦ M(θ′) − O ◦ δ†(θ′)∥2 Σ−1 ε ) � Θ exp(− 1 2∥y − O ◦ M(ˆθ) − O ◦ δ†(ˆθ)∥2 Σ−1 ε )dµθ(ˆθ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The posterior µy,A θ that uses the approximate model M and ignores δ† has the likelihood Θ ∋ θ′ �→ dµy,A θ dµθ (θ′) = exp(− 1 2∥y − O ◦ M(θ′)∥2 Σ−1 ε ) � Θ exp(− 1 2∥y − O ◦ M(ˆθ)∥2 Σ−1 ε )dµθ(ˆθ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, the presence of model error δ† leads to a misspecified likelihood if and only if O ◦ δ†(θ′) is nonzero with positive µθ-probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This suggests that a possible approach to mitigate the 2 effect of model error in Bayesian inverse problems of the type given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) is to choose an observation operator O so that O ◦ δ†(θ′) is close to zero with high µθ-probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The approach of choosing observation operators suggests a connection with experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In Bayesian experimental design, the main task is to select observations in order to maximise information about the parameter to be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' To quantify the information gain, one may use the Kullback–Leibler divergence of the posterior with respect to the prior, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In contrast, we control the Kullback–Leibler divergence between pairs of posteriors defined by the same prior but different likelihoods, by using the L1 µθ difference between the pair of negative log-likelihoods or ‘misfits’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The main task is then to select observations in order to minimise this L1 µθ difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This approach can be seen as experimental design for reducing the impact of likelihood misspecification due to model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In this paper, we consider three deterministic approaches for accounting for model error in Bayesian inference: the ‘enhanced error approach’ of [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' the ‘joint approach’ that infers both θ† and δ†;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' and the marginalisation approach, which integrates out the model error component of the posterior from the joint inference approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For the first two ap- proaches, we compute upper bounds for the L1 µθ difference between the misfit of each approach and the misfit of the best posterior µy,† θ defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' These upper bounds on the L1 µθ difference between misfits yield upper bounds for the symmetrised Kullback–Leibler divergence max{dKL(µy,• θ ∥µy,† θ ), dKL(µy,† θ ∥µy,• θ )} between the posterior µy,• θ that results from each approach and the best posterior µy,† θ defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We repeat the procedure for the approximate posterior µy,A θ instead of µy,† θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For each approach, we express the upper bounds on the L1 µθ differences in terms of the model error δ† and the objects that each approach uses to account for model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' To prove these bounds, we rely on the assumption of additive Gaussian noise, in the form of a lemma concerning the difference of two quadratic forms that are weighted by different matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We prove the upper bounds on the symmetrised Kullback–Leibler divergence between the posteriors by combining the upper bounds on the L1 µθ differences between the misfits with a local Lipschitz stability estimate of posteriors from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' An important advantage of this estimate is that it holds for the Kullback–Leibler topology under the mild assumption of L1 µθ-integrable misfits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The upper bounds on the symmetrised Kullback–Leibler divergence with respect to the best posterior µy,† θ provide sufficient conditions on the observation operator O, the model error δ†, and the approach, in order for the resulting posterior µy,• θ to closely approximate µy,† θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In contrast, the upper bounds on the symmetrised Kullback–Leibler divergence with respect to the approximate posterior µy,A θ provide necessary conditions for the resulting posterior µy,• θ to differ from µy,A θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The first and second set of upper bounds give respectively a set of ‘positive’ and ‘negative’ criteria by which to choose observation operators for Bayesian inverse problems in the presence of model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Overview of related work The importance of accounting for the model error is well-documented in the literature on Bayesian inverse problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [14, 12, 13, 4] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The ‘Bayesian approximation error’ and ‘enhanced error’ approaches due to [13] and [12] respectively have been applied in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For example, the enhanced error approach has been applied with premarginalisation to electrical impedance tomography [19], diffuse optical tomography 3 [15], and inversion for coefficient fields in the presence of uncertain conductivities [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Various methods have been developed to estimate or account for model error in Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For example, the work [5] presented an iterative algorithm to update an estimate of the model error in a model order reduction context, and proved geometric convergence of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The authors of [20, 21] take a different perspective: instead of viewing model errors as additive perturbations to an approximate model, they incorporate these model errors into parametrisations of some phenomenon of interest, and use polynomial chaos expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Infor- mation theory has been used to quantify model error uncertainty or model bias in goal-oriented inference settings [11] and by exploiting concentration inequalities [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Optimal transport was applied to tackle problems due to model errors in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In the context of Bayesian optimal experimental design, model error is also referred to as ‘model uncertainty’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The work [16] considers A-optimal designs for inverse problems in the presence of so-called ‘irreducible’ model uncertainties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' uncertainties that cannot be reduced by collecting more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In contrast, the work [3] considers reducible uncertainties, and describes an A-optimality criterion that involves marginalising out this reducible uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The work [2] combines the Laplace approximation and the Bayesian approximation error approach to find A-optimal designs for nonlinear Bayesian inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' As far as we are aware, the work that is most closely related to our paper consists in [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The work [6] considers Bayesian inverse problems where the observation operator may be nonlinear and the model is approximated by a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, [6, Theorem 1] bounds the Kullback–Leibler divergence between the original and approximated posterior in terms of an Lp norm for p ≥ 2 of the model error itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In contrast, we consider linear observation operators, do not focus on any specific class of approximate models, and bound the Kullback–Leibler divergence in terms of an L1 norm of the observed model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In [8], the main stability estimate [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4] bounds the expected utility of a design in terms of a sequence of likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The focus of [8] is not model error, but on the convergence of the utility of approximate optimal designs corresponding to a convergent sequence of likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In contrast, we focus on choosing observation operators to mitigate the effect of model error on Bayesian inference instead of ‘classical’ experimental design, and compare pairs of likelihoods instead of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Outline We describe the notation that we use in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In Section 2 we define the posteriors that we analyse in this paper and state our main assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In Section 3 we bound the L1 µθ differences between misfits, and the symmetrised Kullback–Leibler divergences between their associated posterior measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We first consider posteriors defined only on parameter space in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1, before we consider the misfit and posterior obtained from jointly inferring the parameter and model error in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We use the Kullback–Leibler bounds to identify conditions on observation operators so as to mitigate the effect of model error on parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We conclude in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Appendix A contains the proofs of lemmas and results that we do not write in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Notation Let P be the probability measure of a probability space that serves as a common domain for all random variables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given a measurable space (E, E) and an E-valued random variable ξ : (Ω, F) → (E, E), we denote the law of ξ by µξ and write ξ ∼ µξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given a metric space (E, dE) with Borel σ-algebra B(E), let M1(E) denote the set of all 4 probability measures µ on (E, B(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, for µ ∈ M1(E) and an E-valued random variable ξ, the expression ξ ∼ ν means that µξ = ν as measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given two measures µ and ν on a common measurable space (E, E), we denote the absolute continuity of µ with respect to ν by µ ≪ ν and the mutual equivalence of µ and ν by µ ∼ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For µ ∈ M1(E), p ≥ 1, and d ∈ N, Lp µ(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Rd) := {f : E → Rd : ∥f∥Lp µ < ∞}, ∥f∥p Lp µ := � E |f(x)|pdµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We will denote a Gaussian measure with mean m and covariance operator Σ by N(m, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The notation a ← b indicates the replacement of a using b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We denote the standard Euclidean inner product and norm on Rd by ⟨·, ·⟩ and ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For d ∈ N and a symmetric positive semidefinite matrix L ∈ Rd×d, the matrix-weighted inner product and norm are ⟨a, b⟩L := a⊤Lb = ⟨L1/2a, L1/2b⟩, ∥a∥L := ⟨a, a⟩1/2 L = ∥L1/2a∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) Below, Θ will denote the parameter space, M will denote the space of models, O will denote the space of observation operators, and D will denote the space of model errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given normed vector spaces (Vi, ∥ · ∥Vi) for i = 1, 2, L (V1, V2) denotes the space of bounded linear operators from V1 to V2, and V ∗ i denotes the continuous dual space of Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We denote the kernel, range, and adjoint of L ∈ L (V1, V2) by ker(L), ran(L), and L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, for a matrix A ∈ Rm×n, ∥A∥ denotes the norm of A : Rn → Rm where both Rn and Rm are equipped with the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Description of assumptions and posterior measures Setup and assumptions Fix a space (Θ, ∥ · ∥Θ) of admissible unknown parameters or ‘pa- rameters’ θ, which we take to be a measurable subset of a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Fix a Banach space (U, ∥ · ∥U) of ‘states’ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We shall refer to a measurable mapping M : Θ → U as a ‘model’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let M denote a fixed collection of admissible models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We shall refer to M as the ‘model space’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In many inverse problems, the state space U is a Banach space of functions on a fixed, bounded spatial or spatiotemporal domain D that take values in a common Euclidean space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' U may be the space (C(D), ∥ · ∥L∞) of continuous functions on a bounded domain, equipped with the supremum norm, or a Sobolev space (Hk(D), ∥ · ∥Hk) for some k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The parameter space is also a function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The model M is often described implicitly, by an ordinary or partial differential equation where one or more coefficients are determined by the parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' There exists a unique best model M† ∈ M and a unique best parameter θ† ∈ Θ such that among all model-unknown pairs (M′, θ′) ∈ M × Θ, the corresponding ‘best state’ u† := M†(θ†) describes the phenomenon of interest most accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The assumption of a unique θ† is commonly made in the context of frequentist statistics, where θ† is often referred to as the ‘true parameter’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' However, in the context of experimental design for statistical inverse problems where observations are assumed to have the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1), the question of whether a parameter is the true parameter makes sense only when a parameter- to-state map or model has been specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Hence, for θ† to have the interpretation of the ‘true parameter’, we must also fix a unique ‘true model’ M†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In this paper, we shall consider the setting where the best model M† in Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' However, one may also consider the best model M† to be a model that is known but is too expensive to use ‘frequently’, where the meaning of ‘frequently’ depends on the 5 context or the target application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The model M can be considered as an emulator, a surrogate, or a reduced order model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' M ideally has the property that it approximates M† reasonably well and is cheaper to evaluate than M†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' There is a fixed collection of measurement functions (ℓi)i∈I indexed by a countable set I ⊂ N, where each ℓi is a continuous linear mapping from U to Rd for some d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In addition, every admissible observation operator O has the form O : U → RNd, u �→ (ℓi1(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' , ℓiN (u)), ij ∈ I, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) where the (ℓij)N j=1 are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Associated to the collection (ℓij)N j=1 of measurement functions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) is a collection of random variables (εij)N j=1 that are independent and identically N(0, Σ0)- distributed, where Σ0 ∈ Rd×d is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The random variables (εij)N j=1 represent additive measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If U = C(D, ∥ · ∥L∞), then pointwise evaluation functionals of the form ℓij(u) := u(xij) for some xij ∈ D give an example of the measurement functions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' An important consequence of the assumptions on the measurement noise (εij)N j=1 in Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3 is that the RNd-valued random variable satisfies ε := (εi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' , εiN ) ∼ N(0, Σε), Σε = diag(Σ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Σ0) ∈ RNd×Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The last equation above means that Σε is a block-diagonal matrix with identical blocks that do not depend on O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, Σε depends on the choice of O only via the number N of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' One could achieve greater generality by allowing the (εij)N j=1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) to be statistically correlated or to have different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This generality would allow Σε to depend not only on N but on O itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In the remainder of this section, we will describe the posterior measures that we shall analyse in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For a measurable misfit Φ : Θ → R, we will write Z(Φ) := � Θ exp(−Φ(θ′))dµθ(θ′) to denote the normalisation constant that makes θ′ �→ exp(−Φ(θ′))Z(Φ)−1 a probability density function with respect to µθ, whenever the normalisation constant belongs to (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Approximate posterior Given O as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) and some model M ∈ M , we assume that an observation y is a noisy observation of state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' a realisation of the Y := RNd-valued random variable Y := O ◦ M(θ†) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) Given Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3, the observation model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2), a prior µθ on θ, the data y and Bayes’ law, we obtain the approximate misfit Φy,A and the approximate posterior Φy,A(θ′) := 1 2∥y − O ◦ M(θ′)∥2 Σ−1 ε , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a) dµy,A θ (θ′) := exp(−Φy,A(θ′)) Z(Φy,A) dµθ(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3b) 6 Best posterior Recall from Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 that u† := M†(θ†) best describes the phenomenon of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For an arbitrary O ∈ O, the ‘best model’ is given by replacing M with M† in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2): Y = O ◦ M†(θ†) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The corresponding best misfit Φy,† and best posterior µy,† θ are defined by Φy,†(θ′) := 1 2∥y − O ◦ M†(θ′)∥2 Σ−1 ε , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4a) dµy,† θ (θ′) := exp(−Φy,†(θ′)) Z(Φy,†) dµθ(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4b) Given M ∈ M , the corresponding ‘model error’ is given by the difference δ† := M† − M ∈ D, D := M† − M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) We refer to D as the ‘model error space’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If the model space M is a vector space, then the model error space D and M coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If M† is not known or too expensive to evaluate, then so is δ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For the unique, fixed θ† in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1, define the corresponding ‘state error’ δ†(θ†) = M†(θ†) − M(θ†) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Rewriting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) as M† = M + δ†, substituting the latter equation into the best observation model, and using the linearity of O, we obtain Y = O ◦ M(θ†) + O ◦ δ†(θ†) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The observation model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) thus corresponds to the assumption of zero observed state error O ◦ δ†(θ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Enhanced noise posterior One approach that aims to account for the observed state error is to group the observed state error O ◦ δ†(θ†) with the noise ε to obtain O ◦ δ†(θ†) + ε, and to model this random variable with an ‘enhanced noise’ random variable [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This is also known as the ‘pre-marginalisation’ approach, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We approximate the unknown state error δ†(θ†) with a random variable u, and make the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The random variable u that approximates the unknown state error δ†(θ†) is Gaussian with mean mu and covariance Σu, and is independent of θ ∼ µθ and ε ∼ N(0, Σε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given the distributional assumptions in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4 and the linearity assumption on O in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3, it follows from the properties of Gaussian random variables that the enhanced noise random variable Ou + ε has the law N(Omu, Σε + OΣuO∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This yields the enhanced noise observation model Y = O ◦ M(θ†) + Ou + ε, which yields the enhanced noise misfit and enhanced noise posterior Φy,E(θ′) := 1 2∥y − O ◦ M(θ′) − Omu∥2 (Σε+OΣuO∗)−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6a) dµy,E θ (θ′) := exp(−Φy,E(θ′)) Z(Φy,E) dµθ(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6b) 7 Joint parameter-error posterior In the enhanced noise approach presented in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6), we account for the uncertainty due to the state error δ†(θ†) by approximating it using a random variable u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The only unknown that we aim to infer is θ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In the joint parameter-error inference approach, one aims to infer (θ†, δ†) jointly, by using a random variable (θ, δ) with prior µθ,δ and using Bayes’ formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The joint prior on the random variable (θ, δ) is a product measure of the form µθ ⊗ µδ, for µθ ∈ M1(Θ) and µδ ∈ M1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The assumption that the prior µθ,δ on (θ, δ) has product structure is equivalent to the as- sumption that θ and δ are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Under the observation model Y = O ◦ M(θ) + O ◦ δ(θ) + ε and under the distributional assumptions on ε in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3, we have the joint misfit and joint posterior Φy,J(θ′, δ′) := 1 2∥y − O ◦ M(θ′) − O ◦ δ′(θ′)∥2 Σ−1 ε , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7a) dµy,J θ,δ(θ′, δ′) := exp(−Φy,J(θ′, δ′)) Z(Φy,J) dµθ ⊗ µδ(θ′, δ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7b) One important disadvantage of jointly inferring the parameter and model error is that the di- mension of the space on which one performs inference increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' this tends to make the inference task more computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' It is also known that the problem of identifiability may arise, but we shall not consider the problem of identifiability here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' On the other hand, jointly inferring the parameter and model error is consistent with the Bayesian approach of treating all unknowns as random variables and updating these distributions using the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In addition, the joint inference approach offers the possibility to improve a possibly incorrect model M ∈ M by posterior estimates of δ†, and thus also the possibility of obtaining better estimates of both the parameter θ† as well as the state u† = M†(θ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Marginal posterior The marginal approach involves first using the joint inference approach to obtain the joint posterior µy,J θ,δ on (θ, δ), and then integrating over all δ′ ∈ D: µy,M θ (S) = � S×D dµy,J θ,δ(θ′, δ′), S ∈ B(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) The marginal posterior µy,M θ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) can be approximated by using Monte Carlo integration of the joint posterior µy,J θ,δ over δ′ ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The marginal approach inherits the problem of high computational cost from the joint inference approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' On the other hand, it has an advantage over the enhanced noise approach, namely that it involves a Bayesian update of the distribution of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Error bounds on misfits and posteriors In this section we compare the approaches presented above, by using some local Lipschitz stability bounds with respect to the Kullback–Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Recall that the Kullback– Leibler divergence between two probability measures µ and ν on a metric space (E, dE) is given 8 by dKL(µ∥ν) := �� E log dµ dν dµ µ ≪ ν +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) Given µ ∈ M1(E) and Φ ∈ L1 µ(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' R), define µΦ ∈ M1(E) by dµΦ dµ (x′) = exp(−Φ(x′)) Z(Φ) , Z(Φ) := � E exp(−Φ(x′))dµ(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For a measure µ on some measurable space (E, E) and a measurable function f : E → R, ess infµf denotes the essential infimum of f with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The following local Lipschitz stability result is due to [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let µ ∈ M1(E), Φ(1) ∈ L1 µ(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' R≥0), and Φ(2) ∈ L1 µ(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Assume that ess infµΦ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then dKL(µΦ(1)∥µΦ(2)) ≤ 2 exp � − min{ess infµΦ(2), 0} + ∥Φ(1)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ � ∥Φ(1) − Φ(2)∥L1µ, and thus µΦ(1) is absolutely continuous with respect to µΦ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, max{dKL(µΦ(1)∥µΦ(2)), dKL(µΦ(2)∥µΦ(1))} ≤ 2 exp � − min{ess infµΦ(2), 0} + 2∥Φ(1)∥L1µ + 2∥Φ(2)∥L1µ � ∥Φ(1) − Φ(2)∥L1µ, and thus µΦ(1) is mutually equivalent to µΦ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The first statement follows by combining [23, Theorem 11] and [23, Proposition 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By the triangle inequality, max{∥Φ(1)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ, ∥Φ(2)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ} ≤ 2∥Φ(1)∥L1µ + 2∥Φ(2)∥L1µ, and thus the second statement follows from the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Below, we will use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 to bound the Kullback–Leibler error between pairs of poste- riors, for the posteriors defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Recall that the Hellinger metric between µ, ν ∈ M1(E) is defined by d2 H(µ, ν) := � E �� dµ dλ − � dν dλ �2 dλ, where λ is any measure such that both µ ≪ λ and ν ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The definition of dH(µ, ν) does not depend on the choice of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The Hellinger metric and Kullback–Leibler divergence satisfy d2 H(µ, ν) ≤ dKL(µ∥ν), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [25, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Hence, the bounds on the Kullback–Leibler error that we present below imply bounds with respect to the Hellinger metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kullback–Leibler error of posteriors on parameter space 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Error with respect to the best posterior Error of approximate posterior with respect to best posterior In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 below, we bound the L1 µθ error between the approximate misfit Φy,A and the best misfit Φy,† defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4a) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We express the bound in terms of the average observed model error O ◦ δ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose Φy,† ∈ L1 µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,A ∈ L1 µθ, then ∥Φy,† − Φy,A∥L1µθ ≤ 2−1/2∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) where ∥∥O ◦ δ†∥2 Σ−1 ε ∥L1µθ ≤ 21/2� ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 for the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) yields ∥Φy,† − Φy,A∥L1µθ ≤ � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Since ∥Φy,†∥L1µθ + ∥Φy,A∥L1µθ ≤ � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ �2, it follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) together are not optimal: they yield a worse bound on ∥Φy,† − Φy,A∥L1µθ than the bound we could obtain using the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' However, the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) is useful, because it bounds ∥Φy,† − Φy,A∥L1µθ in terms of the average observed model error ∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ and quantities that are assumed to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,A, Φy,† ∈ L1 µθ(Θ, R), then max{dKL(µy,A θ ∥µy,† θ ), dKL(µy,† θ ∥µy,A θ )} ≤C∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ for C = C(∥Φy,A∥L1µθ , ∥Φy,†∥L1µθ ) := 21/2 exp(2∥Φy,†∥L1µθ + 2∥Φy,A∥L1µθ ) � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The constant C in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4 is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The main value in defining C is to show that, given the hypotheses, the constant C is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a) of Φy,A, it follows that ess infµΦy,A ≥ 0, so min{ess infµθΦy,A, 0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given that Φy,A, Φy,† ∈ L1 µθ, we may apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, and also the second statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 with Φ(1) ← Φy,†, Φ(2) ← Φy,A, and µ ← µθ, which yields max{dKL(µy,A θ ∥µy,† θ ), dKL(µy,† θ ∥µy,A θ )} ≤2 exp � 2∥Φy,A∥L1µθ + 2∥Φy,†∥L1µθ � ∥Φy,† − Φy,A∥L1µθ ≤21/2 exp � 2∥Φy,A∥L1µθ + 2∥Φy,†∥L1µθ �� ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � ∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 10 If M† is not completely known, then neither are Φy,† nor δ†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Furthermore, it may be difficult to compute ∥Φy,A∥L1µθ exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, it will in general be difficult to compute the constant C in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4 shows that the Kullback–Leibler divergences of the approximate posterior µy,A θ with respect to the best posterior µy,† θ and vice versa are controlled by the average ob- served model error ∥∥O ◦δ†∥2 Σ−1 ε ∥1/2 L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In order for dKL(µy,A θ ∥µy,† θ ) or dKL(µy,† θ ∥µy,A θ ) to be small, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4 suggests that one could choose the observation operator O such that δ† takes values in or near ker(O) with high µθ-probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, if the observation operator O satisfies P(O ◦ δ†(θ) = 0) = 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) then the corresponding approximate posterior and the best posterior coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This is not surprising, since if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) holds then Φy,† = Φy,A coincide µθ-almost everywhere, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, and hence µy,† θ and µy,A θ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) can be useful for guiding the choice of observation operator O even if δ† is not fully known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For example, if the state space U is a Hilbert space, and if one can determine a priori that δ† takes values in some proper subspace V of U without knowing δ† exactly, then any choice of observation operator O such that V ⊆ ker(O) will yield an approximate posterior µy,A θ that agrees with the best posterior µy,† θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The problem then consists in choosing O so that ker(O) is as small as possible, while satisfying the constraint that the model error takes values in ker(O) µθ-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The payoff in choosing O in this way is that Bayesian inference with µy,A θ will be as good as Bayesian inference with µy,† θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, the key idea in choosing observation operators to mitigate the effect of model error on Bayesian inference is to exploit all available knowledge about the model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Recall from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 that we may also interpret M as a reduced-order model or surrogate for a more accurate but costly model M†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The preceding discussion then implies that, for suitably chosen observation operators, Bayesian inference with µy,A θ will be as good as µy,† θ , and have smaller computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Error of enhanced noise posterior with respect to best posterior Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6 below bounds the L1 µθ error between the misfits Φy,† and Φy,E from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6a) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The bound indicates the importance of the shifted observed model error term O ◦ (δ† − mu) and difference Σ−1 ε − (Σε + OΣuO∗)−1 of covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Define the scalar CE := ∥Σ−1/2 ε (Σε + OΣuO∗)1/2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1, CE satisfies ∥z∥Σ−1 ε ≤ CE∥z∥(Σε+OΣuO∗)−1, z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose Φy,† ∈ L1 µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,E ∈ L1 µθ, then for CE as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), ∥Φy,† − Φy,E∥L1µθ ≤2−1/2∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ � ∥Φy,†∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6) + 2−1∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Furthermore, ∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ ≤21/2� ∥Φy,†∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � , ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ ≤(CE + 1)∥2Φy,E∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 11 See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 for the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,E, Φy,† ∈ L1 µθ(Θ, R), then max{dKL(µy,† θ ∥µy,E θ ), dKL(µy,E θ ∥µy,† θ )} ≤C � ∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ � for C = C(∥Φy,E∥L1µθ , ∥Φy,†∥L1µθ , CE), where C := exp � 2∥Φy,†∥L1µθ + 2∥Φy,E∥L1µθ � max �√ 2 � ∥Φy,†∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' As with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4, the importance of the constant C above is that C is finite under the hypotheses of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6a) of Φy,E, it follows that ess infµΦy,E ≥ 0, so min{ess infµθΦy,E, 0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given that Φy,E, Φy,† ∈ L1 µθ, we may apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 with Φ(1) ← Φy,†, Φ(2) ← Φy,E, and µ ← µθ, to obtain max{dKL(µy,† θ ∥µy,E θ ), dKL(µy,E θ ∥µy,† θ )} ≤2 exp � 2∥Φy,E∥L1µθ + 2∥Φy,†∥L1µθ � ∥Φy,† − Φy,E∥L1µθ ≤ exp � 2∥Φy,E∥L1µθ + 2∥Φy,†∥L1µθ � max{ √ 2 � ∥Φy,†∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � , 1} × � ∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ � where the second inequality follows from the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The significance of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7 is similar to that of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The main differences follow from the fact the L1 µθ error between the enhanced noise misfit Φy,E and the best misfit Φy,† — and hence also the Kullback–Leibler error between µy,E θ and µy,† θ — is now controlled by the sum ∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7) Recall from Section 2 that the enhanced noise approach consists in modelling the unknown state correction term δ†(θ†) ∈ U by a Gaussian random variable u ∼ N(mu, Σu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Since Σ−1 ε is invertible by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3, the first term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7) vanishes if and only if δ† − mu ∈ ker(O) µθ-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This condition differs from the sufficient condition for µy,† θ = µy,A θ that was implied by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, namely, that δ† ∈ ker(O) µθ-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The difference consists in the mu term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By recalling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3), the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7) vanishes if and only if P � y − O ◦ M(θ) − Omu ∈ ker � Σ−1 ε − (Σε + OΣuO∗)−1� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) By a rearrangement of the Woodbury formula that we obtained from [7, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3)], we have Σ−1 ε − (Σε + OΣuO∗)−1 = Σ−1 ε OΣuO∗Σ−1 ε (Σε + OΣuO)−1Σ−1 ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The equation above holds for non-invertible OΣuO∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If OΣuO∗ = 0, then both sides of the equation above vanish, and thus the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Since Σu describes the 12 covariance of the U-valued random model u of the state error δ†(θ†), the condition OΣuO∗ = 0 has the equivalent formulation that the RNd-valued random variable Ou is constant P-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Since Ou = O(u − mu) + Omu, the latter condition is equivalent to u − mu ∈ ker(O) P-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' More generally, if OΣuO∗ is nonzero but has non-trivial kernel, then Σ−1 ε − (Σε + OΣuO∗)−1 also has a non-trivial kernel, and it may be possible for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If one knows a priori that the image of Θ under δ† is contained in some affine subspace x + V of U for a linear subspace V of U, then one can exploit this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For example, given the enhanced noise model N(mu, Σu), one should choose the observation operator O so that V ⊆ mu + ker(O) and u takes values in mu + ker(O) P-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In this case, the enhanced noise posterior µy,E θ and the best posterior µy,† θ will coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Error of enhanced noise posterior with respect to approximate posterior Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose Φy,A ∈ L1 µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,E ∈ L1 µθ, then for CE as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), ∥Φy,A − Φy,E∥L1µθ ≤2−1/2∥Omu∥Σ−1 ε � ∥Φy,A∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9) + 2−1∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ where ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ satisfies the bound in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,A, Φy,E ∈ L1 µθ(Θ, R), then max{dKL(µy,A θ ∥µy,E θ ), dKL(µy,E θ ∥µy,A θ )} ≤C � ∥Omu∥Σ−1 ε + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ � for C = C(∥Φy,E∥L1µθ , ∥Φy,A∥L1µθ , CE), where C := exp � 2∥Φy,A∥L1µθ + 2∥Φy,E∥L1µθ � max �√ 2 � ∥Φy,A∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8, max{dKL(µy,A θ ∥µy,E θ ), dKL(µy,E θ ∥µy,A θ )} ≤2 exp � 2∥Φy,E∥L1µθ + 2∥Φy,A∥L1µθ � ∥Φy,A − Φy,E∥L1µθ ≤ exp � 2∥Φy,E∥L1µθ + 2∥Φy,A∥L1µθ � max{ √ 2 � ∥Φy,A∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � , 1} × � ∥Omu∥Σ−1 ε + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9 implies that if, given an enhanced noise model with mean mu and covariance Σu, one chooses the observation operator O so that ∥Omu∥Σ−1 ε = ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ = 0, then the enhanced noise posterior µy,E θ and the approximate posterior µy,A θ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Equiva- lently, Omu = 0 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) together imply that µy,E θ = µy,A θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, such a choice of observation 13 operator yields an enhanced noise posterior µy,E θ that does not account for model error, in which case it would be simpler to use the approximate posterior µy,A θ instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By the contrapositive statement, if µy,E θ and µy,A θ differ, then either Omu does not vanish, or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If OΣuO∗ = 0, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) does not hold if and only if P(y − O ◦ M(θ) − Omu ̸= 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We expect that in many cases, the latter condition will hold — and thus that µy,A θ and µy,E θ will differ — for a large collection of observation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kullback–Leibler error of joint parameter-error posterior Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7) that the joint misfit and the joint posterior are defined by Φy,J(θ′, δ′) := 1 2∥y − O ◦ M(θ′) − O ◦ δ′(θ′)∥2 Σ−1 ε , dµy,J θ,δ(θ′, δ′) := exp(−Φy,J(θ′, δ′)) Z(Φy,J) dµθ ⊗ µδ(θ′, δ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In this section, we shall compare µy,J θ ∈ M1(Θ×D) with the other posterior measures considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' To do this, we need to redefine these posteriors as measures on Θ × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For • ∈ {A, †, E}, define the lifted misfit and lifted posterior by Φy,•(θ′, δ′) := Φy,•(θ′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a) dµy,• θ,δ(θ′, δ′) := exp(−Φy,•(θ′, δ′)) Z(Φy,•) dµθ ⊗ µδ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b) where the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b) follows from Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5, namely that the joint prior is a product measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Note that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a), we use the notation Φy,• to refer to two functions defined on different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The following lemma shows that this abuse of notation is not problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let µθ, µδ, Φy,• and µy,• θ,δ be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then for • ∈ {A, †, E}, dµy,• θ,δ(θ′, δ′) = dµy,• θ (θ′) ⊗ µδ(δ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12) If Φy,• : Θ → R belongs to L1 µθ, then its lifted version Φy,• defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a) belongs to L1 µθ⊗µδ, and for every q > 0, ∥Φy,•∥Lq µθ⊗µδ = ∥Φy,•∥Lq µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The second statement follows immediately from the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For the first statement, observe that � Θ×D exp(−Φy,•(θ′, δ′))dµθ ⊗ µδ(θ′, δ′) = � Θ exp(−Φy,•(θ′))dµθ(θ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, the normalisation constant Z(Φy,•) for the lifted posterior µy,• θ,δ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b) agrees with the corresponding normalisation constant for the posterior µy,• θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 14 Notation: Recall from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3 that P denotes the probability measure in the probability space on which we define all random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In this subsection, we will sometimes write the random variables θ and δ explicitly, and take Lp-norms with respect to P instead of µθ ⊗ µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For example, ∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥L1 P = � Θ×D ∥O ◦ (δ† − δ′)(θ′)∥2 Σ−1 ε dµθ ⊗ µδ(θ′, δ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11 below for an example where we use this notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Error with respect to lifted best posterior Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,† be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a) with • = †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,† ∈ L1 µθ and Φy,J ∈ L1 µθ⊗µδ, then ∥Φy,† − Φy,J∥L1 µθ⊗µδ ≤ 2−1/2∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14) Furthermore, ∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P ≤ 21/2� ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,† and Φy,J be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11 and let µy,† θ,δ be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b) with • = †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then max{dKL(µy,† θ,δ∥µy,J θ,δ), dKL(µy,J θ,δ∥µy,† θ,δ)} ≤ C∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P , where C = 21/2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,†∥L1µθ �� ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 for the proofs of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12, a sufficient condition for µy,J θ,δ to coincide with µy,† θ,δ is P � (δ† − δ)(θ) ∈ ker(O) � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15) For example, suppose that one knows a priori that there exist a vector x ∈ U and a linear subspace V of U such that {δ†(θ′) : θ′ ∈ Θ} ⊆ x + V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose that (D, ⟨·, ·⟩D) is a Hilbert space and δ ∼ µδ = N(mδ, Σδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then mδ + Σ1/2 δ D is the Cameron–Martin space of µδ, and the support supp(µδ) of µδ is the closure of mδ +Σ1/2 δ D with respect to ∥ · ∥D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Assume there exists y ∈ U and a closed linear subspace W ⊆ U such that {δ′(θ′) : θ′ ∈ Θ, δ′ ∈ supp(µδ)} ⊆ y + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If for every v ∈ V and w ∈ W it holds that (x+v)−(y +w) ∈ ker(O), then the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The preceding example suggests that, in general, it may be difficult to choose µδ and O so that µy,† θ,δ and µy,J θ,δ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This is to be expected, since the lifted best posterior measure µy,† θ,δ is a product measure with δ-marginal equal to µδ, whereas for many reasonable choices of µδ the joint posterior measure µy,J θ,δ will not have δ-marginal equal to µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 15 Error with respect to lifted approximate posterior Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,A be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a) with • = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,A ∈ L1 µθ and Φy,J ∈ L1 µθ⊗µδ, then ∥Φy,A − Φy,J∥L1 µθ⊗µδ ≤ 2−1/2∥∥O ◦ δ(θ)∥2 Σ−1 ε ∥1/2 L1 P � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='16) Furthermore, ∥∥O ◦ δ(θ)∥2 Σ−1 ε ∥1/2 L1 P ≤ 21/2� ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,A and Φy,J be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13 and let µy,A θ,δ be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b) with • = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then max{dKL(µy,A θ,δ ∥µy,J θ,δ), dKL(µy,J θ,δ∥µy,A θ,δ )} ≤ C∥∥O ◦ δ(θ)∥2 Σ−1 ε ∥1/2 L1 P , where C = 21/2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,A∥L1µθ �� ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 for the proofs of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14, a sufficient condition for µy,A θ,δ and µy,J θ,δ to coincide is that δ(θ) ∈ ker(O), P-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By the same reasoning that we used when considering µy,† θ,δ and µy,J θ,δ, we do not expect µy,A θ,δ and µy,J θ,δ to coincide, since the former is a product measure with δ-marginal equal to µδ, and the latter will not have this property in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Connection with parametrised background data-weak approach Recall the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) of the observation operator as O := (ℓi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' , ℓiN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose the state space U is a Hilbert space and d = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' suppose that the measurement functions (ℓij)N j=1 are continuous linear functionals on U with Riesz representatives denoted by (Rℓij)N j=1 ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then the condition δ(θ) ∈ ker(O) is equivalent to δ(θ) ∈ (span(Rℓij, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' , N))⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We may reformulate the necessary condition for µy,A θ,δ and µy,J θ,δ to differ, namely that δ(θ) /∈ ker(O) with positive P-probability, as δ(θ) ∈ span(Rℓij, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' , N) with positive P-probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Given the interpretation of δ(θ) as a state correction term, the condition that δ(θ) ∈ span(Rℓij, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' , N) closely resembles the ‘variational update’ from the parametrised background data-weak approach for state inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [17, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' It is possible to state and prove the analogues of the preceding bounds for the lifted enhanced noise posterior µy,E θ,δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' These bounds are not relevant for the main goal of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' However, for the sake of completeness, we state them in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kullback–Leibler error of marginal posterior Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) that the marginal posterior is defined by µy,M θ (S) = � S×D dµy,J θ,δ(θ′, δ′), S ∈ B(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, we bounded the Kullback–Leibler error of the joint posterior µy,J θ,δ with respect to the lifted posteriors µy,• θ,δ for • ∈ {A, †, E} that were defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b), and we observed in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10 that the θ-marginal of the lifted posterior µy,• θ,δ is exactly µy,• θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 16 In this section, we shall bound the Kullback–Leibler error of the marginal posterior µy,M with respect to the θ-marginals of the lifted posteriors that we considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' with respect to µy,† θ , µy,A θ , and µy,E θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Below, µy,• δ|θ denotes the regular version of the posterior distribution of δ conditioned on θ, for ∈ {A, †, E, J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We assume that such regular conditional distributions exist and are unique up to sets of measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,J ∈ L1 µθ⊗µδ and • ∈ {A, †, E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose Φy,• : Θ → R≥0 belongs to L1 µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then dKL(µy,M θ ∥µy,• θ ) =dKL(µy,J θ,δ∥µy,• θ,δ) − � Θ dKL(µy,J δ|θ∥µy,• δ|θ)dµθ, dKL(µy,• θ ∥µy,M θ ) =dKL(µy,• θ,δ∥µy,J θ,δ) − � Θ dKL(µy,• δ|θ∥µy,J δ|θ)dµθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, max{dKL(µy,M θ ∥µy,• θ ), dKL(µy,• θ ∥µy,M θ )} ≤ max{dKL(µy,J θ,δ∥µy,• θ,δ), dKL(µy,• θ,δ∥µy,J θ,δ)} − min � � Θ dKL(µy,J δ|θ∥µy,• δ|θ)dµθ, � Θ dKL(µy,• δ|θ∥µy,J δ|θ)dµθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By nonnegativity of the Kullback–Leibler divergence, the first statement implies the second statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The first statement follows by recalling the chain rule for the Kullback–Leibler divergence, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [26, Exercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2]: dKL(µy,J θ,δ∥µy,• θ,δ) =dKL(µy,M θ ∥µy,• θ ) + � Θ dKL(µy,J δ|θ∥µy,• δ|θ)dµy,M θ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='17) dKL(µy,• θ,δ∥µy,J θ,δ) =dKL(µy,• θ ∥µy,M θ ) + � Θ dKL(µy,• δ|θ∥µy,J δ|θ)dµy,• θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='18) Above, we used the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) of the marginal posterior µy,M θ , and the fact expressed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12), namely that the θ-marginal of µy,• θ,δ is µy,• θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15 is important for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' First, it implies that the Kullback– Leibler error of the marginal posterior µy,M θ with respect to any of the above-mentioned poste- riors on Θ cannot be larger than the Kullback–Leibler error of the joint posterior µy,J θ,δ and the corresponding lifted version of the posterior on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In other words, marginalisation can only reduce the Kullback–Leibler error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' As a result, for • ∈ {A, †, E}, the Kullback–Leibler error of the marginal posterior µy,M θ with respect to µy,• θ satisfies the same bounds as the Kullback– Leibler error of the joint posterior µy,J θ,δ with respect to the lifted posterior µy,• θ,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus, the same statements regarding sufficient conditions for the coincidence of the posteriors on Θ × D that were made after Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14, and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4, also apply to the marginalised versions of the posteriors in the above-mentioned propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Conclusion We considered Bayesian inverse problems in the presence of model error in the following set- ting: the data is finite-dimensional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' the noise is additive, Gaussian, and independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' and the 17 parameter-to-observable map is the composition of a possibly nonlinear model with a linear observation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We assumed that there exists a unique best model and best parameter, such that the resulting best state most accurately describes the phenomenon of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The ‘model error’ is then the difference between the model that one uses and the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We described some existing deterministic approaches for accounting for model error and used the local Lipschitz stability property of posteriors with respect to perturbations in the likelihood to bound the symmetrised Kullback–Leibler error between pairs of posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' These bounds have two important properties: first, they control the Kullback–Leibler error in terms of quantities that depend on the observation operator and the objects used to account for model error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' and second, the other terms in the bounds are finite under mild hypotheses, namely L1-integrability of the misfits with respect to the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The bounds yield sufficient conditions on the observation operator and the model error-aware approach to yield a posterior that performs almost as well as the best posterior that uses the best model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' They also yield necessary conditions for a model error-aware approach to yield a posterior that differs from the posterior yielded by the model error-agnostic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' A recurring theme in the sufficient conditions is the importance of the kernel of the observation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We expect that the design of observation operators that approximately or exactly satisfy the sufficient conditions will be challenging and highly problem-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' It would be of interest to study the setting of nonlinear observation operators and other approaches for accounting for model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Acknowledgements The research of HCL has been partially funded by the DFG — Project-ID 318763901 — SFB1294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Technical lemmas Let d ∈ N and Mi ∈ Rd×d, i = 1, 2 be symmetric and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' For convenience, we state the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Suppose that M1, M2 ∈ Rd×d are symmetric, that M1 is positive definite, and that M2 is nonnegative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then M1+M2 is symmetric positive definite and M−1 1 −(M1+M2)−1 is symmetric nonnegative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In addition, ∥z∥(M1+M2)−1 ∥(M1 + M2)−1/2M1/2 1 ∥ ≤ ∥z∥M−1 1 ≤ ∥M−1/2 1 (M1 + M2)1/2∥∥z∥(M1+M2)−1, z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let λmin(A) denote the smallest eigenvalue of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By the assumptions on M1 and M2, M1 + M2 is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' As the difference of two symmetric matrices, M−1 1 − (M1 + M2)−1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' To show that it is nonnegative, we use the following rearrangement of the Woodbury formula, which we take from [7, Equation (3)]: (M1 + M2)−1 =M−1 1 − M−1 1 M2(I + M−1 1 M2)−1M−1 1 ⇐⇒ M−1 1 − (M1 + M2)−1 =M−1 1 M2(I + M−1 1 M2)−1M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 18 Invertibility of I+M−1 1 M2 = M−1 1 (M1+M2) follows from the invertibility of M−1 1 and M1+M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' From the second equation above, M−1 1 − (M1 + M2)−1 inherits the nonnegative definiteness of M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Recall the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) for a matrix-weighted norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The first inequality of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) follows by ∥z∥(M1+M2)−1 = ∥(M1 + M2)−1/2z∥ = ∥(M1 + M2)−1/2M1/2 1 M−1/2 1 z∥ ≤ ∥(M1 + M2)−1/2M1/2 1 ∥∥M−1/2 1 z∥ = ∥(M1 + M2)−1/2M1/2 1 ∥∥z∥M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The second inequality of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) follows by ∥z∥M−1 1 = ∥M−1/2 1 z∥ = ∥M−1/2 1 (M1 + M2)1/2(M1 + M2)−1/2z∥ ≤ ∥M−1/2 1 (M1 + M2)1/2∥∥(M1 + M2)−1/2z∥ = ∥M−1/2 1 (M1 + M2)1/2∥∥z∥(M1+M2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We will use the following bound in the proofs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let (E, dE) be a metric space, µ ∈ M1(E), and f and g be Rd-valued measurable functions on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let M1 ∈ Rd×d be symmetric and positive definite and M2 ∈ Rd×d be symmetric nonnegative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then ∥∥f∥2 M−1 1 − ∥f + g∥2 M−1 1 ∥L1µ ≤ ∥∥g∥2 M−1 1 ∥1/2 L1µ � ∥∥f + g∥2 M−1 1 ∥1/2 L1µ + ∥∥f∥2 M−1 1 ∥1/2 L1µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) More generally, ∥∥f∥2 M−1 1 − ∥f + g∥2 (M1+M2)−1∥L1µ ≤∥∥g∥2 M−1 1 ∥1/2 L1µ � ∥M−1/2 1 (M1 + M2)1/2∥ · ∥∥f + g∥2 (M1+M2)−1∥1/2 L1µ + ∥∥f∥2 M−1 1 ∥1/2 L1µ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) + ∥∥f + g∥2 M−1 1 −(M1+M2)−1∥L1µ, where ∥f + g∥2 M−1 1 −(M1+M2)−1 := ∥f + g∥2 M−1 1 − ∥f + g∥2 (M1+M2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In addition, ∥∥g∥2 M−1 1 ∥1/2 L1µ ≤ ∥∥f∥2 M−1 1 ∥1/2 L1µ + ∥M−1/2 1 (M1 + M2)1/2∥∥∥f + g∥2 (M1+M2)−1∥1/2 L1µ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) ∥∥f + g∥2 M−1 1 −(M1+M2)−1∥L1µ ≤ � 1 + ∥M−1/2 1 (M1 + M2)1/2∥ � ∥∥f + g∥2 (M1+M2)−1∥L1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' We claim that for arbitrary a, b ∈ Rd, symmetric positive definite M1 ∈ Rd×d, and symmetric nonnegative definite M2 ∈ Rd×d, ∥a∥2 M−1 1 − ∥a + b∥2 (M1+M2)−1 = −⟨b, 2a + b⟩M−1 1 + ∥a + b∥2 M−1 1 −(M1+M2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6) Recall that (M1 + M2)−1 exists and is positive definite, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 19 Using the matrix-weighted inner product and norm notation from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3), ∥a∥2 M−1 1 − ∥a + b∥2 M−1 1 =a⊤M−1 1 a − (a + b)⊤M−1 1 (a + b) =a⊤M−1 1 a − (a⊤M−1 1 a + 2a⊤M−1 1 b + b⊤M−1 1 b) = − b⊤M−1 1 (2a + b) = − ⟨b, 2a + b⟩M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This implies (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6), since ∥a∥2 M−1 1 − ∥a + b∥2 M−1 1 + ∥a + b∥2 M−1 1 − ∥a + b∥2 (M1+M2)−1 = − ⟨b, 2a + b⟩M−1 1 + ∥a + b∥2 M−1 1 − ∥a + b∥2 (M1+M2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Now let f, g, and µ be as in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6) and the triangle inequality, ∥∥f∥2 M−1 1 − ∥f + g∥2 (M1+M2)−1∥L1µ =∥ − ⟨g, 2f + g⟩M−1 1 + ∥f + g∥2 M−1 1 −(M1+M2)−1∥L1µ ≤∥⟨g, 2f + g⟩M−1 1 ∥L1µ + ∥∥f + g∥2 M−1 1 −(M1+M2)−1∥L1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7) Next, ∥⟨g, 2f + g⟩M−1 1 ∥L1µ ≤∥∥g∥M−1 1 ∥2f + g∥M−1 1 ∥L1µ ≤∥∥g∥M−1 1 ∥L2µ∥∥2f + g∥M−1 1 ∥L2µ ≤∥∥g∥M−1 1 ∥L2µ � ∥∥f + g∥M−1 1 ∥L2µ + ∥∥f∥M−1 1 ∥L2µ � =∥∥g∥2 M−1 1 ∥1/2 L1µ � ∥∥f + g∥2 M−1 1 ∥1/2 L1µ + ∥∥f∥2 M−1 1 ∥1/2 L1µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8) The first and second inequalities follow by applying the Cauchy–Schwarz inequality with respect to ⟨·, ·⟩M−1 1 and ∥·∥L1µ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The third inequality and the equation follow from the ∥·∥L2µ- triangle inequality and the definition of the Lp µ norm for p = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8), we bound the first term on the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By using M2 ← 0, the second term on the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Next, we bound the first term inside the parentheses on the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1), ∥∥f + g∥2 M−1 1 ∥1/2 L1µ ≤ ∥M−1/2 1 (M1 + M2)1/2∥∥∥f + g∥2 (M1+M2)−1∥1/2 L1µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using the above bound yields (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' To prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4), ∥∥g∥2 M−1 1 ∥1/2 L1µ =∥∥ − f + (f + g)∥M−1 1 ∥L2µ ≤∥∥f∥M−1 1 ∥L2µθ + ∥∥f + g∥M−1 1 ∥L2µ ≤∥∥f∥2 M−1 1 ∥1/2 L1µθ + ∥M−1/2 1 (M1 + M2)1/2∥∥∥f + g∥2 (M1+M2)−1∥1/2 L1µ 20 where the last inequality uses (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' To prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), observe that ∥∥f + g∥2 M−1 1 −(M1+M2)−1∥L1µ =∥∥f + g∥2 M−1 1 − ∥f + g∥2 (M1+M2)−1∥L1µ ≤∥∥f + g∥2 M−1 1 ∥L1µ + ∥∥f + g∥2 (M1+M2)−1∥L1µ ≤(∥M−1/2 1 (M1 + M2)1/2∥ + 1)∥∥f + g∥2 (M1+M2)−1∥L1µ where the first and second inequality follow from the triangle inequality and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of lemmas in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proofs for Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 Proof of error of approximate posterior with respect to best posterior Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 bounds ∥Φy,†−Φy,A∥Lq µθ in terms of the observed model error O◦δ†, under the hypothesis that Φy,† ∈ L1 µθ and Φy,A ∈ L1 µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The bound is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2): ∥Φy,† − Φy,A∥L1µθ ≤ 2−1/2∥∥O ◦ δ†∥Σ−1 ε ∥L2µθ � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a) that Φy,†(θ′) = 1 2∥y − O ◦ M†(θ′)∥2 Σ−1 ε and Φy,A(θ′) = 1 2∥y − O ◦ M(θ′)∥2 Σ−1 ε respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By these definitions, ∥2 · 1 2∥y − O ◦ M†∥2 Σ−1 ε ∥1/2 L1µθ = ∥2Φy,†∥1/2 L1µθ = √ 2∥Φy,†∥1/2 L1µθ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9) and similarly ∥∥y − O ◦ M∥2 Σ−1 ε ∥1/2 L1µθ = √ 2∥Φy,A∥1/2 L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10) Now set f ← y − O ◦ M†, g ← O ◦ δ†, µ ← µθ, M1 ← Σε, and M2 ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), we have f + g = y − O ◦ (M† − δ†) = y − O ◦ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Hence ∥f∥2 M−1 1 = 2Φy,† and ∥f + g∥2 M−1 1 = 2Φy,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 with these choices yields 2∥Φy,† − Φy,A∥L1µθ ≤∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ � ∥∥y − O ◦ M†∥2 Σ−1 ε ∥1/2 L1µθ + ∥∥y − O ◦ M∥2 Σ−1 ε ∥1/2 L1µθ � =∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ √ 2 � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � , where we used (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10) for the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The bound on ∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ in the statement of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 with the choices above: ∥∥O ◦ δ†∥2 Σ−1 ε ∥1/2 L1µθ ≤ √ 2 � ∥Φy,†∥1/2 L1µθ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of error of enhanced noise posterior with respect to best posterior Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6 bounds ∥Φy,† −Φy,E∥L1µθ in terms of the observed model error O◦δ† and the Gaussian model N(mu, Σδ) 21 of δ†(θ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In particular, under the hypotheses that Φy,† ∈ L1 µθ and Φy,E ∈ L1 µθ, then for CE := ∥Σ−1/2 ε (Σε + OΣuO∗)1/2∥ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6) ∥Φy,† − Φy,E∥L1µθ ≤2−1/2∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ � ∥Φy,†∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � + 2−1∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ , holds, and all terms on the right-hand side are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' In the same way that we proved (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9), we can use the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6a) of Φy,E to prove ∥∥y − O ◦ M − Omu∥2 (Σε+OΣuO∗)−1∥1/2 L1µθ = √ 2∥Φy,E∥1/2 L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11) Let f ← y − O ◦ M†, g ← O ◦ (δ† − mu), µ ← µθ, M1 ← Σε, and M2 ← OΣuO∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then f + g = y − O ◦ (M† − δ†) − Omu = y − O ◦ M − Omu, ∥f∥2 M−1 1 = 2Φy,†, and ∥f + g∥2 (M1+M2)−1 = 2Φy,E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 yields 2∥Φy,† − Φy,E∥L1µθ ≤∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ � CE∥2Φy,E∥1/2 L1µθ + ∥2Φy,†∥1/2 L1µθ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12) + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) and CE as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), ∥∥O ◦ (δ† − mu)∥2 Σ−1 ε ∥1/2 L1µθ ≤ ∥2Φy,†∥1/2 L1µθ + CE∥2Φy,E∥1/2 L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ ≤ (CE + 1)∥2Φy,E∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proofs for Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8 asserts that, under the hypotheses that Φy,A ∈ L1 µθ and Φy,E ∈ L1 µθ, then for CE := ∥Σ−1/2 ε (Σε + OΣuO∗)1/2∥ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5), the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9) ∥Φy,A − Φy,E∥L1µθ ≤2−1/2∥Omu∥Σ−1 ε � ∥Φy,A∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � + 2−1∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ , holds, and all terms on the right-hand side are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let f ← y −O ◦M, g ← −Omu, M1 ← Σ−1 ε , M2 ← OΣuO∗, and µ ← µθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then ∥f∥2 M−1 1 = 2Φy,A and ∥f + g∥2 (M1+M2)−1 = 2Φy,E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 yields 2∥Φy,A − Φy,E∥L1µθ ≤∥Omu∥Σ−1 ε √ 2 � ∥Φy,A∥1/2 L1µθ + CE∥Φy,E∥1/2 L1µθ � + ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ , for CE := ∥Σ−1/2 ε (Σε + OΣuO∗)1/2∥ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Next, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 yields ∥∥y − O ◦ M − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1µθ ≤ 2 � CE + 1)∥Φy,E∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proofs for Kullback–Leibler error of joint parameter-error posterior Proof of error of joint parameter-error posterior with respect to lifted best posterior In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11, one assumes that Φy,† as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4a) belongs to L1 µθ, and also that Φy,J ∈ L1 µθ⊗µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The resulting bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14) is ∥Φy,† − Φy,J∥L1 µθ⊗µδ ≤ 2−1/2∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let f ← y − O ◦ M†(θ), g ← O ◦ (δ† − δ)(θ), M1 ← Σε, M2 ← 0, and µ ← P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then ∥f∥2 M−1 1 = 2Φy,†(θ, δ) and ∥f + g∥2 M−1 1 = 2Φy,J(θ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using the same argument that we used to prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9), it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7a) that ∥∥y − O ◦ M(θ) − O ◦ δ(θ)∥2 Σ−1 ε ∥1/2 L1 P = √ 2∥Φy,J∥1/2 L1 µθ⊗µδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13) By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10, ∥∥y − O ◦ M†(θ)∥2 Σ−1 ε ∥1/2 L1 P = √ 2∥Φy,†∥1/2 L1µθ = √ 2∥Φy,†∥1/2 L1 µθ⊗µδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14) Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 with these choices yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14): 2∥Φy,A − Φy,J∥L1 µθ⊗µδ ≤∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P √ 2 � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Next, ∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P ≤ √ 2 � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1µθ � , follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 with the choices stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,† are nonnegative, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11 yields max{dKL(µy,† θ,δ∥µy,J θ,δ), dKL(µy,J θ,δ∥µy,† θ,δ)} ≤2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,†∥L1 µθ⊗µδ � ∥Φy,† − Φy,J∥L1 µθ⊗µδ ≤21/2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,†∥L1 µθ⊗µδ �� ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,†∥1/2 L1 µθ⊗µδ � × ∥∥O ◦ (δ† − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using the definition of C given in the statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='12 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of error of joint parameter-error posterior with respect to lifted approximate posterior In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13, one assumes that Φy,A as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a) belongs to L1 µθ, and also that Φy,J ∈ L1 µθ⊗µδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The resulting bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='16) is ∥Φy,A − Φy,J∥L1 µθ⊗µδ ≤ 2−1/2∥∥O ◦ δ(θ)∥2 Σ−1 ε ∥1/2 L1 P � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13 is very similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 23 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let f ← y −O ◦M(θ), g ← O ◦(−δ)(θ), M1 ← Σε, M2 ← 0, and µ ← P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then ∥f + g∥2 M−1 1 = 2Φy,J(θ, δ) and ∥f∥2 M−1 1 = 2Φy,A(θ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Analogously to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14), we have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10 that ∥∥y − O ◦ M(θ)∥2 Σ−1 ε ∥1/2 L1 P = √ 2∥Φy,A∥1/2 L1µθ = √ 2∥Φy,A∥1/2 L1 µθ⊗µδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15) Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 with the choices above yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='16): 2∥Φy,A − Φy,J∥L1 µθ⊗µδ ≤∥∥O ◦ (−δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P √ 2 � ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,A∥1/2 L1µθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Next, ∥∥O ◦ (−δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P ≤ √ 2 � ∥Φy,A∥1/2 L1µθ + ∥Φy,J∥1/2 L1 µθ⊗µδ � follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,A are nonnegative, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13 yields max{dKL(µy,A θ,δ ∥µy,J θ,δ), dKL(µy,J θ,δ∥µy,A θ,δ )} ≤2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,A∥L1 µθ⊗µδ � ∥Φy,A − Φy,J∥L1 µθ⊗µδ ≤21/2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,A∥L1 µθ⊗µδ �� ∥Φy,J∥1/2 L1 µθ⊗µδ + ∥Φy,A∥1/2 L1 µθ⊗µδ � × ∥∥O ◦ (−δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using the definition of C given in the statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='15) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of error of joint parameter-error posterior with respect to lifted enhanced noise posterior For the sake of completeness, we compare the joint posterior with the lifted enhanced noise posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,E be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11a) with • = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' If Φy,E ∈ L1 µθ and Φy,J ∈ L1 µθ⊗µδ, then ∥Φy,E − Φy,J∥L1 µθ⊗µδ ≤2−1/2∥∥O ◦ (δ(θ) − mu)∥2 Σ−1 ε ∥1/2 L1 P � CE∥Φy,E∥1/2 L1µθ + ∥Φy,J∥1/2 L1 µθ⊗µδ � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='16) + 2−1∥∥y − O ◦ M(θ) − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Furthermore, ∥∥O ◦ (δ(θ) − mu)∥2 Σ−1 ε ∥1/2 L1 P ≤ √ 2 � CE∥Φy,E∥1/2 L1µθ + ∥Φy,J∥1/2 L1 µθ⊗µδ � ∥∥y − O ◦ M(θ) − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1 P ≤ 2(CE + 1)∥Φy,E∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let f ← y − O ◦ (M(θ) + δ(θ)), g ← O ◦ (δ(θ) − mu), M1 ← Σε, M2 ← OΣuO∗, and µ ← P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then ∥f∥2 M−1 1 = 2Φy,J(θ, δ) and ∥f + g∥2 (M1+M2)−1 = 2Φy,E(θ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Analogously to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='14), we have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10 that ∥∥y − O ◦ M(θ) − Omu∥2 (Σε+OΣuO∗)−1∥1/2 L1 P = √ 2∥Φy,E∥1/2 L1µθ = √ 2∥Φy,E∥1/2 L1 µθ⊗µδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='17) 24 Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3) of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2 with the choices above yields (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='16): 2∥Φy,E − Φy,J∥L1 µθ⊗µδ ≤∥∥O ◦ (δ(θ) − mu)∥2 Σ−1 ε ∥1/2 L1 P � CE∥2Φy,E∥1/2 L1µθ + ∥2Φy,J∥1/2 L1 µθ⊗µδ � + ∥∥y − O ◦ M(θ) − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Next, we apply (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='5) from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='2, and use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='17): ∥∥O ◦ (δ(θ) − mu)∥2 Σ−1 ε ∥1/2 L1 P ≤ √ 2 � CE∥Φy,E∥1/2 L1µθ + ∥Φy,J∥1/2 L1 µθ⊗µδ � ∥∥y − O ◦ M(θ) − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1 P ≤ 2(CE + 1)∥Φy,E∥L1µθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Let Φy,E and Φy,J be as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3, and let µy,E θ,δ be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='11b) with = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Then max{dKL(µy,E θ,δ ∥µy,J θ,δ), dKL(µy,J θ,δ∥µy,E θ,δ )} ≤ C∥∥O ◦ (mu − δ)(θ)∥2 Σ−1 ε ∥1/2 L1 P , where C = exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,E∥L1µθ � max{21/2� CE∥Φy,E∥1/2 L1 µθ⊗µδ + ∥Φy,J∥1/2 L1 µθ⊗µδ � , 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,E are nonnegative, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='7a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='1 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='3 yields max{dKL(µy,E θ,δ ∥µy,J θ,δ), dKL(µy,J θ,δ∥µy,E θ,δ )} ≤2 exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,E∥L1 µθ⊗µδ � ∥Φy,E − Φy,J∥L1 µθ⊗µδ ≤ exp � 2∥Φy,J∥L1 µθ⊗µδ + 2∥Φy,E∥L1 µθ⊗µδ � max{21/2� CE∥Φy,E∥1/2 L1 µθ⊗µδ + ∥Φy,J∥1/2 L1 µθ⊗µδ � , 1} × � ∥∥O ◦ (mu − δ)(θ)∥2 Σ−1 ε ∥L1 P + ∥∥y − O ◦ M(θ) − Omu∥2 Σ−1 ε −(Σε+OΣuO∗)−1∥L1 P � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Using the definition of C given in the statement of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='4 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='13) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='10 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' References [1] Assyr Abdulle and Giacomo Garegnani, Random time step probabilistic methods for un- certainty quantification in chaotic and geometric numerical integration, Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 30 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 4, 907–932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [2] Alen Alexanderian, Ruanui Nicholson, and No´emi Petra, Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty, 2022, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='03952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [3] Alen Alexanderian, No´emi Petra, Georg Stadler, and Isaac Sunseri, Optimal design of large-scale Bayesian linear inverse problems under reducible model uncertainty: Good to know what you don’t know, SIAM/ASA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Uncertainty Quantif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 9 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 1, 163–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [4] Jenn´y Brynjarsd´ottir and Anthony O’Hagan, Learning about physical parameters: the im- portance of model discrepancy, Inverse Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 30 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 11, 114007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 25 [5] Daniela Calvetti, Matthew Dunlop, Erkki Somersalo, and Andrew Stuart, Iterative updat- ing of model error for Bayesian inversion, Inverse Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 34 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 2, 025008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [6] Lianghao Cao, Thomas O’Leary-Roseberry, Prashant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Jha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Tinsley Oden, and Omar Ghattas, Residual-based error correction for neural operator accelerated infinite- dimensional Bayesian inverse problems, 2022, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='03008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [7] Feng Cheng Chang, Inversion of a perturbed matrix, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 19 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 2, 169–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [8] Duc-Lam Duong, Tapio Helin, and Jose Rodrigo Rojo-Garcia, Stability estimates for the expected utility in Bayesian optimal experimental design, 2022, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='04399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [9] Subhashis Ghosal and Aad van der Vaart, Fundamentals of Nonparametric Bayesian In- ference, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 44, Cambridge: Cambridge University Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [10] Konstantinos Gourgoulias, Markos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Katsoulakis, Luc Rey-Bellet, and Jie Wang, How biased is your model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Concentration inequalities, information and model bias, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Theory 66 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 5, 3079–3097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [11] Eric Joseph Hall and Markos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Katsoulakis, Robust information divergences for model- form uncertainty arising from sparse data in random PDE, SIAM/ASA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Quan- tif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 6 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 4, 1364–1394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [12] Jari Kaipio and Erkki Somersalo, Statistical and computational inverse problems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 160, Springer Science & Business Media, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [13] , Statistical inverse problems: discretization, model reduction and inverse crimes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 198 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 2, 493–504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [14] Marc C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kennedy and Anthony O’Hagan, Bayesian calibration of computer models, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=', Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' B, Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 63 (2001), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 3, 425–464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [15] Ville Kolehmainen, Tanja Tarvainen, Simon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Arridge, and Jari P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kaipio, Marginalization of uninteresting distributed parameters in inverse problems – application to diffuse optical tomography, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Quantif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 1 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 1, 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [16] Karina Koval, Alen Alexanderian, and Georg Stadler, Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs, Inverse Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 36 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 7, 075007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [17] Yvon Maday and Tommaso Taddei, Adaptive PBDW approach to state estimation: noisy observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' user-defined update spaces, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 41 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 4, b669– b693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [18] Ruanui Nicholson, No´emi Petra, and Jari P Kaipio, Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field, Inverse Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 34 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 11, 115005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Nissinen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Heikkinen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kolehmainen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kaipio, Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography, Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 20 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 10, 105504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 26 [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Sargsyan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Najm, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Ghanem, On the statistical calibration of physical models, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Kinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 47 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 4, 246–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [21] Khachik Sargsyan, Xun Huan, and Habib N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Najm, Embedded model error representation for Bayesian model calibration, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Quantif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 9 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 4, 365–394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [22] Andrea Scarinci, Michael Fehler, and Youssef Marzouk, Bayesian inference under model misspecification using transport-Lagrangian distances: an application to seismic inversion, 2021, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content='07027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [23] Bj¨orn Sprungk, On the local Lipschitz stability of Bayesian inverse problems, Inverse Probl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 36 (2020), 055015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [24] Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Stuart, Inverse problems: A Bayesian perspective, Acta Numerica 19 (2010), 451–559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [25] Alexandre B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Tsybakov, Introduction to nonparametric estimation, Springer series in statis- tics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 160, Springer Science & Business Media, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' [26] Martin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' Wainwright, High-dimensional statistics, Cambridge Series in Statistical and Probabilistic Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 48, Cambridge University Press, Cambridge, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfCwtu/content/2301.04863v1.pdf'} diff --git a/8tAzT4oBgHgl3EQf-v68/content/tmp_files/2301.01939v1.pdf.txt b/8tAzT4oBgHgl3EQf-v68/content/tmp_files/2301.01939v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2517be256d1db0ff0f60b2edf61b6fcceac97eec --- /dev/null +++ b/8tAzT4oBgHgl3EQf-v68/content/tmp_files/2301.01939v1.pdf.txt @@ -0,0 +1,1976 @@ +Thomsen-type parameters and attenuation coefficients for constant-Q +transverse isotropy +Qi Haoa,∗, Ilya Tsvankinb +aCollege of Geoexploration Science and Technology, Jilin University, Changchun, 130026, P. R. China +bDepartment of Geophysics, Colorado School of Mines, Golden, 80401, USA +Abstract +Transversely isotropic (TI) media with the frequency-independent quality-factor elements (also called “constant-Q” +transverse isotropy) are often used to describe attenuation anisotropy in sedimentary rocks. The attenuation coef- +ficients in constant-Q TI models can be conveniently defined in terms of the Thomsen-type attenuation-anisotropy +parameters. +Recent research indicates that not all those parameters for such constant-Q media are frequency- +independent. Here, we present concise analytic formulae for the Thomsen-type attenuation parameters for Kjar- +tansson’s constant-Q TI model and show that one of them (δQ) varies with frequency. The analytic expression +for δQ helps evaluate the frequency dependence of the normalized attenuation coefficients of P- and SV-waves by +introducing the newly defined “dispersion factors”. Viscoacoustic constant-Q transverse isotropy is also discussed as +a special case, for which the elliptical condition and simplified expressions for the parameters δ and δQ are derived. +Our results show that in the presence of significant absorption the attenuation coefficients of the “constant-Q” +model vary with frequency for oblique propagation with respect to the symmetry axis. This variation needs to be +taken into account when applying the spectral-ratio method and other attenuation-analysis techniques. +Keywords: +seismic, attenuation, anisotropy, wave, Q +1. Introduction +The frequency-independent quality factor (called “constant-Q” for brevity) provides a useful phenomenologi- +cal description of seismic attenuation in rocks and is widely used in seismic attenuation analysis. Among such +constant-Q dissipative models are those proposed by [1] and [2]. For isotropic media, the Kjartansson model pro- +duces the constant-Q factor for all frequencies, whereas the Kolsky model leads to nearly constant Q-values. The +complex moduli for the Kolsky model represent the first-order Maclaurin series expansion with respect to 1/Q of +the corresponding moduli for the Kjartansson model [3, 4]. +As an extension of non-dissipative transverse isotropy, the constant-Q TI model can be used to process seismic +attenuation data for most sedimentary rocks, such as shale formations. +The constant-Q assumption facilitates +estimation of the quality factor and attenuation anisotropy [e.g., 5, 6, 7, 8]. Ultrasonic measurements demonstrate +that attenuation anisotropy generally is stronger than velocity anisotropy for rock samples [9, 10, 11]. +Velocity anisotropy for TI media can be efficiently described by the Thomsen anisotropy parameters [12, 13]. +Likewise, attenuation anisotropy for dissipative TI media is convenient to study using the Thomsen-type nota- +tion introduced by [14]. The combination of the velocity- and attenuation-related Thomsen-type parameters [14] +completely defines the complex stiffness matrix at a specified frequency for a general dissipative TI model with +a vertical symmetry axis (VTI medium). The generic Thomsen velocity parameters depend on the real parts of +the stiffness coefficients (cij), whereas the Thomsen-type attenuation parameters are defined by both the real and +imaginary parts of cij. [15] found that some Thomsen-type parameters in constant-Q VTI media are frequency +∗Corresponding author +Email addresses: xqi.hao@gmail.com (Qi Hao), ilya@mines.edu (Ilya Tsvankin) +January 6, 2023 +arXiv:2301.01939v1 [physics.geo-ph] 5 Jan 2023 + +dependent. This phenomenon is not entirely surprising, because all stiffness coefficients of constant-Q TI media, +which are involved in the definition of the Thomsen-type parameters, are functions of frequency. Investigating +the frequency variations of these parameters can facilitate understanding of such key signatures in TI media as +velocities, traveltimes, attenuation coefficients, and polarization vectors. However, to our knowledge, there are no +analytic expressions for the frequency-dependent Thomsen-type attenuation parameters in constant-Q dissipative +VTI media. +Here, we derive analytic formulae for the Thomsen-type parameters using Kjartansson’s constant-Q VTI model. +A set of the corresponding reference parameters is defined at a specified frequency and used to obtain those +parameters for the entire frequency range. We also present a formula for the frequency-dependent anellipticity +and define the condition for elliptical anisotropy in constant-Q TI media. The newly proposed formulae for the +Thomsen-type parameters allow us to study the normalized plane-wave attenuation coefficients in constant-Q media +with weak attenuation anisotropy and define the “dispersion factors” for P- and SV-waves. Numerical examples +are used to analyze the accuracy of the obtained expressions for the Thomsen-type parameters, the validity of the +elliptical condition, and the frequency dependence of the attenuation coefficients. +2. Thomsen-type parameters of constant-Q VTI media +2.1. Constant-Q dissipative VTI model +Referring to [14] and [16], the complex stiffness (or modulus) matrix M for viscoelastic VTI media is given by: +M = +� +� +� +� +� +� +� +� +M11 +M11 − 2M66 +M13 +0 +0 +0 +M11 − 2M66 +M11 +M13 +0 +0 +0 +M13 +M13 +M33 +0 +0 +0 +0 +0 +0 +M55 +0 +0 +0 +0 +0 +0 +M55 +0 +0 +0 +0 +0 +0 +M66 +� +� +� +� +� +� +� +� +, +(1) +where Mij = M R +ij − i sgn(f)M I +ij denote the complex stiffness coefficients for the frequency f, and the minus sign in +front of i sgn(f)M I +ij follows from the definition of the Fourier transform in [17] and [3]. Both the real and imaginary +parts of Mij generally are frequency dependent. +For the [2] model (also called the constant-Q model), the nonzero independent elements in equation 1 are +expressed as: +Mij = +˜ +M R +ij +cos(πγij) +� +−i f +f0 +�2γij +, +(2) +with +γij = 1 +π tan−1 +� 1 +Qij +� +, +(3) +where Qij ≡ M R +ij /M I +ij, f0 is the reference frequency, and ˜ +M R +ij denote the real parts of Mij at f0: ˜ +M R +ij = Re(Mij)|f=f0. +By design, the quality-factor elements Qij for the Kjartansson model are independent of frequency. As follows from +equation 2, the complex stiffness coefficients Mij for a given frequency can be expressed in terms of ˜ +M R +ij and Qij. +2.2. Thomsen-type parameterization +[14] and [18] show that dissipative VTI media can be conveniently parameterized by the Thomsen-type attenu- +ation parameters. The [12] velocity parameters [see 13] are defined in the nonattenuative reference VTI medium. +The parameter VP 0 is the vertical velocity of P-waves: +VP 0 ≡ +� +M R +33 +ρ , +(4) +where ρ denotes density. +2 + +The parameter VS0 is the vertical velocity of S-waves: +VS0 ≡ +� +M R +55 +ρ . +(5) +The parameter ϵ is approximately equal to the fractional difference between the horizontal and vertical velocities +of P-waves: +ϵ ≡ M R +33 − M R +11 +2M R +33 +. +(6) +The parameter δ determines the second derivative of the P-wave phase velocity at vertical incidence and is given +by: +δ ≡ +� +M R +13 + M R +55 +�2 − +� +M R +33 − M R +55 +�2 +2M R +33(M R +33 − M R +55) +. +(7) +The parameter γ is approximately equal to the fractional difference between the horizontal and vertical velocities +of SH-waves: +γ ≡ M R +66 − M R +55 +2M R +55 +. +(8) +The Thomsen-type attenuation parameters [14] can be used to define the normalized phase attenuation coefficient +A ≡ |kI|/|kR| for P-, SV-, and SH-waves, which is generally supposed to be frequency-independent in constant-Q +models. For more details about A, see the section “Plane-wave attenuation in constant-Q VTI media” below. +The parameter AP 0 is the vertical attenuation coefficient of P-waves: +AP 0 ≡ Q33 +�� +1 + +1 +Q2 +33 +− 1 +� +≈ +1 +2Q33 +. +(9) +The parameter AS0 is the vertical attenuation coefficient of S-waves: +AS0 ≡ Q55 +�� +1 + +1 +Q2 +55 +− 1 +� +≈ +1 +2Q55 +. +(10) +The parameter ϵQ is close to the fractional difference between the horizontal and vertical attenuation coefficients +of P-waves: +ϵQ ≡ Q33 − Q11 +Q11 +. +(11) +The parameter δQ controls the second derivative of the P-wave attenuation coefficient at vertical incidence and +is expressed as [14]: +δQ ≡ +Q33−Q55 +Q55 +M R +55 +(M R +13+M R +33) +2 +M R +33−M R +55 ++ 2 Q33−Q13 +Q13 +M R +13(M R +13 + M R +55) +M R +33(M R +33 − M R +55) +. +(12) +The parameter γQ is close to the fractional difference between the horizontal and vertical attenuation coefficients +of SH-waves: +γQ ≡ Q55 − Q66 +Q66 +. +(13) +3 + +3. Analytic description of Thomsen-type parameters +In this section, we represent the Thomsen velocity parameters and Thomsen-type attenuation parameters in +terms of their reference values defined at f = f0: +˜VP 0 = VP 0|f=f0, ˜VS0 = VS0|f=f0, ˜ϵ = ϵ|f=f0, ˜δ = δ|f=f0, +˜ +AP 0 = AP 0|f=f0, +˜ +AS0 = AS0|f=f0, ˜ϵQ = ϵQ|f=f0, and ˜δQ = δQ|f=f0. These parameters are used to find the +real parts of the reference stiffness coefficients ( ˜ +M R +ij ), the quality-factor elements Qij (see Appendix A), and the +frequency-dependent stiffness matrix M. +According to equations 4–7 and 12, the Thomsen-type parameters involve the coefficients M R +ij , where ij=11, 13, +33, 55 and 66. Using equations 2 and 3, M R +ij are approximately expressed as: +M R +ij ≈ ˜ +M R +ij +� +1 + 2 +π Q−1 +ij ln +���� +f +f0 +���� + 2 +π2 Q−2 +ij ln2 +���� +f +f0 +���� +� +, +(14) +where we use the approximation tan−1(Q−1 +ij ) ≈ Q−1 +ij because typically Qij ≫ 1. +Substitution of equation 14 into equations 4–7 and 12 allows us to derive approximate expressions for the +frequency-dependent Thomsen-type parameters, which are discussed in the following two subsections. +3.1. Velocity parameters +The second-order approximations for the Thomsen velocity parameters with respect to ln|f/f0| are given by: +VP 0 = ˜VP 0 +� +1 + 1 +π Q−1 +33 ln +���� +f +f0 +���� + +1 +2π2 Q−2 +33 ln2 +���� +f +f0 +���� +� +, +(15) +VS0 = ˜VS0 +� +1 + 1 +π Q−1 +55 ln +���� +f +f0 +���� + +1 +2π2 Q−2 +55 ln2 +���� +f +f0 +���� +� +, +(16) +ϵ = ˜ϵ + 1 +π (1 + 2˜ϵ) Q−1 +33 ˜ϵQ ln +���� +f +f0 +���� + 1 +π2 (1 + 2˜ϵ) Q−2 +33 ˜ϵ2 +Q ln2 +���� +f +f0 +���� , +(17) +δ = ˜δ + 1 +π Q−1 +33 ˜δQ ln +���� +f +f0 +���� + 1 +π2 Q−2 +33 ζQ ln2 +���� +f +f0 +���� , +(18) +γ = ˜γ + 1 +π (1 + 2˜γ) Q−1 +55 ˜γQ ln +���� +f +f0 +���� + 1 +π2 (1 + 2˜γ) Q−2 +55 ˜γ2 +Q ln2 +���� +f +f0 +���� , +(19) +where the P- and S-wave inverse vertical quality factors Q33 and Q55 (respectively) are: +Q−1 +33 = +˜ +AP 0 +2(1 − ˜ +A2 +P 0) +, +(20) +Q−1 +55 = +˜ +AS0 +2(1 − ˜ +A2 +S0) +. +(21) +The coefficient ζQ in equation 18 is defined as: +ζQ = d0 (1 − gQ)2 + d1 (1 − gQ) ˜δQ + d2 ˜δ2 +Q , +(22) +with +gQ ≡ Q33 +Q55 +, +(23) +4 + +d0 = +g(1 − g + χ)2 � +(1 + 2˜δ) χ − (1 + 2˜δ) g + (1 + ˜δ) g2� +(1 − g)2(χ − g)χ2 +, +(24) +d1 = +2g +� +1 + 2˜δ + χ − (2 + ˜δ + χ) g + g2� +(χ − g)χ2 +, +(25) +d2 = +2χ − g +2(1 + 2˜δ − g)(χ − g) +; +(26) +g ≡ +˜V 2 +S0 +˜V 2 +P 0 +, +(27) +χ = +� +(1 − g)(1 + 2˜δ − g) . +(28) +In equations 15–19, the first-order terms with respect to ln|f/f0| are scaled by Q−1 +33 or Q−1 +55 , whereas the second- +order terms by Q−2 +33 or Q−2 +55 . Because Q33 and Q55 typically are much greater than unity, the frequency dependence +of the velocity parameters is mostly determined by the first-order terms. Equations 15–19 indicate that: (1) VP 0 +and VS0 always monotonically increase with frequency; (2) ϵ, δ and γ also monotonically increase with f, if ˜ϵQ > 0, +˜δQ > 0, and ˜γQ > 0, respectively. Overall, the frequency dependence of VP 0, VS0, ϵ, δ, and γ for realistic values of +Q33 and Q55 (Q33 ≫ 1 and Q55 ≫ 1) remains weak, as illustrated by the numerical examples below. +Note that phase and group velocities in strongly dissipative TI media are influenced by attenuation and do +not represent the same functions of the Thomsen parameters as in purely elastic models [14, 13]. For sedimentary +formations, both gQ and g vary within a limited range. +In particular, according to [9], for relatively shallow +sedimentary rocks 0.5 < gQ ≤ 3 (Figure 1). +gQ= 1 +2 +gQ=1 +gQ=2 +gQ=3 +0 +20 +40 +60 +80 +100 +0 +50 +100 +150 +Q55 +Q33 +Figure 1: Vertical quality factors Q33 and Q55 in dissipative VTI rocks. The black dots are the data from Table 3 +of [9]; gQ ≡ Q33/Q55. +Using equations 17 and 18 for ϵ and δ, the anellipticity parameter η [19] can be approximately obtained as: +η ≡ ϵ − δ +1 + 2δ = η0 + η1 Q−1 +33 ln +���� +f +f0 +���� + η2 Q−2 +33 ln2 +���� +f +f0 +����, +(29) +5 + +where Q33 is given by equation 20, and +η0 = ˜η = ˜ϵ − ˜δ +1 + 2˜δ +, +(30) +η1 = +1 + 2˜ϵ +(1 + 2˜δ)2 +� +(1 + 2˜δ)˜ϵQ − ˜δQ +� +, +(31) +η2 = 1 + 2˜ϵ +1 + 2˜δ +� +r0 + +r1 +1 + 2˜δ ++ +r2 +(1 + 2˜δ)2 +� +, +(32) +with +r0 = ˜ϵ2 +Q, +(33) +r1 = −ζQ − 2˜ϵQ ˜δQ, +(34) +r2 = 2˜δ2 +Q. +(35) +The parameter η controls (along with the zero-dip normal-moveout velocity) all P-wave time-domain signatures for +laterally homogeneous VTI media above a horizontal or dipping target reflector [19, 13]. +3.2. Attenuation parameters +The following Thomsen-type attenuation parameters are expressed directly through the elements Qij and, there- +fore, are frequency-independent in constant-Q VTI media: +AP 0 = ˜ +AP 0, +(36) +AS0 = ˜ +AS0, +(37) +ϵQ = ˜ϵQ, +(38) +γQ = ˜γQ. +(39) +The attenuation parameter δQ, however, also depends on the coefficients M R +ij (equation 12), which vary with +frequency. The second-order approximation for δQ with respect to ln |f/f0| is: +δQ = ˜δQ + 2 +π Q−1 +33 ζQ ln +���� +f +f0 +���� + 2 +π2 Q−2 +33 ξQ ln2 +���� +f +f0 +����, +(40) +where ζQ is defined in equation 22, and +ξQ = s0(1 − gQ)3 + s1(1 − gQ)2 ˜δQ + s2(1 − gQ) ˜δ2 +Q + s3 ˜δ3 +Q. +(41) +The explicit expressions for the coefficients si are given in Appendix B. +Because for Q33 ≫ 1 the influence of the second-order term in equation 40 is insignificant, the frequency variation +of δQ is largely controlled by the coefficient ζQ. For ζQ > 0, δQ monotonically increases with frequency. As follows +from equations 22 and 24–28, ζQ is a function of g (equation 27), gQ (equation 23), ˜δ, and ˜δQ. +3.3. Numerical analysis +Here, we analyze the above expressions for the Thomsen-type parameters numerically. The reference frequency +is set as f0 = 40 Hz and the frequency range as [1, 200] Hz for all examples below. +First, we test the accuracy of the equations 15 and 16 for the vertical velocities and their first-order versions +(i.e., those without the second-order term with respect to ln|f/f0|). As demonstrated by Figure 2, the first-order +approximations for VP 0 and VS0 are sufficiently accurate even for strong attenuation in a wide frequency range. +Overall, the frequency dependence of the vertical velocities is almost negligible, except for very low frequencies. +6 + +exact +1st +2nd +0 +50 +100 +150 +200 +2.92 +2.94 +2.96 +2.98 +3.00 +3.02 +3.04 +f (Hz) +vP0 (km/s) +(a) +exact +1st +2nd +0 +50 +100 +150 +200 +1.42 +1.44 +1.46 +1.48 +1.50 +1.52 +1.54 +f (Hz) +vS0 (km/s) +(b) +Figure 2: Frequency-dependent vertical velocities (a) VP 0 and (b) VS0. “Exact” in the legend refers to the exact +values, whereas “1st” and “2nd” denote the first- and second-order approximations with respect to ln |f/f0|, re- +spectively. On plot (a), ˜VP 0 = 3.0 km/s and Q33 = 40 ( ˜ +AP 0 = 0.0125); on plot (b), ˜VS0 = 1.5 km/s and Q55 = 20 +( ˜ +AS0 = 0.025). +Table 1: Medium parameters for two constant-Q VTI models at the reference frequency f0 = 40 Hz. +[H] +Model +˜VP 0 +˜VS0 +˜ϵ +˜δ +˜γ +˜ +AP 0 (Q33) +˜ +AS0 (Q55) +˜ϵQ +˜δQ +˜γQ +1 +3.0 +1.5 +0.3 +-0.1 +0.1 +0.0125 (40) +0.0167 (30) +-0.3 +-1.91 +0.5 +2 +3.0 +1.5 +0.3 +-0.1 +0.2 +0.0250 (20) +0.0333 (15) +0.3 +0.98 +-0.2 +Figures 3, 4 and 5 show that the first-order versions of equations 17–19 can accurately describe the variations of +the anisotropy parameters ϵ, δ, and γ with frequency. Comparison of Figures 3, 4, and 5 confirms that the reference +parameters ˜ϵQ, ˜δQ, and ˜γQ govern the frequency dependence of ϵ, δ, and γ. For example, if ˜ϵQ > 0, ϵ increases +with frequency. As is the case for VP 0 and VS0, the anisotropy coefficients vary with frequency primarily in the +low-frequency range. +exact +1st +2nd +0 +50 +100 +150 +200 +0.295 +0.300 +0.305 +0.310 +0.315 +f (Hz) +ϵ +(a) +exact +1st +2nd +0 +50 +100 +150 +200 +0.27 +0.28 +0.29 +0.30 +0.31 +f (Hz) +ϵ +(b) +Figure 3: Variation of the Thomsen parameter ϵ with frequency for (a) Model 1 and (b) Model 2 from Table 1. +The legend is the same as in Figure 2. +Next, we investigate the only frequency-dependent attenuation-anisotropy parameter, δQ, by comparing the +exact equation for δQ with its first- and second-order approximations. The first-order equation accurately models +δQ in a wide frequency range, whereas contribution of the second-order term is practically negligible (Figure 6). +As mentioned above, the coefficient ζQ in equation 40 is largely responsible for the frequency variation of δQ for +7 + +exact +1st +2nd +0 +50 +100 +150 +200 +-0.12 +-0.11 +-0.10 +-0.09 +-0.08 +-0.07 +f (Hz) +δ +(a) +exact +1st +2nd +0 +50 +100 +150 +200 +-0.16 +-0.14 +-0.12 +-0.10 +-0.08 +f (Hz) +δ +(b) +Figure 4: Same as Figure 3 but for the parameter δ. +exact +1st +2nd +0 +50 +100 +150 +200 +0.075 +0.080 +0.085 +0.090 +0.095 +0.100 +0.105 +0.110 +f (Hz) +γ +(a) +exact +1st +2nd +0 +50 +100 +150 +200 +0.190 +0.195 +0.200 +0.205 +0.210 +0.215 +0.220 +f (Hz) +γ +(b) +Figure 5: Same as Figure 3 but for the parameter γ. +exact +1st +2nd +0 +50 +100 +150 +200 +-2.3 +-2.2 +-2.1 +-2.0 +-1.9 +-1.8 +-1.7 +-1.6 +f (Hz) +δQ +(a) +exact +1st +2nd +0 +50 +100 +150 +200 +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +f (Hz) +δQ +(b) +Figure 6: Frequency-dependent Thomsen-type attenuation parameter δQ for (a) Model 1 and (b) Model 2 from +Table 1. The legend is the same as in Figure 2. +8 + +a specified value of Q33. Equation 22 shows that ζQ is a function of the parameters g = ˜V 2 +S0/ ˜V 2 +P 0, gQ = Q−1 +55 /Q−1 +33 , +˜δ and ˜δQ. Using the results from Figure 1, we restrict gQ to the range 0.5 ≤ gQ ≤ 3. Figures 7 and 8 show that +the smallest absolute value of ζQ corresponds to gQ = 1, and |ζQ| increases with the deviation of gQ from unity. +As a result, the parameter δQ is almost independent of frequency for gQ = 1 (Figure 9). Overall, the frequency +dependence of δQ becomes noticeable for large |gQ − 1| (e.g., gQ = 3; Figure 9), but it is also influenced by the +parameters ˜δ and ˜δQ. For the most common values of gQ considered here, the parameter δQ significantly varies +with f only for low frequencies. +-0.1 0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +δ˜ +δ˜ +Q +ζQ +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +(a) +-0.1 0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +δ˜ +δ˜ +Q +ζQ +2 +4 +6 +8 +10 +(b) +-0.1 0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +δ˜ +δ˜ +Q +ζQ +0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(c) +-0.1 0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +δ˜ +δ˜ +Q +ζQ +0 +1 +2 +3 +4 +5 +6 +(d) +Figure 7: Contour plots of the coefficient ζQ as a function of ˜δ and ˜δQ. The parameter g = ˜V 2 +S0/ ˜V 2 +P 0 = 0.3. The +parameter gQ = Q−1 +55 / ˜Q−1 +P 0 is defined as (a) gQ = 3, (b) gQ = 2, (c) gQ = 1, and (d) gQ = 0.5. +9 + +g=0 +g=0.1 +g=0.3 +g=0.5 +0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 +-10 +-5 +0 +5 +10 +15 +gQ +ζQ +(a) +g=0 +g=0.1 +g=0.3 +g=0.5 +0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 +-6 +-4 +-2 +0 +2 +4 +6 +8 +gQ +ζQ +(b) +g=0 +g=0.1 +g=0.3 +g=0.5 +0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 +0 +10 +20 +30 +40 +50 +gQ +ζQ +(c) +g=0 +g=0.1 +g=0.3 +g=0.5 +0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 +0 +10 +20 +30 +gQ +ζQ +(d) +Figure 8: Variation of the coefficient ζQ with gQ for different values of g. (a) ˜δ = −0.2 and ˜δQ = −0.6; (b) ˜δ = −0.2 +and ˜δQ = 0; (c) ˜δ = 0.2 and ˜δQ = −0.4; (d) ˜δ = 0.2 and ˜δQ = 0.98. +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +f (Hz) +δQ +(a) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +f (Hz) +δQ +(b) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +f (Hz) +δQ +(c) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +f (Hz) +δQ +(d) +Figure 9: Variation of the attenuation parameter δQ with frequency for different gQ and g = 0.3. The parameters +˜δ and ˜δQ are the same as in Figure 8. +10 + +4. Viscoacoustic constant-Q transverse isotropy +4.1. Simplified parameter expressions +Next, we consider the so-called “viscoacoustic” constant-Q media described by the Thomsen-type notation. The +acoustic approximation is implemented by setting ˜VS0 = AS0 = 0 in equations 18, 40, and 29 [20, 21, 22] . The +parameters δ, η, and δQ then reduce to: +δ = ˜δ + 1 +π Q−1 +33 ˜δQ ln +���� +f +f0 +���� + 1 +π2 Q−2 +33 +˜δ2 +Q +1 + 2˜δ +ln2 +���� +f +f0 +����, +(42) +η = η0 + η1 Q−1 +33 ln +���� +f +f0 +���� + +� +˜ϵQ − +˜δQ +1 + 2˜δ +� +η1 Q−2 +33 ln2 +���� +f +f0 +����, +(43) +δQ = ˜δQ + 2 +π Q−1 +33 +˜δ2 +Q +1 + 2˜δ +ln +���� +f +f0 +���� + 2 +π2 Q−2 +33 +˜δ3 +Q +(1 + 2˜δ)2 ln2 +���� +f +f0 +����. +(44) +Setting η (equation 43) to zero, which requires η0 = η1 = 0 (see equations 30 and 31), we obtain the elliptical +conditions: +˜ϵ = ˜δ, +(45) +˜ϵQ = +˜δQ +1 + 2˜δ +. +(46) +Equations 45 and 46 make the parameters of viscoacoustic constant-Q media satisfy the same conditions at all +frequencies: +ϵ = δ, +(47) +ϵQ = +δQ +1 + 2δ , +(48) +which follows from equations 17, 31, 42, and 44. Equation 47 implies that the elliptical conditions at the reference +frequency ensure that η = 0 at all frequencies. +For viscoelastic constant-Q media discussed earlier, equation 47 remains approximately valid (i.e., the model is +elliptical at all frequencies), if equations 45 and 46 are satisfied (see equations 29–31). +4.2. Numerical validation +Here, we verify the elliptical conditions (equations 45 and 46) by computing the anellipticity parameter η. The +exact η is calculated using equations 6, 7, 11 and 12 along with equations 2 and 3 under the acoustic approximation +( ˜VS0 = 0 and Q−1 +55 = 0). The first-order approximation for η is given by equation 43 without the second-order term +with respect to ln|f/f0|. +Figure 10 shows that for models that satisfy equations 45 and 46 the exact anellipticity parameter is negligibly +small for all frequencies (on the order of 10−7 for both models), which confirms that the elliptical conditions at the +reference frequency lead to equation 47. In addition, our testing confirms that the difference between the left and +right sides of equation 48 is negligible, if equations 45 and 46 are satisfied. +11 + +exact +1st +2nd +0 +50 +100 +150 +200 +-4.×10-7 +-2.×10-7 +0 +2.×10-7 +4.×10-7 +f (Hz) +η +(a) +exact +1st +2nd +0 +50 +100 +150 +200 +-5.×10-7 +0 +5.×10-7 +1.×10-6 +f (Hz) +η +(b) +Figure 10: Variation of the anellipticity parameter η with frequency under the elliptical conditions (equations 45 +and 46). The P-wave quality factor and reference vertical velocity at f0 = 40 Hz are Q33 = 40 and ˜VP 0 = 3 km/s. +The parameters ˜ϵ and ˜ϵQ are (a) ˜ϵ = 0.3 and ˜ϵQ = −0.33; (b) ˜ϵ = 0.2 and ˜ϵQ = 0.4. +5. Plane-wave attenuation in constant-Q VTI media +In this section, we apply the obtained expressions for the Thomsen-type parameters to study the normalized +plane-wave attenuation coefficients in constant-Q VTI media. +The normalized phase attenuation coefficient is +defined as A ≡ |kI|/|kR|, where kR and kI denote the real and imaginary parts of the complex wave vector [14]. +The words “phase” and “normalized” are omitted below for brevity. The angle between kR and kI is called the +“inhomogeneity” angle, which is not defined in plane-wave propagation (i.e., it is a free parameter that can vary +within certain bounds). The coefficient A corresponding to kR ∥ kI is approximately equal to the group attenuation +coefficient, which can be estimated from seismic data, for a wide range of “inhomogeneity” angles [23, 18]. +5.1. Attenuation coefficients +[14] and [18] show that the approximate attenuation coefficients of plane waves in viscoelastic constant-Q VTI +media are given by: +AP = AP 0 (1 + δQ sin2 θ cos2 θ + ϵQ sin4 θ), +(49) +ASV = AS0 (1 + σQ sin2 θ cos2 θ), +(50) +ASH = AS0 (1 + γQ sin2 θ), +(51) +where the subscripts P, SV, and SH denote the wave types, and θ is the phase angle measured from the vertical. +The quantity σQ in equation 52 is defined as [14]: +σQ = 2V 2 +P 0 +V 2 +S0 +�Q33 +Q55 +− 1 +� +(ϵ − δ) + V 2 +P 0 Q55 +V 2 +S0 Q33 +(ϵQ − δQ). +(52) +Equations 49–51 are derived under the assumption of weak attenuation and weak anisotropy (in both velocity +and attenuation). Note that the effective quality factor, assumed to be frequency-independent in constant-Q TI +media, is proportional to the inverse of the attenuation coefficient [14]. +Substitution of the Thomsen parameters from equations 15–19 and 36–40 into equations 49–52 allows us to sepa- +rate the frequency-dependent parts of the attenuation coefficients. The approximate P-wave attenuation coefficient +then becomes (only the linear term in ln |f/f0| is retained): +AP = ˜ +AP 0 +� +1 + ˜δQ sin2 θ cos2 θ + ˜ϵQ sin4 θ + RP ln +���� +f +f0 +���� +� +, +(53) +12 + +where RP controls the derivative of AP with respect to ln |f/f0|, +RP = 1 +π +˜ +AP 0 ζQ sin2 θ cos2 θ; +(54) +ζQ is defined in equation 22. +For SV-waves, +ASV = ˜ +AS0 +� +1 + ˜σQ sin2 θ cos2 θ + RSV ln +���� +f +f0 +���� +� +, +(55) +with +˜σQ = 2˜σ (gQ − 1) + +1 +g gQ +(˜ϵQ − ˜δQ), +(56) +RSV = 1 +π +˜ +AS0 σ′ +Q sin2 θ cos2 θ, +(57) +σ′ +Q = 2(1 − gQ) +g g2 +Q +� +(1 − gQ)(˜ϵ − ˜δ) − ˜δQ + (1 + ˜ϵ)˜ϵQ +� +− ζQ +g g2 +Q +, +(58) +where g and gQ are given by equations 27 and 23, respectively. The factor RSV controls the derivative of ASV with +respect to ln |f/f0|. +The terms ˜ +AP 0 RP and ˜ +AS0 RSV define the rate of the P- and SV-wave attenuation-coefficient change (increase +or decrease) with respect to ln |f/f0|. +The larger RP and RSV are, the stronger is the dispersion (frequency +dependence) of AP and ASV . +Therefore, RP and RSV can be called the P- and SV-wave dispersion factors, +respectively. +The SH-wave attenuation coefficient (equation 51) is independent of frequency, with γQ = ˜γQ: +ASH = ˜ +AS0(1 + ˜γQ sin2 θ). +(59) +5.2. Numerical dispersion analysis +Here, we evaluate the frequency dependence of the attenuation coefficients of P- and SV-waves, starting with the +dispersion factors RP and RSV (equations 54 and 57). As before, we restrict gQ to the realistic range 0.5 < gQ ≤ 3 +(Figure 1). Figures 11 and 12 show that gQ = 1 yields the smallest values of RP and RSV ; the dispersion factors +and the magnitude of their variation with angle increase with the deviation of gQ from unity. +Next, we use the medium parameters from Figures 11d and 12d to calculate the exact attenuation coefficients +for P- and SV-waves (respectively) at three frequencies. For the reference frequency f0 = 40 Hz, the term ln |f/f0| +in equations 54 and 57 is close to −1 at f = 15 Hz and 1 at f = 109 Hz. In agreement with equations 53 and 55, +the variation of AP with ln |f/f0| between 15 Hz and 40 Hz (and 40 Hz and 109 Hz) is approximately proportional +to RP , and the corresponding variation of ASV is approximately proportional to RSV . +Figures 13 and 14 show that the frequency dependence of the P- and SV-wave attenuation coefficients AP and +ASV is generally mild. However, they may become noticeable for propagation angles close to 45◦ as illustrated in +Figures 15 and 16. Both AP and ASV exhibit a more significant variation with frequency for strongly attenuative +media (Q33=Q55=20) when gQ ≥ 2 (for P-waves) and gQ ≤ 0.5 (for SV-waves). +13 + +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +0 +1 +2 +3 +4 +5 +6 +θ (degrees) +RP (%) +(a) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +0 +1 +2 +3 +4 +5 +θ (degrees) +RP (%) +(b) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +0 +1 +2 +3 +4 +5 +6 +θ (degrees) +RP (%) +(c) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +0 +1 +2 +3 +4 +5 +θ (degrees) +RP (%) +(d) +Figure 11: Variation of the P-wave dispersion factor RP (equation 54) with the phase angle for different gQ. The +reference parameters defined at f0 = 40 Hz are ˜VP 0 = 3.0 km/s, g = 0.3, ˜ϵ = 0.2, ˜ +AP 0 = 0.0125 (corresponding to +Q33 = 40), ˜ϵQ = −0.1 and ˜δQ = −0.2. (a) ˜δ = 0.1 and ˜δQ = −0.2; (b) ˜δ = 0.1 and ˜δQ = 0.2; (c) ˜δ = −0.1 and +˜δQ = −0.2; (d) ˜δ = −0.1 and ˜δQ = 0.2. +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +θ (degrees) +RSV (%) +(a) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +0 +θ (degrees) +RSV (%) +(b) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +θ (degrees) +RSV (%) +(c) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +20 +40 +60 +80 +-8 +-6 +-4 +-2 +0 +θ (degrees) +RSV (%) +(d) +Figure 12: Variation of the SV-wave dispersion factor RSV (equation 57) with the phase angle for different gQ. The +reference parameters defined at f0 = 40 Hz are ˜VP 0 = 3.0 km/s, g = 0.3, ˜ϵ = 0.2, ˜ +AS0 = 0.0125 (corresponding to +Q55 = 40), ˜ϵQ = −0.1 and ˜ϵQ = −0.2. The parameters ˜δ and ˜δQ are the same as in Figure 11. +14 + +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0115 +0.0120 +0.0125 +0.0130 +0.0135 +θ (degrees) +AP +(a) +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0115 +0.0120 +0.0125 +0.0130 +θ (degrees) +AP +(b) +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0115 +0.0120 +0.0125 +0.0130 +θ (degrees) +AP +(c) +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0115 +0.0120 +0.0125 +0.0130 +θ (degrees) +AP +(d) +Figure 13: Variation of the P-wave normalized phase attenuation coefficient with the phase angle at different +frequencies. The medium parameters are the same as in Figure 11d, and (a) gQ = 3; (b) gQ = 2; (c) gQ = 1; (d) +gQ = 0.5. +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.009 +0.010 +0.011 +0.012 +θ (degrees) +ASV +(a) +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0095 +0.0100 +0.0105 +0.0110 +0.0115 +0.0120 +0.0125 +θ (degrees) +ASV +(b) +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0100 +0.0105 +0.0110 +0.0115 +0.0120 +0.0125 +θ (degrees) +ASV +(c) +f=15 Hz +f=40 Hz +f=109 Hz +0 +20 +40 +60 +80 +0.0110 +0.0115 +0.0120 +0.0125 +0.0130 +θ (degrees) +ASV +(d) +Figure 14: Variation of the SV-wave attenuation coefficient with the phase angle at different frequencies. The +medium parameters are the same as in Figure 12d, and (a) gQ = 3; (b) gQ = 2; (c) gQ = 1; (d) gQ = 0.5. +15 + +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +0.0120 +0.0125 +0.0130 +0.0135 +f (Hz) +AP +(a) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +0.024 +0.025 +0.026 +0.027 +0.028 +0.029 +f (Hz) +AP +(b) +Figure 15: Variation of the P-wave attenuation coefficient with frequency at θ = 45◦ for different gQ. Except for +˜ +AP 0, the medium parameters are the same as in Figures 11d and 13. On plot (a), ˜ +AP 0 = 0.0125 (corresponding to +Q33 = 40); on plot (b), ˜ +AP 0 = 0.025 (corresponding to Q33 = 20). +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +0.009 +0.010 +0.011 +0.012 +0.013 +0.014 +f (Hz) +ASV +(a) +gQ=3 +gQ=2 +gQ=1 +gQ= 1 +2 +0 +50 +100 +150 +200 +0.018 +0.020 +0.022 +0.024 +f (Hz) +ASV +(b) +Figure 16: Variation of the SV-wave attenuation coefficient with frequency at θ = 45◦ for different gQ. Except for +˜ +AS0, the medium parameters are the same as in Figures 12d and 14. On plot (a), ˜ +AS0 = 0.0125 (corresponding to +Q55 = 40); on plot (b), ˜ +AS0 = 0.025 (corresponding to Q55 = 20). +16 + +6. Conclusions +We obtained concise analytic expressions for the Thomsen-type parameters of constant-Q TI media. All Thomsen +velocity parameters (VP 0, VS0, ϵ, δ and γ) are frequency dependent, with the reference attenuation parameters ˜ +AP 0 +(proportional to 1/Q33) and ˜ +AS0 (proportional to 1/Q55) controlling the dispersion (frequency dependence) of the +vertical velocities VP 0 and VS0, respectively. The reference attenuation parameters ˜ϵQ, ˜δQ, and ˜γQ govern the +variations of the anisotropy parameters ϵ, δ, and γ with frequency. +However, the frequency dependence of all +Thomsen velocity parameters is weak in a wide frequency range, even for strong attenuation. In viscoacoustic +constant-Q TI media, the elliptical conditions at the reference frequency ensure that the anellipticity parameter η +vanishes for all frequencies. +Despite the fact that all Qij elements in constant-Q TI media are frequency independent, one of the Thomsen- +type attenuation parameters (δQ) does vary with frequency. The frequency dependence of δQ is controlled by the +newly defined coefficient ζQ and can be substantial when ζQ has a large magnitude. As a result, the frequency +variation of the P- and SV-wave attenuation coefficients may be non-negligible at oblique propagation angles with +the symmetry axis. That variation is highly sensitive to the ratio of the vertical quality factors gQ = Q33/Q55. +Both attenuation coefficients are insensitive to frequency for gQ = 1, whereas their frequency dependence is most +substantial for gQ ≥ 3 (for P-waves) and gQ ≤ 0.5 (for SV-waves). In contrast, the SH-wave attenuation coefficient +in constant-Q TI media is frequency-independent. +The constant-Q assumption is often made in attenuation analysis because the effective attenuation coefficients +estimated from seismic data (e.g., using the spectral-ratio method) become linear functions of frequency. However, +our results show that this linear dependence may not hold for constant-Q TI models, which can cause errors in the +inversion for the attenuation parameters. +Appendix A. Appendix A: Complex stiffness coefficients expressed in terms of the Thomsen-type +parameters +The stiffness coefficients for the constant-Q dissipative VTI model (equations 1–3) can be found at the reference +frequency as Mij|f=f0 = ˜ +M R +ij (1 − i/Qij). Using the parameter definitions in equations 4–12, we express ˜ +M R +ij and +Qij in terms of the reference Thomsen-type parameters as follows: +˜ +M R +33 = ρ ˜V 2 +P 0, +(A.1) +˜ +M R +55 = ρ ˜V 2 +S0, +(A.2) +˜ +M R +11 = ρ ˜V 2 +P 0(1 + 2˜ϵ), +(A.3) +˜ +M R +66 = ρ ˜V 2 +S0(1 + 2˜γ), +(A.4) +˜ +M R +13 = −ρ ˜V 2 +S0 + ρ +� +( ˜V 2 +P 0 − ˜V 2 +S0) +� +(1 + 2˜δ) ˜V 2 +P 0 − ˜V 2 +S0 +� +, +(A.5) +Q−1 +33 = +2 ˜ +AP 0 +1 − ˜ +A2 +P 0 +, +(A.6) +Q−1 +55 = +2 ˜ +AS0 +1 − ˜ +A2 +S0 +, +(A.7) +Q−1 +11 = Q−1 +33 (1 + ˜ϵQ), +(A.8) +Q−1 +66 = Q−1 +55 (1 + ˜γQ) +(A.9) +Q−1 +13 = ˜Q−1 +33 +� +1 + ˜δQf1 + f2 +� +− Q−1 +55 f2, +(A.10) +17 + +with +f1 = +˜ +M R +33 ( ˜ +M R +33 − ˜ +M R +55) +2 ˜ +M R +13( ˜ +M R +13 + ˜ +M R +55) +, +(A.11) +f2 = +˜ +M R +55 ( ˜ +M R +13 + ˜ +M R +33)2 +2 ˜ +M R +13( ˜ +M R +13 + ˜ +M R +55)( ˜ +M R +33 − ˜ +M R +55) +. +(A.12) +Appendix B. Appendix B: Explicit expressions for sn +Here, we provide explicit expressions for the coefficients sn in equation 41. +The coefficient s0 is given by: +s0 = g(1 − g + χ)2(h0 + h1g + h2g2 + h3g3 + h4g4 + h5g5) +(1 − g)3χ3(g − χ)2 +, +(B.1) +where +h0 = −(1 + 2˜δ)2χ, +(B.2) +h1 = (1 + 2˜δ)(5 + 10˜δ + 2χ), +(B.3) +h2 = (1 + 2˜δ)(2(˜δ − 3)χ − 13˜δ − 14), +(B.4) +h3 = ˜δ(7˜δ + 9χ + 30) + 7χ + 15, +(B.5) +h4 = −˜δ2 − 2(˜δ + 1)χ − 11˜δ − 8, +(B.6) +h5 = 2(1 + 2˜δ). +(B.7) +For the coefficient s1 we have: +s1 = 3g(g − χ − 1)(k0 + k1g + k2g2 + k3g3 + k4g4) +2(1 − g)χ3(g − χ)2 +, +(B.8) +where +k0 = −2(1 + 2˜δ)2, +(B.9) +k1 = 2 +� +1 + χ + 4˜δ(˜δ + χ + 1) +� +, +(B.10) +k2 = 2(χ + 1) − ˜δ(χ + 3), +(B.11) +k3 = −(˜δ + 2χ + 4), +(B.12) +k4 = 2; +(B.13) +Finally, the coefficient s2 has the form: +s2 = +3g +� +3 + 6˜δ + 2χ − 3g(3˜δ + 2χ + 3) + 3g2(˜δ + χ + 3) − 3g3� +2χ3(g − χ)2 +; +(B.14) +s3 = (1 − g)2(4χ − 3g) +4χ3(g − χ)2 +. +(B.15) +The quantities g and χ are defined in equations 27 and 28, respectively. +18 + +References +[1] H. Kolsky, The propagation of stress pulses in viscoelastic solids, Philosophical magazine 1 (8) (1956) 693–710. +[2] Kjartansson, Constant Q-wave propagation and attenuation, Journal of Geophysical Research 84 (1979) 4737– +4748. +[3] Q. Hao, S. Greenhalgh, Nearly constant Q models of the generalized standard linear solid type and the corre- +sponding wave equations, Geophysics 86 (4) (2021) T239–T260. +[4] Q. Hao, S. Greenhalgh, Nearly constant Q dissipative models and wave equations for general viscoelastic +anisotropy, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477 (2251) +(2021) 20210170. +[5] J. Behura, I. Tsvankin, Estimation of interval anisotropic attenuation from reflection data, Geophysics 74 (6) +(2009) A69–A74. +[6] B. Shekar, I. Tsvankin, Estimation of shear-wave interval attenuation from mode-converted data, Geophysics +76 (6) (2011) D11–D19. +[7] B. Shekar, I. Tsvankin, Anisotropic attenuation analysis of crosshole data generated during hydraulic fracturing, +The Leading Edge 31 (5) (2012) 588–593. +[8] J. Behura, I. Tsvankin, E. Jenner, A. Calvert, Estimation of interval velocity and attenuation anisotropy from +reflection data at coronation field, The Leading Edge 31 (5) (2012) 580–587. +[9] A. I. Best, J. Sothcott, C. McCann, A laboratory study of seismic velocity and attenuation anisotropy in +near-surface sedimentary rocks, Geophysical Prospecting 55 (5) (2007) 609–625. +[10] Y. Zhu, I. Tsvankin, P. Dewangan, K. van Wijk, Physical modeling and analysis of p-wave attenuation +anisotropy in transversely isotropic media, Geophysics 72 (1) (2007) D1–D7. +[11] A. Zhubayev, M. E. Houben, D. M. Smeulders, A. Barnhoorn, Ultrasonic velocity and attenuation anisotropy +of shales, Whitby, United Kingdom, Geophysics 81 (1) (2016) D45–D56. +[12] L. Thomsen, Weak elastic anisotropy, Geophysics 51 (10) (1986) 1954–1996. +[13] I. Tsvankin, Seismic signatures and analysis of reflection data in anisotropic media, Elsevier Science Ltd., 2001. +[14] Y. Zhu, I. Tsvankin, Plane-wave propagation in attenuative transversely isotropic media, Geophysics 71 (2) +(2006) T17–T30. +[15] Q. Hao, S. Greenhalgh, X. Huang, H. Li, Viscoelastic wave propagation for nearly constant Q transverse +isotropy, Geophysical Prospecting 70 (7) (2022) 1176–1192. +[16] J. M. Carcione, Wave fields in real media: Theory and numerical simulation of wave propagation in anisotropic, +anelastic, porous and electromagnetic media: Handbook of Geophysical exploration (3rd ed.), Elsevier, 2014. +[17] V. ˇCerven´y, I. Pˇsenc´ık, Perturbation hamiltonians in heterogeneous anisotropic weakly dissipative media, +Geophysical Journal International 178 (2) (2009) 939–949. +[18] I. Tsvankin, V. Grechka, Seismology of azimuthally anisotropic media and seismic fracture characterization, +Society of Exploration Geophysicists, 2011. +[19] T. Alkhalifah, I. Tsvankin, Velocity analysis for transversely isotropic media, Geophysics 60 (5) (1995) 1550– +1566. +19 + +[20] Q. Hao, T. Alkhalifah, An acoustic eikonal equation for attenuating transversely isotropic media with a vertical +symmetry axis, Geophysics 82 (1) (2017) C9–C20. +[21] Q. Hao, T. Alkhalifah, An acoustic eikonal equation for attenuating orthorhombic media, Geophysics 82 (4) +(2017) WA67–WA81. +[22] Q. Hao, T. Alkhalifah, Viscoacoustic anisotropic wave equations, Geophysics 84 (6) (2019) C323–C337. +[23] J. Behura, I. Tsvankin, Role of the inhomogeneity angle in anisotropic attenuation analysis, Geophysics 74 (5) +(2009) WB177–WB191. +20 + diff --git a/8tAzT4oBgHgl3EQf-v68/content/tmp_files/load_file.txt b/8tAzT4oBgHgl3EQf-v68/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6df2e454e4414e6f683cf0dec92b4d42ac6b5f32 --- /dev/null +++ b/8tAzT4oBgHgl3EQf-v68/content/tmp_files/load_file.txt @@ -0,0 +1,711 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf,len=710 +page_content='Thomsen-type parameters and attenuation coefficients for constant-Q transverse isotropy Qi Haoa,∗, Ilya Tsvankinb aCollege of Geoexploration Science and Technology, Jilin University, Changchun, 130026, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' China bDepartment of Geophysics, Colorado School of Mines, Golden, 80401, USA Abstract Transversely isotropic (TI) media with the frequency-independent quality-factor elements (also called “constant-Q” transverse isotropy) are often used to describe attenuation anisotropy in sedimentary rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The attenuation coef- ficients in constant-Q TI models can be conveniently defined in terms of the Thomsen-type attenuation-anisotropy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Recent research indicates that not all those parameters for such constant-Q media are frequency- independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Here, we present concise analytic formulae for the Thomsen-type attenuation parameters for Kjar- tansson’s constant-Q TI model and show that one of them (δQ) varies with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The analytic expression for δQ helps evaluate the frequency dependence of the normalized attenuation coefficients of P- and SV-waves by introducing the newly defined “dispersion factors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Viscoacoustic constant-Q transverse isotropy is also discussed as a special case, for which the elliptical condition and simplified expressions for the parameters δ and δQ are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Our results show that in the presence of significant absorption the attenuation coefficients of the “constant-Q” model vary with frequency for oblique propagation with respect to the symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' This variation needs to be taken into account when applying the spectral-ratio method and other attenuation-analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Keywords: seismic, attenuation, anisotropy, wave, Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Introduction The frequency-independent quality factor (called “constant-Q” for brevity) provides a useful phenomenologi- cal description of seismic attenuation in rocks and is widely used in seismic attenuation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Among such constant-Q dissipative models are those proposed by [1] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For isotropic media, the Kjartansson model pro- duces the constant-Q factor for all frequencies, whereas the Kolsky model leads to nearly constant Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The complex moduli for the Kolsky model represent the first-order Maclaurin series expansion with respect to 1/Q of the corresponding moduli for the Kjartansson model [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As an extension of non-dissipative transverse isotropy, the constant-Q TI model can be used to process seismic attenuation data for most sedimentary rocks, such as shale formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The constant-Q assumption facilitates estimation of the quality factor and attenuation anisotropy [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Ultrasonic measurements demonstrate that attenuation anisotropy generally is stronger than velocity anisotropy for rock samples [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Velocity anisotropy for TI media can be efficiently described by the Thomsen anisotropy parameters [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Likewise, attenuation anisotropy for dissipative TI media is convenient to study using the Thomsen-type nota- tion introduced by [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The combination of the velocity- and attenuation-related Thomsen-type parameters [14] completely defines the complex stiffness matrix at a specified frequency for a general dissipative TI model with a vertical symmetry axis (VTI medium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The generic Thomsen velocity parameters depend on the real parts of the stiffness coefficients (cij), whereas the Thomsen-type attenuation parameters are defined by both the real and imaginary parts of cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [15] found that some Thomsen-type parameters in constant-Q VTI media are frequency ∗Corresponding author Email addresses: xqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='hao@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='com (Qi Hao), ilya@mines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='edu (Ilya Tsvankin) January 6, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='01939v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='geo-ph] 5 Jan 2023 dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' This phenomenon is not entirely surprising, because all stiffness coefficients of constant-Q TI media, which are involved in the definition of the Thomsen-type parameters, are functions of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Investigating the frequency variations of these parameters can facilitate understanding of such key signatures in TI media as velocities, traveltimes, attenuation coefficients, and polarization vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' However, to our knowledge, there are no analytic expressions for the frequency-dependent Thomsen-type attenuation parameters in constant-Q dissipative VTI media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Here, we derive analytic formulae for the Thomsen-type parameters using Kjartansson’s constant-Q VTI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' A set of the corresponding reference parameters is defined at a specified frequency and used to obtain those parameters for the entire frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' We also present a formula for the frequency-dependent anellipticity and define the condition for elliptical anisotropy in constant-Q TI media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The newly proposed formulae for the Thomsen-type parameters allow us to study the normalized plane-wave attenuation coefficients in constant-Q media with weak attenuation anisotropy and define the “dispersion factors” for P- and SV-waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Numerical examples are used to analyze the accuracy of the obtained expressions for the Thomsen-type parameters, the validity of the elliptical condition, and the frequency dependence of the attenuation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Thomsen-type parameters of constant-Q VTI media 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Constant-Q dissipative VTI model Referring to [14] and [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' the complex stiffness (or modulus) matrix M for viscoelastic VTI media is given by: M = � � � � � � � � M11 M11 − 2M66 M13 0 0 0 M11 − 2M66 M11 M13 0 0 0 M13 M13 M33 0 0 0 0 0 0 M55 0 0 0 0 0 0 M55 0 0 0 0 0 0 M66 � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (1) where Mij = M R ij − i sgn(f)M I ij denote the complex stiffness coefficients for the frequency f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' and the minus sign in front of i sgn(f)M I ij follows from the definition of the Fourier transform in [17] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Both the real and imaginary parts of Mij generally are frequency dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For the [2] model (also called the constant-Q model), the nonzero independent elements in equation 1 are expressed as: Mij = ˜ M R ij cos(πγij) � −i f f0 �2γij , (2) with γij = 1 π tan−1 � 1 Qij � , (3) where Qij ≡ M R ij /M I ij, f0 is the reference frequency, and ˜ M R ij denote the real parts of Mij at f0: ˜ M R ij = Re(Mij)|f=f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' By design, the quality-factor elements Qij for the Kjartansson model are independent of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As follows from equation 2, the complex stiffness coefficients Mij for a given frequency can be expressed in terms of ˜ M R ij and Qij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Thomsen-type parameterization [14] and [18] show that dissipative VTI media can be conveniently parameterized by the Thomsen-type attenu- ation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The [12] velocity parameters [see 13] are defined in the nonattenuative reference VTI medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameter VP 0 is the vertical velocity of P-waves: VP 0 ≡ � M R 33 ρ , (4) where ρ denotes density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 2 The parameter VS0 is the vertical velocity of S-waves: VS0 ≡ � M R 55 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (5) The parameter ϵ is approximately equal to the fractional difference between the horizontal and vertical velocities of P-waves: ϵ ≡ M R 33 − M R 11 2M R 33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (6) The parameter δ determines the second derivative of the P-wave phase velocity at vertical incidence and is given by: δ ≡ � M R 13 + M R 55 �2 − � M R 33 − M R 55 �2 2M R 33(M R 33 − M R 55) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (7) The parameter γ is approximately equal to the fractional difference between the horizontal and vertical velocities of SH-waves: γ ≡ M R 66 − M R 55 2M R 55 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (8) The Thomsen-type attenuation parameters [14] can be used to define the normalized phase attenuation coefficient A ≡ |kI|/|kR| for P-, SV-, and SH-waves, which is generally supposed to be frequency-independent in constant-Q models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For more details about A, see the section “Plane-wave attenuation in constant-Q VTI media” below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameter AP 0 is the vertical attenuation coefficient of P-waves: AP 0 ≡ Q33 �� 1 + 1 Q2 33 − 1 � ≈ 1 2Q33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (9) The parameter AS0 is the vertical attenuation coefficient of S-waves: AS0 ≡ Q55 �� 1 + 1 Q2 55 − 1 � ≈ 1 2Q55 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (10) The parameter ϵQ is close to the fractional difference between the horizontal and vertical attenuation coefficients of P-waves: ϵQ ≡ Q33 − Q11 Q11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (11) The parameter δQ controls the second derivative of the P-wave attenuation coefficient at vertical incidence and is expressed as [14]: δQ ≡ Q33−Q55 Q55 M R 55 (M R 13+M R 33) 2 M R 33−M R 55 + 2 Q33−Q13 Q13 M R 13(M R 13 + M R 55) M R 33(M R 33 − M R 55) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (12) The parameter γQ is close to the fractional difference between the horizontal and vertical attenuation coefficients of SH-waves: γQ ≡ Q55 − Q66 Q66 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (13) 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Analytic description of Thomsen-type parameters In this section, we represent the Thomsen velocity parameters and Thomsen-type attenuation parameters in terms of their reference values defined at f = f0: ˜VP 0 = VP 0|f=f0, ˜VS0 = VS0|f=f0, ˜ϵ = ϵ|f=f0, ˜δ = δ|f=f0, ˜ AP 0 = AP 0|f=f0, ˜ AS0 = AS0|f=f0, ˜ϵQ = ϵQ|f=f0, and ˜δQ = δQ|f=f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' These parameters are used to find the real parts of the reference stiffness coefficients ( ˜ M R ij ), the quality-factor elements Qij (see Appendix A), and the frequency-dependent stiffness matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' According to equations 4–7 and 12, the Thomsen-type parameters involve the coefficients M R ij , where ij=11, 13, 33, 55 and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Using equations 2 and 3, M R ij are approximately expressed as: M R ij ≈ ˜ M R ij � 1 + 2 π Q−1 ij ln ���� f f0 ���� + 2 π2 Q−2 ij ln2 ���� f f0 ���� � , (14) where we use the approximation tan−1(Q−1 ij ) ≈ Q−1 ij because typically Qij ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Substitution of equation 14 into equations 4–7 and 12 allows us to derive approximate expressions for the frequency-dependent Thomsen-type parameters, which are discussed in the following two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Velocity parameters The second-order approximations for the Thomsen velocity parameters with respect to ln|f/f0| are given by: VP 0 = ˜VP 0 � 1 + 1 π Q−1 33 ln ���� f f0 ���� + 1 2π2 Q−2 33 ln2 ���� f f0 ���� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (15) VS0 = ˜VS0 � 1 + 1 π Q−1 55 ln ���� f f0 ���� + 1 2π2 Q−2 55 ln2 ���� f f0 ���� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (16) ϵ = ˜ϵ + 1 π (1 + 2˜ϵ) Q−1 33 ˜ϵQ ln ���� f f0 ���� + 1 π2 (1 + 2˜ϵ) Q−2 33 ˜ϵ2 Q ln2 ���� f f0 ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (17) δ = ˜δ + 1 π Q−1 33 ˜δQ ln ���� f f0 ���� + 1 π2 Q−2 33 ζQ ln2 ���� f f0 ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (18) γ = ˜γ + 1 π (1 + 2˜γ) Q−1 55 ˜γQ ln ���� f f0 ���� + 1 π2 (1 + 2˜γ) Q−2 55 ˜γ2 Q ln2 ���� f f0 ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (19) where the P- and S-wave inverse vertical quality factors Q33 and Q55 (respectively) are: Q−1 33 = ˜ AP 0 2(1 − ˜ A2 P 0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (20) Q−1 55 = ˜ AS0 2(1 − ˜ A2 S0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (21) The coefficient ζQ in equation 18 is defined as: ζQ = d0 (1 − gQ)2 + d1 (1 − gQ) ˜δQ + d2 ˜δ2 Q , (22) with gQ ≡ Q33 Q55 , (23) 4 d0 = g(1 − g + χ)2 � (1 + 2˜δ) χ − (1 + 2˜δ) g + (1 + ˜δ) g2� (1 − g)2(χ − g)χ2 , (24) d1 = 2g � 1 + 2˜δ + χ − (2 + ˜δ + χ) g + g2� (χ − g)χ2 , (25) d2 = 2χ − g 2(1 + 2˜δ − g)(χ − g) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (26) g ≡ ˜V 2 S0 ˜V 2 P 0 , (27) χ = � (1 − g)(1 + 2˜δ − g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (28) In equations 15–19, the first-order terms with respect to ln|f/f0| are scaled by Q−1 33 or Q−1 55 , whereas the second- order terms by Q−2 33 or Q−2 55 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Because Q33 and Q55 typically are much greater than unity, the frequency dependence of the velocity parameters is mostly determined by the first-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Equations 15–19 indicate that: (1) VP 0 and VS0 always monotonically increase with frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (2) ϵ, δ and γ also monotonically increase with f, if ˜ϵQ > 0, ˜δQ > 0, and ˜γQ > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Overall, the frequency dependence of VP 0, VS0, ϵ, δ, and γ for realistic values of Q33 and Q55 (Q33 ≫ 1 and Q55 ≫ 1) remains weak, as illustrated by the numerical examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Note that phase and group velocities in strongly dissipative TI media are influenced by attenuation and do not represent the same functions of the Thomsen parameters as in purely elastic models [14, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For sedimentary formations, both gQ and g vary within a limited range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' In particular, according to [9], for relatively shallow sedimentary rocks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 < gQ ≤ 3 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' gQ= 1 2 gQ=1 gQ=2 gQ=3 0 20 40 60 80 100 0 50 100 150 Q55 Q33 Figure 1: Vertical quality factors Q33 and Q55 in dissipative VTI rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The black dots are the data from Table 3 of [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' gQ ≡ Q33/Q55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Using equations 17 and 18 for ϵ and δ, the anellipticity parameter η [19] can be approximately obtained as: η ≡ ϵ − δ 1 + 2δ = η0 + η1 Q−1 33 ln ���� f f0 ���� + η2 Q−2 33 ln2 ���� f f0 ����, (29) 5 where Q33 is given by equation 20, and η0 = ˜η = ˜ϵ − ˜δ 1 + 2˜δ , (30) η1 = 1 + 2˜ϵ (1 + 2˜δ)2 � (1 + 2˜δ)˜ϵQ − ˜δQ � , (31) η2 = 1 + 2˜ϵ 1 + 2˜δ � r0 + r1 1 + 2˜δ + r2 (1 + 2˜δ)2 � , (32) with r0 = ˜ϵ2 Q, (33) r1 = −ζQ − 2˜ϵQ ˜δQ, (34) r2 = 2˜δ2 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (35) The parameter η controls (along with the zero-dip normal-moveout velocity) all P-wave time-domain signatures for laterally homogeneous VTI media above a horizontal or dipping target reflector [19, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Attenuation parameters The following Thomsen-type attenuation parameters are expressed directly through the elements Qij and, there- fore, are frequency-independent in constant-Q VTI media: AP 0 = ˜ AP 0, (36) AS0 = ˜ AS0, (37) ϵQ = ˜ϵQ, (38) γQ = ˜γQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (39) The attenuation parameter δQ, however, also depends on the coefficients M R ij (equation 12), which vary with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The second-order approximation for δQ with respect to ln |f/f0| is: δQ = ˜δQ + 2 π Q−1 33 ζQ ln ���� f f0 ���� + 2 π2 Q−2 33 ξQ ln2 ���� f f0 ����, (40) where ζQ is defined in equation 22, and ξQ = s0(1 − gQ)3 + s1(1 − gQ)2 ˜δQ + s2(1 − gQ) ˜δ2 Q + s3 ˜δ3 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (41) The explicit expressions for the coefficients si are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Because for Q33 ≫ 1 the influence of the second-order term in equation 40 is insignificant, the frequency variation of δQ is largely controlled by the coefficient ζQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For ζQ > 0, δQ monotonically increases with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As follows from equations 22 and 24–28, ζQ is a function of g (equation 27), gQ (equation 23), ˜δ, and ˜δQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Numerical analysis Here, we analyze the above expressions for the Thomsen-type parameters numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The reference frequency is set as f0 = 40 Hz and the frequency range as [1, 200] Hz for all examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' First, we test the accuracy of the equations 15 and 16 for the vertical velocities and their first-order versions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', those without the second-order term with respect to ln|f/f0|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As demonstrated by Figure 2, the first-order approximations for VP 0 and VS0 are sufficiently accurate even for strong attenuation in a wide frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Overall, the frequency dependence of the vertical velocities is almost negligible, except for very low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 6 exact 1st 2nd 0 50 100 150 200 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='94 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='04 f (Hz) vP0 (km/s) (a) exact 1st 2nd 0 50 100 150 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='54 f (Hz) vS0 (km/s) (b) Figure 2: Frequency-dependent vertical velocities (a) VP 0 and (b) VS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' “Exact” in the legend refers to the exact values, whereas “1st” and “2nd” denote the first- and second-order approximations with respect to ln |f/f0|, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' On plot (a), ˜VP 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 km/s and Q33 = 40 ( ˜ AP 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' on plot (b), ˜VS0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 km/s and Q55 = 20 ( ˜ AS0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Table 1: Medium parameters for two constant-Q VTI models at the reference frequency f0 = 40 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [H] Model ˜VP 0 ˜VS0 ˜ϵ ˜δ ˜γ ˜ AP 0 (Q33) ˜ AS0 (Q55) ˜ϵQ ˜δQ ˜γQ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 (40) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0167 (30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0250 (20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0333 (15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 Figures 3, 4 and 5 show that the first-order versions of equations 17–19 can accurately describe the variations of the anisotropy parameters ϵ, δ, and γ with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Comparison of Figures 3, 4, and 5 confirms that the reference parameters ˜ϵQ, ˜δQ, and ˜γQ govern the frequency dependence of ϵ, δ, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For example, if ˜ϵQ > 0, ϵ increases with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As is the case for VP 0 and VS0, the anisotropy coefficients vary with frequency primarily in the low-frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='315 f (Hz) ϵ (a) exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='31 f (Hz) ϵ (b) Figure 3: Variation of the Thomsen parameter ϵ with frequency for (a) Model 1 and (b) Model 2 from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The legend is the same as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Next, we investigate the only frequency-dependent attenuation-anisotropy parameter, δQ, by comparing the exact equation for δQ with its first- and second-order approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The first-order equation accurately models δQ in a wide frequency range, whereas contribution of the second-order term is practically negligible (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As mentioned above, the coefficient ζQ in equation 40 is largely responsible for the frequency variation of δQ for 7 exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='07 f (Hz) δ (a) exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='08 f (Hz) δ (b) Figure 4: Same as Figure 3 but for the parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='110 f (Hz) γ (a) exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='220 f (Hz) γ (b) Figure 5: Same as Figure 3 but for the parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' exact 1st 2nd 0 50 100 150 200 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6 f (Hz) δQ (a) exact 1st 2nd 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='05 f (Hz) δQ (b) Figure 6: Frequency-dependent Thomsen-type attenuation parameter δQ for (a) Model 1 and (b) Model 2 from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The legend is the same as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 8 a specified value of Q33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Equation 22 shows that ζQ is a function of the parameters g = ˜V 2 S0/ ˜V 2 P 0, gQ = Q−1 55 /Q−1 33 , ˜δ and ˜δQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Using the results from Figure 1, we restrict gQ to the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 ≤ gQ ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Figures 7 and 8 show that the smallest absolute value of ζQ corresponds to gQ = 1, and |ζQ| increases with the deviation of gQ from unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As a result, the parameter δQ is almost independent of frequency for gQ = 1 (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Overall, the frequency dependence of δQ becomes noticeable for large |gQ − 1| (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', gQ = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Figure 9), but it is also influenced by the parameters ˜δ and ˜δQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For the most common values of gQ considered here, the parameter δQ significantly varies with f only for low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 δ˜ δ˜ Q ζQ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 δ˜ δ˜ Q ζQ 2 4 6 8 10 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 δ˜ δ˜ Q ζQ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 δ˜ δ˜ Q ζQ 0 1 2 3 4 5 6 (d) Figure 7: Contour plots of the coefficient ζQ as a function of ˜δ and ˜δQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameter g = ˜V 2 S0/ ˜V 2 P 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameter gQ = Q−1 55 / ˜Q−1 P 0 is defined as (a) gQ = 3, (b) gQ = 2, (c) gQ = 1, and (d) gQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 9 g=0 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 10 5 0 5 10 15 gQ ζQ (a) g=0 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 6 4 2 0 2 4 6 8 gQ ζQ (b) g=0 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0 10 20 30 40 50 gQ ζQ (c) g=0 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0 10 20 30 gQ ζQ (d) Figure 8: Variation of the coefficient ζQ with gQ for different values of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (a) ˜δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 and ˜δQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (b) ˜δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 and ˜δQ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (c) ˜δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 and ˜δQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (d) ˜δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 and ˜δQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 f (Hz) δQ (a) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 f (Hz) δQ (b) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 f (Hz) δQ (c) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 f (Hz) δQ (d) Figure 9: Variation of the attenuation parameter δQ with frequency for different gQ and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameters ˜δ and ˜δQ are the same as in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Viscoacoustic constant-Q transverse isotropy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Simplified parameter expressions Next, we consider the so-called “viscoacoustic” constant-Q media described by the Thomsen-type notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The acoustic approximation is implemented by setting ˜VS0 = AS0 = 0 in equations 18, 40, and 29 [20, 21, 22] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameters δ, η, and δQ then reduce to: δ = ˜δ + 1 π Q−1 33 ˜δQ ln ���� f f0 ���� + 1 π2 Q−2 33 ˜δ2 Q 1 + 2˜δ ln2 ���� f f0 ����, (42) η = η0 + η1 Q−1 33 ln ���� f f0 ���� + � ˜ϵQ − ˜δQ 1 + 2˜δ � η1 Q−2 33 ln2 ���� f f0 ����, (43) δQ = ˜δQ + 2 π Q−1 33 ˜δ2 Q 1 + 2˜δ ln ���� f f0 ���� + 2 π2 Q−2 33 ˜δ3 Q (1 + 2˜δ)2 ln2 ���� f f0 ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (44) Setting η (equation 43) to zero, which requires η0 = η1 = 0 (see equations 30 and 31), we obtain the elliptical conditions: ˜ϵ = ˜δ, (45) ˜ϵQ = ˜δQ 1 + 2˜δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (46) Equations 45 and 46 make the parameters of viscoacoustic constant-Q media satisfy the same conditions at all frequencies: ϵ = δ, (47) ϵQ = δQ 1 + 2δ , (48) which follows from equations 17, 31, 42, and 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Equation 47 implies that the elliptical conditions at the reference frequency ensure that η = 0 at all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For viscoelastic constant-Q media discussed earlier, equation 47 remains approximately valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', the model is elliptical at all frequencies), if equations 45 and 46 are satisfied (see equations 29–31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Numerical validation Here, we verify the elliptical conditions (equations 45 and 46) by computing the anellipticity parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The exact η is calculated using equations 6, 7, 11 and 12 along with equations 2 and 3 under the acoustic approximation ( ˜VS0 = 0 and Q−1 55 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The first-order approximation for η is given by equation 43 without the second-order term with respect to ln|f/f0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Figure 10 shows that for models that satisfy equations 45 and 46 the exact anellipticity parameter is negligibly small for all frequencies (on the order of 10−7 for both models), which confirms that the elliptical conditions at the reference frequency lead to equation 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' In addition, our testing confirms that the difference between the left and right sides of equation 48 is negligible, if equations 45 and 46 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 11 exact 1st 2nd 0 50 100 150 200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-7 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-7 f (Hz) η (a) exact 1st 2nd 0 50 100 150 200 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-7 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='×10-6 f (Hz) η (b) Figure 10: Variation of the anellipticity parameter η with frequency under the elliptical conditions (equations 45 and 46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The P-wave quality factor and reference vertical velocity at f0 = 40 Hz are Q33 = 40 and ˜VP 0 = 3 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameters ˜ϵ and ˜ϵQ are (a) ˜ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3 and ˜ϵQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (b) ˜ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2 and ˜ϵQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Plane-wave attenuation in constant-Q VTI media In this section, we apply the obtained expressions for the Thomsen-type parameters to study the normalized plane-wave attenuation coefficients in constant-Q VTI media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The normalized phase attenuation coefficient is defined as A ≡ |kI|/|kR|, where kR and kI denote the real and imaginary parts of the complex wave vector [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The words “phase” and “normalized” are omitted below for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The angle between kR and kI is called the “inhomogeneity” angle, which is not defined in plane-wave propagation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', it is a free parameter that can vary within certain bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The coefficient A corresponding to kR ∥ kI is approximately equal to the group attenuation coefficient, which can be estimated from seismic data, for a wide range of “inhomogeneity” angles [23, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Attenuation coefficients [14] and [18] show that the approximate attenuation coefficients of plane waves in viscoelastic constant-Q VTI media are given by: AP = AP 0 (1 + δQ sin2 θ cos2 θ + ϵQ sin4 θ), (49) ASV = AS0 (1 + σQ sin2 θ cos2 θ), (50) ASH = AS0 (1 + γQ sin2 θ), (51) where the subscripts P, SV, and SH denote the wave types, and θ is the phase angle measured from the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The quantity σQ in equation 52 is defined as [14]: σQ = 2V 2 P 0 V 2 S0 �Q33 Q55 − 1 � (ϵ − δ) + V 2 P 0 Q55 V 2 S0 Q33 (ϵQ − δQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (52) Equations 49–51 are derived under the assumption of weak attenuation and weak anisotropy (in both velocity and attenuation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Note that the effective quality factor, assumed to be frequency-independent in constant-Q TI media, is proportional to the inverse of the attenuation coefficient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Substitution of the Thomsen parameters from equations 15–19 and 36–40 into equations 49–52 allows us to sepa- rate the frequency-dependent parts of the attenuation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The approximate P-wave attenuation coefficient then becomes (only the linear term in ln |f/f0| is retained): AP = ˜ AP 0 � 1 + ˜δQ sin2 θ cos2 θ + ˜ϵQ sin4 θ + RP ln ���� f f0 ���� � , (53) 12 where RP controls the derivative of AP with respect to ln |f/f0|, RP = 1 π ˜ AP 0 ζQ sin2 θ cos2 θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (54) ζQ is defined in equation 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For SV-waves, ASV = ˜ AS0 � 1 + ˜σQ sin2 θ cos2 θ + RSV ln ���� f f0 ���� � , (55) with ˜σQ = 2˜σ (gQ − 1) + 1 g gQ (˜ϵQ − ˜δQ), (56) RSV = 1 π ˜ AS0 σ′ Q sin2 θ cos2 θ, (57) σ′ Q = 2(1 − gQ) g g2 Q � (1 − gQ)(˜ϵ − ˜δ) − ˜δQ + (1 + ˜ϵ)˜ϵQ � − ζQ g g2 Q , (58) where g and gQ are given by equations 27 and 23, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The factor RSV controls the derivative of ASV with respect to ln |f/f0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The terms ˜ AP 0 RP and ˜ AS0 RSV define the rate of the P- and SV-wave attenuation-coefficient change (increase or decrease) with respect to ln |f/f0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The larger RP and RSV are, the stronger is the dispersion (frequency dependence) of AP and ASV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Therefore, RP and RSV can be called the P- and SV-wave dispersion factors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The SH-wave attenuation coefficient (equation 51) is independent of frequency, with γQ = ˜γQ: ASH = ˜ AS0(1 + ˜γQ sin2 θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (59) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Numerical dispersion analysis Here, we evaluate the frequency dependence of the attenuation coefficients of P- and SV-waves, starting with the dispersion factors RP and RSV (equations 54 and 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As before, we restrict gQ to the realistic range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 < gQ ≤ 3 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Figures 11 and 12 show that gQ = 1 yields the smallest values of RP and RSV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' the dispersion factors and the magnitude of their variation with angle increase with the deviation of gQ from unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Next, we use the medium parameters from Figures 11d and 12d to calculate the exact attenuation coefficients for P- and SV-waves (respectively) at three frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' For the reference frequency f0 = 40 Hz, the term ln |f/f0| in equations 54 and 57 is close to −1 at f = 15 Hz and 1 at f = 109 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' In agreement with equations 53 and 55, the variation of AP with ln |f/f0| between 15 Hz and 40 Hz (and 40 Hz and 109 Hz) is approximately proportional to RP , and the corresponding variation of ASV is approximately proportional to RSV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Figures 13 and 14 show that the frequency dependence of the P- and SV-wave attenuation coefficients AP and ASV is generally mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' However, they may become noticeable for propagation angles close to 45◦ as illustrated in Figures 15 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Both AP and ASV exhibit a more significant variation with frequency for strongly attenuative media (Q33=Q55=20) when gQ ≥ 2 (for P-waves) and gQ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 (for SV-waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 13 gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 0 1 2 3 4 5 6 θ (degrees) RP (%) (a) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 0 1 2 3 4 5 θ (degrees) RP (%) (b) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 0 1 2 3 4 5 6 θ (degrees) RP (%) (c) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 0 1 2 3 4 5 θ (degrees) RP (%) (d) Figure 11: Variation of the P-wave dispersion factor RP (equation 54) with the phase angle for different gQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The reference parameters defined at f0 = 40 Hz are ˜VP 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 km/s, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3, ˜ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2, ˜ AP 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 (corresponding to Q33 = 40), ˜ϵQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 and ˜δQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (a) ˜δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 and ˜δQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (b) ˜δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 and ˜δQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (c) ˜δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 and ˜δQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (d) ˜δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 and ˜δQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 θ (degrees) RSV (%) (a) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 7 6 5 4 3 2 1 0 θ (degrees) RSV (%) (b) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 θ (degrees) RSV (%) (c) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 20 40 60 80 8 6 4 2 0 θ (degrees) RSV (%) (d) Figure 12: Variation of the SV-wave dispersion factor RSV (equation 57) with the phase angle for different gQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The reference parameters defined at f0 = 40 Hz are ˜VP 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0 km/s, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3, ˜ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2, ˜ AS0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 (corresponding to Q55 = 40), ˜ϵQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1 and ˜ϵQ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The parameters ˜δ and ˜δQ are the same as in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 14 f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0135 θ (degrees) AP (a) f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0130 θ (degrees) AP (b) f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0130 θ (degrees) AP (c) f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0130 θ (degrees) AP (d) Figure 13: Variation of the P-wave normalized phase attenuation coefficient with the phase angle at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The medium parameters are the same as in Figure 11d, and (a) gQ = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (b) gQ = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (c) gQ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (d) gQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='012 θ (degrees) ASV (a) f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 θ (degrees) ASV (b) f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 θ (degrees) ASV (c) f=15 Hz f=40 Hz f=109 Hz 0 20 40 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0130 θ (degrees) ASV (d) Figure 14: Variation of the SV-wave attenuation coefficient with the phase angle at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The medium parameters are the same as in Figure 12d, and (a) gQ = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (b) gQ = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (c) gQ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (d) gQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 15 gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0135 f (Hz) AP (a) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='029 f (Hz) AP (b) Figure 15: Variation of the P-wave attenuation coefficient with frequency at θ = 45◦ for different gQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Except for ˜ AP 0, the medium parameters are the same as in Figures 11d and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' On plot (a), ˜ AP 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 (corresponding to Q33 = 40);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' on plot (b), ˜ AP 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='025 (corresponding to Q33 = 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='014 f (Hz) ASV (a) gQ=3 gQ=2 gQ=1 gQ= 1 2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='024 f (Hz) ASV (b) Figure 16: Variation of the SV-wave attenuation coefficient with frequency at θ = 45◦ for different gQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Except for ˜ AS0, the medium parameters are the same as in Figures 12d and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' On plot (a), ˜ AS0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='0125 (corresponding to Q55 = 40);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' on plot (b), ˜ AS0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='025 (corresponding to Q55 = 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Conclusions We obtained concise analytic expressions for the Thomsen-type parameters of constant-Q TI media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' All Thomsen velocity parameters (VP 0, VS0, ϵ, δ and γ) are frequency dependent, with the reference attenuation parameters ˜ AP 0 (proportional to 1/Q33) and ˜ AS0 (proportional to 1/Q55) controlling the dispersion (frequency dependence) of the vertical velocities VP 0 and VS0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The reference attenuation parameters ˜ϵQ, ˜δQ, and ˜γQ govern the variations of the anisotropy parameters ϵ, δ, and γ with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' However, the frequency dependence of all Thomsen velocity parameters is weak in a wide frequency range, even for strong attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' In viscoacoustic constant-Q TI media, the elliptical conditions at the reference frequency ensure that the anellipticity parameter η vanishes for all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Despite the fact that all Qij elements in constant-Q TI media are frequency independent, one of the Thomsen- type attenuation parameters (δQ) does vary with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The frequency dependence of δQ is controlled by the newly defined coefficient ζQ and can be substantial when ζQ has a large magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' As a result, the frequency variation of the P- and SV-wave attenuation coefficients may be non-negligible at oblique propagation angles with the symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' That variation is highly sensitive to the ratio of the vertical quality factors gQ = Q33/Q55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Both attenuation coefficients are insensitive to frequency for gQ = 1, whereas their frequency dependence is most substantial for gQ ≥ 3 (for P-waves) and gQ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5 (for SV-waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' In contrast, the SH-wave attenuation coefficient in constant-Q TI media is frequency-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The constant-Q assumption is often made in attenuation analysis because the effective attenuation coefficients estimated from seismic data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', using the spectral-ratio method) become linear functions of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' However, our results show that this linear dependence may not hold for constant-Q TI models, which can cause errors in the inversion for the attenuation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Appendix A: Complex stiffness coefficients expressed in terms of the Thomsen-type parameters The stiffness coefficients for the constant-Q dissipative VTI model (equations 1–3) can be found at the reference frequency as Mij|f=f0 = ˜ M R ij (1 − i/Qij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Using the parameter definitions in equations 4–12, we express ˜ M R ij and Qij in terms of the reference Thomsen-type parameters as follows: ˜ M R 33 = ρ ˜V 2 P 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1) ˜ M R 55 = ρ ˜V 2 S0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2) ˜ M R 11 = ρ ˜V 2 P 0(1 + 2˜ϵ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3) ˜ M R 66 = ρ ˜V 2 S0(1 + 2˜γ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4) ˜ M R 13 = −ρ ˜V 2 S0 + ρ � ( ˜V 2 P 0 − ˜V 2 S0) � (1 + 2˜δ) ˜V 2 P 0 − ˜V 2 S0 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5) Q−1 33 = 2 ˜ AP 0 1 − ˜ A2 P 0 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6) Q−1 55 = 2 ˜ AS0 1 − ˜ A2 S0 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='7) Q−1 11 = Q−1 33 (1 + ˜ϵQ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='8) Q−1 66 = Q−1 55 (1 + ˜γQ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='9) Q−1 13 = ˜Q−1 33 � 1 + ˜δQf1 + f2 � − Q−1 55 f2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='10) 17 with f1 = ˜ M R 33 ( ˜ M R 33 − ˜ M R 55) 2 ˜ M R 13( ˜ M R 13 + ˜ M R 55) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='11) f2 = ˜ M R 55 ( ˜ M R 13 + ˜ M R 33)2 2 ˜ M R 13( ˜ M R 13 + ˜ M R 55)( ˜ M R 33 − ˜ M R 55) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='12) Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Appendix B: Explicit expressions for sn Here, we provide explicit expressions for the coefficients sn in equation 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' The coefficient s0 is given by: s0 = g(1 − g + χ)2(h0 + h1g + h2g2 + h3g3 + h4g4 + h5g5) (1 − g)3χ3(g − χ)2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='1) where h0 = −(1 + 2˜δ)2χ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='2) h1 = (1 + 2˜δ)(5 + 10˜δ + 2χ), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='3) h2 = (1 + 2˜δ)(2(˜δ − 3)χ − 13˜δ − 14), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='4) h3 = ˜δ(7˜δ + 9χ + 30) + 7χ + 15, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='5) h4 = −˜δ2 − 2(˜δ + 1)χ − 11˜δ − 8, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='6) h5 = 2(1 + 2˜δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='7) For the coefficient s1 we have: s1 = 3g(g − χ − 1)(k0 + k1g + k2g2 + k3g3 + k4g4) 2(1 − g)χ3(g − χ)2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='8) where k0 = −2(1 + 2˜δ)2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='9) k1 = 2 � 1 + χ + 4˜δ(˜δ + χ + 1) � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='10) k2 = 2(χ + 1) − ˜δ(χ + 3), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='11) k3 = −(˜δ + 2χ + 4), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='12) k4 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='13) Finally, the coefficient s2 has the form: s2 = 3g � 3 + 6˜δ + 2χ − 3g(3˜δ + 2χ + 3) + 3g2(˜δ + χ + 3) − 3g3� 2χ3(g − χ)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='14) s3 = (1 − g)2(4χ − 3g) 4χ3(g − χ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content='15) The quantities g and χ are defined in equations 27 and 28, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 18 References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Kolsky, The propagation of stress pulses in viscoelastic solids, Philosophical magazine 1 (8) (1956) 693–710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [2] Kjartansson, Constant Q-wave propagation and attenuation, Journal of Geophysical Research 84 (1979) 4737– 4748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [3] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Greenhalgh, Nearly constant Q models of the generalized standard linear solid type and the corre- sponding wave equations, Geophysics 86 (4) (2021) T239–T260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [4] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Greenhalgh, Nearly constant Q dissipative models and wave equations for general viscoelastic anisotropy, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477 (2251) (2021) 20210170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Behura, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Estimation of interval anisotropic attenuation from reflection data, Geophysics 74 (6) (2009) A69–A74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Shekar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Estimation of shear-wave interval attenuation from mode-converted data, Geophysics 76 (6) (2011) D11–D19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Shekar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Anisotropic attenuation analysis of crosshole data generated during hydraulic fracturing, The Leading Edge 31 (5) (2012) 588–593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Behura, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Jenner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Calvert, Estimation of interval velocity and attenuation anisotropy from reflection data at coronation field, The Leading Edge 31 (5) (2012) 580–587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Best, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Sothcott, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' McCann, A laboratory study of seismic velocity and attenuation anisotropy in near-surface sedimentary rocks, Geophysical Prospecting 55 (5) (2007) 609–625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Zhu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Dewangan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' van Wijk, Physical modeling and analysis of p-wave attenuation anisotropy in transversely isotropic media, Geophysics 72 (1) (2007) D1–D7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Zhubayev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Houben, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Smeulders, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Barnhoorn, Ultrasonic velocity and attenuation anisotropy of shales, Whitby, United Kingdom, Geophysics 81 (1) (2016) D45–D56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Thomsen, Weak elastic anisotropy, Geophysics 51 (10) (1986) 1954–1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Seismic signatures and analysis of reflection data in anisotropic media, Elsevier Science Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=', 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Zhu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Plane-wave propagation in attenuative transversely isotropic media, Geophysics 71 (2) (2006) T17–T30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [15] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Greenhalgh, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Li, Viscoelastic wave propagation for nearly constant Q transverse isotropy, Geophysical Prospecting 70 (7) (2022) 1176–1192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Carcione, Wave fields in real media: Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media: Handbook of Geophysical exploration (3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' ), Elsevier, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [17] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' ˇCerven´y, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Pˇsenc´ık, Perturbation hamiltonians in heterogeneous anisotropic weakly dissipative media, Geophysical Journal International 178 (2) (2009) 939–949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Grechka, Seismology of azimuthally anisotropic media and seismic fracture characterization, Society of Exploration Geophysicists, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Alkhalifah, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Velocity analysis for transversely isotropic media, Geophysics 60 (5) (1995) 1550– 1566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 19 [20] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Hao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Alkhalifah, An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis, Geophysics 82 (1) (2017) C9–C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [21] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Hao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Alkhalifah, An acoustic eikonal equation for attenuating orthorhombic media, Geophysics 82 (4) (2017) WA67–WA81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [22] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Hao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Alkhalifah, Viscoacoustic anisotropic wave equations, Geophysics 84 (6) (2019) C323–C337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Behura, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' Tsvankin, Role of the inhomogeneity angle in anisotropic attenuation analysis, Geophysics 74 (5) (2009) WB177–WB191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQf-v68/content/2301.01939v1.pdf'} diff --git a/ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/2301.11607v1.pdf.txt b/ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/2301.11607v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6cd45c0be30ae2687aa6a7ad22d09ac3f693b55 --- /dev/null +++ b/ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/2301.11607v1.pdf.txt @@ -0,0 +1,1627 @@ +Work flux and efficiency at maximum power of a triply squeezed engine +Manash Jyoti Sarmah and Himangshu Prabal Goswami∗ +Department of Chemistry, Gauhati University, Jalukbari, Guwahati-781014, Assam, India +(Dated: January 30, 2023) +We explore the effects of quantum mechanical squeezing on the nonequilibrium thermodynamics +of a coherent heat engine with squeezed reservoirs coupled to a squeezed cavity. We observe that +the standard known phenomenon of flux- optimization beyond the classical limit with respect to +quantum coherence is destroyed in presence of squeezing. Under extreme nonequilibrium conditions, +the flux is rendered independent of squeezing. The efficiency at maximum power (EMP) obtained +by optimizing the cavity’s squeezing parameter is greater than what was predicted by Curzon and +Ahlborn even in the absence of reservoir squeezing. The EMP with respect to the either of reservoirs’ +squeezing parameters is surprisingly equal and linear in ηC with a slope unequal to the universally +accepted slope, 1/2. The slope is found to be proportional to the dissipation into the cavity mode +and an intercept equal to a specific numerical value of the engine’s efficiency. +I. +INTRODUCTION +One or more quantum systems that operate between +two separate reservoirs make up a Quantum Heat En- +gine (QHE). QHEs have the primary function of convert- +ing heat into work [1–7]. Apart from traditional thermal +reservoirs, the use of non-thermal baths, which are con- +structed reservoirs with correlated characteristics, have +provided a thorough setting for examining the relation- +ship between quantum effects and thermodynamic quan- +tities [8–12]. +Squeezed states or non-canonical initial +states [13–15] are such non-thermal baths which allow +additional control over any quantum systems’ dynamics +garnering tremendous interest off late in the context of +open quantum systems [11, 15, 16]. +Current technologies permit experimental realization +of such states [17] and its effects on the thermodynamics +are experimentally realizable through recently designed +experimental quantum heat engines (QHE)[18–22]. In- +tense efforts have been made to interrogate QHEs on +the role of coherence, correlations or entanglement on +the underlying dynamics [23–26]. +It has already been +demonstrated that certain quantum resources can be ex- +ploited to bend the limits of classical thermodynamics +[16, 27, 28]. Coherence enhanced power and efficiency +and optimization of the flux via quantum coherences in +QHEs are well studied and established phenomena [7, 29– +32]. +Squeezed thermal baths too have proven crucial, +especially in the light of a proof-of-concept experiment +based on a nanobeam heat engine[18]. Efficiency greater +than that of Carnot has also been predicted [20]. +On the theoretical front, quantum thermodynamic +analysis of QHEs s are performed by combining principles +from quantum optics and nonequilibrium statistical me- +chanics [9, 33–35]. In quantum optics, squeezing [36, 37] +generally leads to less observation of quantum noise than +thermal states [38]. +Squeezing alters the entropy flow +associated with the heat exchanged with the system and +∗ hpg@gauhati.ac.in +introduces an additional term proportional to the second- +order coherences which determines the asymmetry in the +second-order moments of the mode quadratures, which +takes into account both the relative variance shape and +the relative optical phase space displacements[33]. This +manifests in an increased efficiency, even surpassing the +Carnot bound [10, 18, 23, 39, 40]. To account for a re- +alistic performance of such QHEs, usually a finite time +assessment is performed by evaluating the efficiency at +maximum power (EMP), originally introduced in a clas- +sical context [41]. From a nonequilibrium quantum sta- +tistical point of view, the near equilibrium EMP is univer- +sally accepted to be ηC/2 [42], with ηC being the stan- +dard Carnot efficiency of a classical heat engine. +Re- +cently, this robust expression has been showed to be in- +valid if the engine is locally optimized [43]. The EMP +has been shown to be modified into several forms as +one keeps changing or introducing or optimizing addi- +tional system parameters [39, 44]. One particularly in- +teresting form of the EMP has been predicted recently +which holds in the presence of squeezed reservoirs, be- +ing equal to η2 +m/(ηm − (1 − ηm) ln(1 − ηm)). Here, ηm +being a squeezing-dependent effective Carnot efficiency +[11]. +However, the validity of such robust thermody- +namic expressions remains questionable when engines op- +erate in presence of both quantum coherences and quan- +tum squeezing since the general framework on which such +studies were based didn’t take such effects into account. +The current work is motivated on this latter aspect. +In this work, we address how the thermodynamics of +a QHE coupled to squeezed cavity respond to reservoir +squeezing in presence of coherences using a quantum mas- +ter equation technique. +Such a technique is standard +and has already been used in nonequilibrium quantum +transport studies with squeezed reservoirs [45–47]. Un- +squeezed dynamics of the engine that we cosider has +also been well studied [7, 31, 48]. +In Sec.(II), we in- +troduce our triple squeezed QHE model and its dynam- +ics. In Sec.(III), we explore the effects of squeezing on +the flux into the cavity mode, which we call the work- +flux. In Sec.(IV), we evaluate the EMP with respect to +three squeezing parameters and a system parameter after +arXiv:2301.11607v1 [quant-ph] 27 Jan 2023 + +2 +which we conclude. +II. +SQUEEZED ENGINE DYNAMICS +The QHE model consists of four quantum levels cou- +pled asymmetrically to two squeezed baths with the up- +per two levels coupled to a squeezed unimodal cavity as +shown schematically in Fig.(1a). Experimentally, similar +QHEs have been realized in cold Rb and Cs atoms us- +ing magneto optical traps [21, 49]. The squeezed density +matrices of the QHE can be written as[46, 50], +¯ρℓ = 1 +Zℓ +exp{−βℓ ˆSℓ ˆHℓ ˆS† +ℓ}, +(1) +¯ρν = 1 +Zν +exp{−βν ˆSν ˆHν ˆS† +ν}, ν = h, c, +(2) +with βz = (kBTz)−1, z = ℓ, h, c being the inverse temper- +atures of the cavity, hot and cold reservoirs respectively. +ˆS( ˆSν) is the squeezing operator on the squeezed cavity’s + + + + +0 +1 +2 +3 +4 +5 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +xc +Ρ12 +ss +0 +1 +2 +3 +4 +5 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +xh +Ρ12 +ss +0 +1 +2 +3 +4 +5 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +x +Ρ12 +ss +a +b +d +c +Th + +Tc +xh +xc +x +(a) +(b) +(c) +(d) +FIG. 1. (Color online) a) Level scheme of the model quan- +tum heat engine. A pair of degenerate levels |1⟩ , |2⟩ is reso- +nantly coupled to two excited levels |a⟩ and |b⟩ by two ther- +mally populated squeezed field modes with hot (Th) and cold +(Tc) temperatures. +Levels |a⟩ and |b⟩ are coupled through +a squeezed cavity mode of frequency νℓ . Emission of pho- +tons into this squeezed cavity is the work done by the QHE. +The engine parameters are fixed through out the manuscript +at E1 = E2 = 0.1, Eb = 0.4, Ea = 1.5, g = 1, r = 0.7 and +τ = 0.5 in the unit of kB → 1 and ¯h → 1. b) The solid (dot- +ted) curves represent the steadystate coherence, ρss +12 (solved +by setting the RHS of Eq.(8)=0) as a function of the b) cold +bath squeezing parameter xc evaluated at different values of +xh = 0, 0.5, 1, 2, bottom to top with x = 1 (x = 0), c) hot +squeezing parameter, xh with xc = 0, 0.5, 1, 2, bottom to top +and x = 1 (x = 0), d) cavity squeezing,x with the solid curves +(bottom to top) evaluated at xh = 0, xc = 0, 0.5, 1, 2. The +dotted ones represent xc = 0, xh = 0, 0.5, 1, 2. +mode (reservoirs’ modes) given by : +ˆSℓ = e +1 +2 (xˆa†2 +ℓ −h.c), +(3) +ˆSν = +� +k +e +1 +2 (λ∗ +kνˆa†2 +kν−h.c), +(4) +λkν = xkνeiθkν, xkν > 0. +(5) +θkν and xkν are the squeezing parameters of the reservoirs +and x is the squeezing parameter [46, 47, 50, 51]. ˆHℓ = +ϵℓˆa† +ℓˆaℓ is the Hamiltonian for the cavity mode and ˆHν = +� +k ϵkνˆa† +kνˆakν is the Hamiltonian for the ν-th reservoir. +The total Hamiltonian of the four level QHE is ˆHT = +� +ν = 1,2,a,b Eν|ν⟩⟨ν|+ ˆHℓ+ ˆHν+ ˆVsb+ ˆVsc, with the system- +reservoir and system-cavity coupling Hamiltonians given +by, +ˆVsb = +� +k ∈ h.c +� +i = 1,2 +� +x = a,b +rikˆak|x⟩⟨i| + h.c +(6) +ˆVsc = gˆa† +ℓ|b⟩⟨a| + h.c. +(7) +ϵk, ϵℓ and Eν denote the energy of the kth mode of the +two thermal reservoirs, the unimodal cavity and system’s +νth energy level respectively. The system-reservoir cou- +pling of the ith state with the kth mode of the reservoirs +is denoted by rik. +ˆa†(ˆa) are the bosonic creation (an- +nihilation) operators. +The radiative decay originating +from the transition |a⟩ → |b⟩ is the work done by the +engine. +Unsqueezed version of such a QHE has been +thoroughly studied using a Markovian quantum mas- +ter equation [7, 30, 31, 48, 52]. Following such a stan- +dard procedure to derive of a quantum master equation +[46, 48] for the matrix elements of the reduced density +matrix ρ (supplementary information) has four popula- +tions, ρii, i = 1, 2, a, b coupled to the real part of a co- +herence term, ρ12. The coherence ρ12 between states |1⟩ +and |2⟩ arise due to interactions with the hot and the +cold baths. This thermally induced coherence couples to +populations due to transition involving the states |1⟩ and +|2⟩. Under the symmetric coupling regime, we can now +write down five coupled first order differential equations +describing the time-evolution of the four populations and +the coherence (under symmetric coupling, r), given by +˙ρ12 = −ry +2 ρ11 − ry +2 ρ22 + rph ˜Nhρaa + rpc ˜Ncρbb +− r(n + τ)ρ12 +(8) +˙ρii = −rnρii + r ˜Nhρaa + ˜Ncρbb − ryρ12, i = 1, 2 (9) +˙ρbb = rNcρ11 + rNcρ22 + g2 ˜Nℓρaa +− (g2Nℓ + 2r ˜Nc)ρbb + 2rpcNcρ12 +(10) +˙ρaa = rNhρ11 + rNhρ22 − (g2 ˜Nℓ + 2r ˜Nh)ρaa ++ g2Nℓρbb + 2rphNhρ12 +(11) +with, � +i ρii = 1, i = 1, 2, a, b and n = Nc + Nh, y = +Ncpc + Nhph, with the reorganized occupation factors + +3 + + +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +t +Ρij +�a� +Ρ11,Ρ22�black� +Ρaa�green� +Ρbb�blue� +Ρ12�brown� +0 +1 +2 +3 +0.10 +0.15 +0.20 +0.25 +0.30 +x +Ρij +ss +x +0 +1 +2 +3 +1.00 +1.01 +1.02 +1.03 +1.04 +1.05 +1.06 +1.07 +x +Ρbb +ss �Ρaa +ss +x +0 +1 +0.96 +0.97 +0.98 +0.99 +1.00 +1.01 +1.02 +1.03 +ph +j�jo +ph +�d� +(b) +(c) +(a) +(b) +(d) +(c) +FIG. 2. +a) The solid (dotted) curves represent time evo- +lution of ρij with, x = 2 (without, x = 0) squeezing ob- +tained by solving Eq.(8-11) for Th = 2, Tc = 0.5, Tl = 0.9. +b) Steady state values as a function of the squeezing pa- +rameters for the same parameters as (a) c) Ratio of the +steady state values between states |b⟩ and |a⟩ reaching unity +highlighting the equipopulated nature under high squeezing; +pc = 0.2, 0.3, 0.5, 0.7, 0.8 from the top to the bottom curves. +(d) Optimization of the flux ratio as a function of hot coher- +ence parameter, ph for different squeezing parameters under +far from equilibrium conditions and pc = 1 (top to bottom: +x = 0, π/6, π/π/2, 2π/3, 5π/6, π, 3π/2). Other parameters are +same as Fig.(1a). +given by +Nz = cosh(2xz)(nz + 1 +2) − 1 +2, z = h, c, +(12) +Nℓ = cosh(2x)(nℓ + 1 +2) − 1 +2. +(13) +Here, nc, nh andnl are the Bose-Einstein distributions +for the cold reservoir, hot reservoir and the cavity re- +spectively. These factors are now squeezing dependent +via the dimensionless parameters, xh, xc and x represent- +ing the extent of squeezing in the hot, cold reservoirs +and the cavity respectively. pν = | cos φν|, ν = h, c are +two dimensionless parameters that governs the strength +of coherences and whose values are dictated by the an- +gles of relative orientation (φν) of the ν−th bath induced +transition in the system [7, 48, 52]. A phenomenological +dimensionless rate τ has been added to take care of the +dephasing. Setting ˙ρ = 0, at the steady state, we can +solve for the steady state values of ρaa, ρbb, ρ11, ρ22, and +ρ12 and obtain these analytically (supplementary text). +The steadystate value of the coherence term ρss +12 as a +function of the squeezing parameters, xh, xc and x are +shown in Fig.(1b,c,d)) for different engine parameters. +The different curves in Fig.(1b) represent ρss +12 evaluated +for different xh and x values as a function of xc. The +solid (dotted) lines represent ρss +12 when xh ̸= 0(xh = 0) +and x = 0(x ̸= 0). At high xc values, the coherence is re- +duced and saturates to a lower value in comparison to ρss +12 +values of lower xc. At high xh values (black curve), ρss +12 +steadily increases and reaches a maximum value around +some intermediate xc value and then sharply drops as +0 +1 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +j� jo +ph +0 +1 +1 +ph +j� jo +ph +FIG. 3. +Failure of coherence to optimize the flux beyond +classical values (j/j0 > 1) under high squeezing as given by +Eq.(17). Inset: Linear dependence of the flux ratio on p − j +under high squeezing (x ≫ 0) and Tl ≫ 0 given by Eq.(18) +evaluated at pc = 1, Tc = 0.5, Th = 1. +The square boxes +represent linear fit. +xc keeps increasing. This behavior is however absent for +lower xh values. Fig.(1c) represent ρss +12 evaluated for dif- +ferent xc and x values as a function of xh. +The solid +(dotted) lines represent ρss +12 when xc ̸= 0(xc = 0) and +x = 0(x ̸= 0). At high xh values, the steady state values +of the coherence term increases and saturates to a higher +value in comparison to coherence at lower xh values. We +can rationalize that, xc(xh) tend to reduce (increase) the +steadystate values of the coherences as we keep squeez- +ing the baths more and more. The same however cannot +be said for ρss +12 vs x as seen from Fig. (1d). The solid +(dotted) lines represent the behavior at xh = 0(xh ̸= 0) +for finite xc values. +The time evolution of each of the equations (Eq.(8-11)) +for various engine parameters for xh = xc = 0 and x = 2 +is shown in Fig.(2a). In Fig.(2b), the steadystate values +of the populations as a function of x is shown where solid +(dotted) curves represent cavity-squeezed, x ̸= 0 (cavity- +unsqueezed, x = 0) evolutions. +Note that under high +squeezing of the cavity mode, the steady state values, +ρss +aa and ρss +bb equipopulate giving, +lim +x→∞ +ρss +bb +ρss +aa += 1 +(14) +and is shown numerically in Fig.(2c) for different values +of the hot coherence parameter, ph. The analytical ex- +pressions for the steadystate values are provided in the +supplementary information. +III. +WORK FLUX +We interprete the emission of photons into the +squeezed cavity as the work done by the engine. This +photon exchange process between the levels |a⟩, |b⟩ with +the squeezed cavity is quantified by the rate of photon +exchange with the cavity which we refer to as the work +flux, j = +d +dt⟨a† +ℓaℓ⟩, where the trace is with respect to + +4 +0 +1 +0.6 +0.7 +0.8 +0.9 +1.0 +ph +j�jo +ph +�a� +0.7 +0.8 +0.9 +1 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +pc +ph +� +Tc�Colour +0.1�Pink +0.2�Blue +0.3�Red +0.4�Orange +0.5�Blue +1.0�Black +pc +�b� +0 +1 +2 +3 +0.6 +1 +1.5 +x +ph +� +x +�c� +0 +1 +2 +3 +0.2 +0.4 +0.6 +0.8 +1.0 +x +Ζ�Ζx�0 +Tl�10 +Tl�0.5 +Tl�1 +Tl�2.5 +x +�d� +FIG. 4. +a) Loss of optimization of flux as a function +of ph for different squeezing parameters, near equilibrium +(Tc = 0.9, Th = 1, Tl = 10). (b) Loss of linear dependence of +pc on the optimal value p∗ +h as given by Eq.(21). The topmost +curve represents Eq.(22). (c) Plot showing breakdown of the +coherent optimization of the flux as a function of squeezing +parameter. The shaded region is not allowed since the maxi- +mum possible value of p∗ +h is unity. Under far from equilibrium +condition p∗ +h exists which saturates (bottom curve) at higher +values of x given by Eq.(21). The top curve shows the behav- +ior of p∗ +h near equilibrium which is nonexistent after a certain +squeezing value. (d) (d) Lowering of thermodynamic affinity +as a function of squeezing evaluated at Tc = 0.1, Th = 2 and +xc = xh = 0. +the squeezed cavity density matrix. +Following a stan- +dard procedure to second order in the coupling as devel- +oped in[30, 48] we get, j = g2( ˜Nℓρss +aa − Nℓρss +bb). We can +substitute the values of the steadystate populations to +obtain an analytical expression for the flux (supplemen- +tary information). When, the hot and the cold coherence +parameters individually go to zero (pc = ph = 0), the +coherence vanishes (ρss +12=0) and we obtain a coherence - +unaffected value of the flux, which we denote as jo. Note +that, jo depends on the squeezing parameters x, xh and +xc. In the absence of squeezing (xh = xc = x = 0), jo +shall be denoted by j0 +o, which we refer to as the classical +value of the flux. There are no effects of coherence or +squeezing on j0 +o. It is a well known phenomena that, in +absence of squeezing, j > jo can be achieved as a func- +tion of coherence parameter, ph [7, 29]. We plot the ratio +j/jo in shown in Fig.(2d) for different squeezing values +of the cavity for xc = xh = 0. As the cavity squeezing +parameter is increased the optimal value of the flux grad- +ually decreases and the ph value that optimizes the ratio +(denoted as p∗ +h) shifts towards larger ph values. We now +attempt to explore the dependence of the flux in presence +of squeezing on the coherences in detail. Since the ana- +lytical expressions of j and j0 +o are too lengthy we focus +on some limiting cases. +Under high cavity squeezing, (x → ∞), we obtain +ρaa +aa = ρss +bb as seen from Eq.(14). The expression for the +flux in this case is simply given by, +lim +x→∞ j = g2( lim +x→∞ ρss +aa), +(15) +which under the condition pc = 0, ph = 0 in Eq.(15) is, +lim +x→∞ jo = r(Nh − Nc) +2(n + 1) +. +(16) +Eq. (15), with pc = 1 can be expressed as, +lim +x→∞ j|pc=1 = r(Nh − Nc) +� +Nh +� +1 − p2 +h +� ++ t +� +(1 − ph)fn + 2τ(n + 1) +(17) +with fn = 4NcNh + n(2Nh(ph + 1) + ph + 2). The RHS +of Eq.(16) is always greater than RHS of Eq.(17) as seen +from the numerical result in Fig (3). The physical in- +terpretation is that the coherences are no longer able to +increase the flux beyond the non coherence values. Un- +der this condition, the ratio is bounded below unity as +seen in Fig.(3). We can analytically prove this by invok- +ing a few conditions. In Eq.(16) and (17), if τ = 0 and +Nh = zNc, z being a positive integer), the ratio between +the two fluxes becomes, +lim +x→∞ j|pc=1 +lim +x→∞ jo +���� +Nh=zNc += +2z(ph+1)(Ncz+ Nc+ 1) +2Nc(ph+1)z2+z(4Nc+ ph+2)+1 +(18) +which is a rational fraction of two linear terms of ph. +Eq.(18) can be shown to have a linear dependence on ph +for some appropriate conditions of the coefficients which +is graphically shown as an inset in Fig.(3). In Eq.(18), +for z = 1 and Nh = Nc (no bias), we see a flux value that +solely depends on only the coherence value, given by +lim +x→∞ +j +jo +|Nh=Nc = 2(1 + ph) +(3 + ph) +(19) +≤ 1 +(20) +and is linear in ph for small values as seen in the inset +of Fig.(3) and in Fig.(4a). +In Fig.(4a), the flux ratio +j/jo is plotted for different squeezing parameters. The +squeezing decreases from top to bottom. For smaller ph, +the linearity is prominent, but for higher ph values, the +linearity is gradually less apparent as the squeezing pa- +rameter increases. +It has been previously reported that p∗ +h increases lin- +early in pc under the unsqueezed case [30]. In the current +case, we observe that under an extremely biased scenario +(Nh ≫ 0) and high squeezing, x ≫ 0, the linear depen- +dence is lost as shown graphically in Fig.(4b) and the +dependence of p∗ +h on the cold coherence parameter, pc is +given by the nonlinear function, +p∗ +h| = +� +(1−p2c) (4N 2c (1−p2c)+4Nc+1)+2Nc +� +p2 +c+1 +� ++1 +4Ncpc + pc +(21) +which reduces to unity when pc = 1 as seen in the Fig. +(4b). The nonlinear dependence takes a simplistic form +when Tc → 0, where the above expression reduces to, +p∗ +h|Tc=0 = 1 − +� +1 − p2c +pc +(22) + +5 +which is shown as the topmost curve in Fig.(4b). The +RHS of Eq.(21) also has a strange dependence on the cav- +ity squeezing parameter. p∗ +h increases as a function of x +and saturates at higher x values as shown in the bottom- +most curve of Fig.(4c). However under extremely biased +conditions, p∗ +h sharply rises beyond unity and goes to the +shaded region. The shaded region is not allowed as the +maximum value of p∗ +h is unity. Since an analytical expres- +sion of p∗ +h as a function of x is beyond the scope of sim- +plistic analysis, the exact identification of this numerical +fallout range is not possible. We simply speculate that +such a breakdown happens when the cavity temperature +Tℓ is set to be very high. Since nℓ is a function of Tℓ, the +numerics blows when there is competition between x and +Tℓ to dominate the behavior. The upper dashed curve in +the shaded portion also corresponds to an unrealistic p∗ +h +evaluated at a high cavity temperature. In Fig.(4d), we +plot the thermodynamic force as a function of squeezing. +The force can be identified from the analytical expression +of the flux (supplementary text) and is given by, +ζ = +˜Nc ˜ +NℓNh +Nc ˜NhNℓ +. +(23) +When ζ > (<)1, j > (<)1. In Fig.(4d), we plot the ratio +between the thermodynamic forces in presence and ab- +sence of squeezing for different cavity temperatures. As +squeezing increases, the ratio decreases for a fixed set +of engine parameters and then saturates. This leads to +lower magnitude of the flux in comparison to the un- +squeezed case and is more prominent when the cavity +temperature is low. +In Fig.(5a,b and c), we plot the ratio between the total +flux j and the classical flux j0 +o as a function of xc, xh and +x respectively for the same parameters as Fig.(2). +As +a function of both the baths’ squeezing parameters, the +increase of the total flux is quite large in comparison to +the classical case. All of the curves show saturation be- +havior. Particularly interesting is the ratio’s dependence +on xh where the saturation value of the ratio is always +greater than unity. +We now focus on an extreme biased case (Th ≫ Tc, a +limit which we invoke by taking Th → ∞ and Tc → 0), +a scenario when the temperature gradient is very high. +This case is different from a standard extreme nonequilib- +rium case where the thermodynamic force must be very +high (ζ ≫ 0). Under the high temperature gradient sce- +nario, the steadystate populations of the upper two states +are given by, +lim +Th≫Tc ρss +aa = +� +p2 +h + 1 +� � +g2Nℓ + 2r +� +g2 (4Nℓ + p2 +h + 1) − 2 (p2 +h − 3) r +(24) +lim +Th≫Tc ρss +bb = +g2 ˜Nℓ +� +p2 +h + 1 +� +g2 (4Nℓ + p2 +h + 1) − 2 (p2 +h − 3) r, +(25) +which no longer depends on the squeezing parameters of +the two baths. Using these above values the flux can be +(a) +(b) +(d) +(c) +FIG. 5. Giant increase of the total flux (in presence of squeez- +ing as well as coherence) in comparison to the classical case. +The solid (dotted) lines represent the ratio between the to- +tal flux j and the classical flux j0 +o as a function of a) xc +evaluated at x = 0(1), xh = 0, 0.5, 1, 2, b) xh evaluated at +x = 0(1), xc = 0, 0.5, 1, 2. c) Solid (dotted) curves indicate the +total flux ratio as a function of cavity squeezing x evaluated +at Tc = 0.5(0.1) with {xh, xc} = {0.5, 0.1}, {0.1, 0.5}, {0, 0} +(top to bottom). (d) Change in the sign of the thermody- +namic affinity, A = log ζ as function of cavity squeezing pa- +rameter evaluated at{xh, xc} = {1, 0.1} (upper curve) and +{0.1, 1} (lower curve). The sign change happens at x∗ given +by Eq.(32). +recast as, +lim +Th≫Tc j = +2g2r ˜Nℓ(1 + p2 +h) +g2(1 + 4Nl + p2 +h) − 2r(p2 +h − 3) +(26) +while the coherence-unaffected value of the flux is simply, +lim +Th≫Tc jo = +2g2 ˜Nℓr +g2(1 + 4Nℓ) + 6r +(27) +It is interesting to note that, in this highly biased sce- +nario, the flux expression (RHS of Eq.(26)) doesn’t de- +pend on the cold coherence parameter any more. In the +above two expressions, if we invoke the high squeezing +scenario (x → ∞), we can write down the ratio between +the two fluxes as, +lim +x→∞ +lim +Th≫Tc j +lim +Th≫Tc jo += (1 + p2 +h) +(28) +Note that, the above expression is bound, 1 ≤ 1 + p2 +h ≤ +2. +In this limit with ph = 1(pc ̸= 1), coherences can +double the value of the flux from its zero coherence value. +Likewise, the ratio between the flux in this limit and the +classical value of the flux can be written as, +lim +x→∞ +lim +Th≫Tc j +lim +Th≫Tc j0 +o += (1 + p2 +h)(1 + 6r − 3g2 +4g2˜nℓ +) +(29) +≥ 1. +(30) +As long as r > g2/2 and pc ̸= ph, within the high bias +scenario and maximal cavity-squeezing, the flux is always +greater than unity in comparison to the classical case. + +20 +15 +15 +10 +10 +5 +5 +0 +0 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +Xc +Xh +3.0 +1.5 +2.5 +1.0 +0, 2.0 +0.5 +A +0.0 +1.0 +0.5 +0.5 +0 +1 +2 +3 +4 +5 +6 +0.0 0.5 1.0 1.5 2.0 2.5 3.0 +X +X6 +0 +0.7 1.2 +2 +2.5 +3 +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 +1.30 +x +W�Wo +Tl�0.5 +Tl�0.4 +Tl�1.0 +Tl�10 +x +�a� +0 +0.7 1.2 +2 +2.5 +3 +�1 +0 +1 +2 +3 +x +W�Wo +Tc�0.1 +Tc�2.0 +Tc�0.4 +Tc�0.7 +x +�b� +0 +0.2 +0.7 +1 +0.990 +0.995 +1.000 +1.005 +ph +ΗEa +� �Ηo +� +x�0.5 +x�0 +x�1 +x�2Π +�c� +ph +0 +1 +2 +3 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +x +ΗEa +� �Ηo +� +Tl�0.9 +Tl�1.0 +Tl�1.2 +Tl�1.7 +x +�d� +FIG. 6. +(a) Squeezing induced increase of the work done +beyond classical limits (Tc = 0.1, Th = 1). The increase is +larger when the cavity temperature is lower. +(b) Negative +work done as a function of squeezing for different Tc( Th = +2, Tl = 1). (c) EMP with respect to Ea as a function of ph +for different squeezing values. (d) EMP with respect to Ea +for the range of squeezing at different cavity temperatures +(pc = 0.1, ph = 1). +IV. +EFFICIENCY AT MAXIMUM POWER +We now move to perform a thorough analysis on the +efficiency at maximum power (EMP or η∗). In a stan- +dard context, the EMP is calculated by maximizing the +efficiency with respect to a system parameter. +In our +QHE model, the efficiency is defined as η = W/Qh with +Qh = (Ea −E1), and the useful work done (W) is defined +as, +W = Ea − Eb − WdissTc, +(31) +with Wdiss = kBln +˜ +Nℓ +Nℓ is the dissipation into the cavity +mode [30, 48]. W doesn’t depend on the squeezing pa- +rameters of the two squeezed reservoirs or the noise in- +duced coherences. In Fig. (6a), we show the variation of +W/Wo (Wo being the useful work in absence of squeez- +ing, x = 0) as a function of x for several values of the +cavity temperature, Tl. As can be seen, the work done +increases as Tl is lowered and saturates at higher values +of x and is always greater than unity as long as Tc > Tℓ. +When Tc < Tℓ (Fig.(6)b), the work done is negative. In +general, the work changes its sign at x = x∗, given by +x∗ = 1 +2ℜ +� +cosh−1 +� +˜NcNh + Nc ˜Nh +(2nℓ + 1)(Nc − Nh) +�� +. +(32) +Although W and η are independent of coherences and +the reservoir squeezing parameters, the EMP however de- +pends on these parameters. The EMP obtained by max- +imizing P with respect to any system parameter puts an +implicit dependence via the optimized value of the chosen +parameter. We choose the three squeezing parameters +xc, xh, x and Ea to optimize the EMP and denote these +by η∗ +xc, η∗ +xh, η∗ +x and η∗ +Ea respectively. The squeezing unaf- +fected values of the EMP are denoted by η∗ +o. In Fig.(6c), + + +0 +0.2 +0.8 +1 +0.990 +0.995 +1.000 +1.005 +1.010 +ph +Ηx +��Ηo +� +�a� +0.6 +0.8 +0.40 +0.45 +0.50 +0.55 +0.60 +Ηx +� +�b� r � g +Η�2 +Η��2�Η� +ΗC +Ηc +0.6 +0.75 +0.9 +0.40 +0.45 +0.50 +0.55 +0.60 +Ηx +� +�c� r � g +Η��2�Η� +ΗC +Ηc +0.6 +0.7 +0.8 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +ΗEa +� +�d� +ΗC +Η��2�Η� +(c) +(a) +(b) +(b) +(d) +FIG. 7. (Color online)(a) EMP with respect to squeezing as a +function of ph fr various pc. In a), b) and c), the black curves +(overlayed with red color) represent the evaluated EMP of +our QHE. The green dashed curve is the upper bound on +the EMP, η∗∗. The brown dashed line represents ηCA. The +dotted line represent ηL. (b) and (c) EMP with respect to x +as a function of ηC with r = 0.7, g = 1) and r = 0.1, g = 3 +respectively. When r ≈ g, η∗ +x > ηCA as seen in (b). (d) EMP +with respect to Ea as a function of ηC with r = 0.7, g = 1). +Here, η∗ +Ea > ηCA with x = 1(xc = xh = 0). +we show the dependence of the ratio η∗ +Ea/η∗ +o as a function +of ph for several x-values evaluated at xc = xh = 0 and +pc = 0.9. The dependence of this ratio on ph is extremely +nonlinear and is unity at ph = 0.8 where effects of coher- +ence vanish. At lower (higher) squeezing values, the ratio +decreases (increases) to unity and then sharply increases +beyond unity as a function of ph. We can theorize that, +lower ph values (under the condition ph < pc), smaller +values of cavity squeezing favor increasing the EMP be- +yond classical values while for larger ph (ph > pc), high +squeezing favor increase of the EMP beyond classical val- +ues. In Fig.(6d), we plot the same ratio as a function of +cavity squeezing parameter for different cavity tempera- +tures, Tl. There is an optimization of the EMP at lower +values of x and the hump keeps shifting leftward to even +smaller values as Tl is increased and the EMP ratio keeps +decreasing. From Fig.(6d), we can conclude that lower +values of Tℓ yield very high values of EMP with respect +to Ea under moderate squeezing conditions of the cavity. +In Fig. (7a), we plot η∗ +x as a function of ph for differ- +ent combinations of xc and xh for a fixed pc value (0.5). +Here, for a fixed set of engine parameters, when xc < xh +leads to a larger optimized value (around ph = 0.5) of the +EMP with respect to x (blue curve in the figure). How- +ever as ph approaches unity, there is a sharper fall in the +EMP and goes below unity. For the case when xc = xh, +the behavior is similar (dotted curve) but the increase is +not as high as the previous case. When squeezed to the +limits, xc → ∞, xh → ∞, the EMP with respect to x no +longer depends on the coherence (dashed curve). This +is due to the fact that, under this scenario, the power +cannot be optimized with respect to x and the maximum +value occurs at x = 0. +In general, the EMP has a universally accepted for- +mula, the Curzon-Ahlborn EMP, ηCA = 1 − √1 − ηC + +7 + + +0 +0.3 +0.6 0.8 +1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ηc +Ηxc +� +Ηc +0 +0.3 +0.6 0.8 +1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ηc +Ηxh +� +Ηc +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.55 +0.60 +0.65 +0.70 +0.75 +Tc +Η,Η� +0.0 +0.2 +0.4 +0.6 +0.8 +0.65 +0.70 +0.75 +0.80 +0.85 +Ηc +ΗEa +� +(a) +(b) +(d) +(c) +FIG. 8. +Linear dependence of η∗ +xc(a) and η∗ +xh(b) as a func- +tion of ηC, governed by Eq.(33) evaluated at x = ∞, 1 and 0 +(top to bottom ). Note that η∗ +xh = η∗ +xc with the upper (mid- +dle) curves having a slope of m = 0.02(0.19) and intercept +of c = 0.76(0.59). c) Solid line represents the EMP, given by +Eq.(34) while the dotted line is simply the normal efficiency, +η = W/Qh. d) Appearance of a quadratic term and an in- +tercept for η∗ +Ea as a function of ηC, evaluated at x = 1.5(∞) +denoted by lower (upper) curves. The fit parameters for the +upper (lower) curves are a1 = 0.85(0.76), a2 = 1.7(0.75), a3 = +0.05(−0.28), a4 = 0.07(−0.28). +[41, 53] and is represented by the dashed curves in +Fig.(7b,c and d). As a function of ηC, the EMP is bound +between ηC/2 ≤ η∗ ≤ η∗∗, where the upper bound is +η∗∗ = +ηC +2−ηC [54]. In Fig.(7b,c and d), we show the be- +havior of our engine’s EMP as a function of the Carnot +efficiency, ηC. +The solid (topmost green) curve repre- +sent the upper bound η∗∗. The EMP of the QHE op- +timized with respect to x for xc = xh = 0 is repre- +sented by the solid line highlighted with red dots. +In +Fig.(7b,c), η∗ +x ≥ (<)ηCA is observed under the condition +r ≥ (<)g. +Values of EMP larger than ηCA has been +previously reported with squeezed reservoirs [17, 20]. In +our case, one can have EMP more than the predicted +ηCA just by squeezing the cavity even in the absence +of squeezed reservoirs. In Fig.(7d), for nonzero values of +cavity-squeezing, η∗ +Ea > ηCA is shown (solid black curve). +This result is valid irrespective of r and g values. The +upper bound is always obeyed in presence of squeezing +as evident from Fig.(8b,c and d). The EMP of the QHE +is always lower than the upper dashed curve (η∗∗). Note +that the universal slope of 1/2 (any EMP = ηC/2 near +equilibrium)[42] is maintained in all the curves for smaller +values of ηC when maximized with respect to x. +We now move to discuss a rather interesting finding +observed when the EMP is maximized with respect to +a reservoir squeezing parameter. As can be seen from +Fig.(8a and b), both η∗ +xh and η∗ +xc are found to be linear +in ηC with a slope which is not equal to the universally +predicted value of 1/2[53]. By a linear curve fitting tech- +nique, we infer that the EMP with respect to xc or xh is +dictated by the equation, +η∗ +xh = η∗ +xc = mηC + c. +(33) +Our numerical results reveal that the slope, m is equal +to the numerical value of Wdiss/Qh and the intercept, c +being given by the numerical value of the quantity, (Eab− +Wdiss)/Qh. This intercept is interestingly the efficiency +of the engine albeit with Tc = 1. Note that, η∗ +xc = η∗ +xh +and is shown as two identical plots in Fig.(8a,b). In these +two figures. The numerical plots reveal that the m ̸= +1/2. Such a breakdown of the universality of the linear +coefficient has also been observed in presence of geometric +phaselike effects [52, 55]. +Since Wdiss > 1, the EMP +increases as x is increased (for fixed Tℓ) to a maximum +value of Eab/Qh at ηC = 1. The efficiency of the QHE, +η = W/Qh is always less than η∗ +ν and is shown as a +function of Tc in Fig.(8c). +This linear dependence doesn’t exist for η∗ +Ea for finite +x as seen from the numerical results in Fig.(8d) for x = 1 +and x → ∞. It has been previously reported that such +a nonlinear dependence of the EMP on the squeezing +parameter x takes the form η∗ +∗ = 1 − +� +sech(2x)√1 − ηC +[56]. We assess the validity if this expression by defining +two curve fitting equations, +η∗ +Ea ≈ a1 − +� +sech(a2x)√a3 − a4ηC +(34) +≈ a5ηC + a6η2 +C + c +(35) +that can best represent the EMP with respect to the sys- +tem parameter Ea. Here, ai-s are fit parameters. We +observe that a1 ̸= a3 ̸= a4 ̸= 1 and a3 ̸= 2 result- +ing in η∗ +Ea ̸= η∗ +∗ and is shown in Fig.(8d). Further, in +Eq.(35), a5 ̸= 1/2 and a6 ̸= 1/8. In this engine, it is al- +ready known that the quadratic coefficient is not 1/8 [30]. +Both the above equations are good fits (solid curves) on +the numerically evaluated η∗ +Ea (dots) as function of ηC +as seen in Fig.(8d). It is interesting to note that the in- +tercept of η∗ +Ea as a function of ηC in Eq.(35) is the same +numerical value of the engine’s efficiency of the engine, +η = W/Qh similar to what was observed in Eq.(33). This +lets us rationalize that Eq.(35) is a better representation +of η∗ +Ea vs ηC than Eq.(34). At ηC = 1, η∗ +Ea again reaches +a maximum value of Eab/Qh. For x = 0, m = 1/2 is +recovered. Further for x = 0, the intercept in Eq.(34) +also vanishes by mixing with the quadratic term. Since +we cannot derive analytical expressions for these coeffi- +cients, we demonstrated it this numerically shown as the +bottom-most dotted line in Fig.(7d)). +The EMP also has other interesting logarithmic +expressions[43, 57, 58], one particularly claimed to be +valid for squeezed states[11], η∗ +L = η2 +m/{1−(1−ηm) ln(1− +ηm)}. ηm is a modified Carnot efficiency given by ηm = +1−Tc/T m +h . T m +h is a modified but fictitious reservoir tem- +perature and is directly proportional to the energy of the +squeezed mode and inversely proportional to the logarith- +mic ratio of the squeezed mode’s occupation factor. By +an analogy with this previous work [11], we can express +the modified temperature in our QHE to be, +T m +h = Ea − E1 +ln 1+Nh +Nh +. +(36) + +8 +0 +0.3 +0.6 +0.8 +1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ηc +ΗEa +� +Ηc +FIG. 9. +Disagreement between the QHE’s EMP optimized +with respect to Ea and the predicted EMP, η∗ +L for the same +parameters. η∗ +L is evaluated using the definition in Eq.(36). +The dotted (dashed) curves represent η∗ +Ea(η∗ +L). Parameters +used are Th = 3, xc = xh = 0.1, x = 0.6 (top dotted), Th = +4, xc = xh = 0.2, x = 0.5 (middle dotted) and Th = 6, xc = +xh = 0.2, x = 2π. +We numerically evaluate ηE∗ +a for different squeezing pa- +rameters and Th values and plot it in Fig.(9) along side +the corresponding η∗ +L values. As can be seen, η∗ +Ea ̸= η∗ +L. +Further since η∗ +xh and η∗ +xc is found to be linear in ηC, these +anyway don’t agree with the predicted value η∗ +L. +Un- +der extremely low squeezing conditions of the hot bath, +ηm +C → ηC in the expression for η∗ +L. Under this condition, +η∗ +L has been high lighted as dotted curves in Fig.(7b,c +and d) and is seen to be unequal to η∗ +x. +V. +CONCLUSION +By deriving a coherence-population coupled quantum +master equation, we carried out a comprehensive study +of the thermodynamics of quantum heat engine coupled +to two squeezed reservoirs and a squeezed unimodal cav- +ity. +We showed that the steadystate value of the co- +herence term of the density matrix vanishes (saturates) +under maximal squeezing of the cold (hot) bath. Under +high squeezing conditions of the cavity, the two upper +states of the engine equipopulate. We showed that under +high squeezing of the cavity, the quantum coherence can +no longer optimize the flux beyond the classical values. +We also showed how the flux can be linearized with re- +spect to coherences under high squeezing conditions and +equal Bose-Einstein distributions for the hot and cold +baths. We also showed that larger EMP favors lower val- +ues of cavity temperatures and lower values of squeezing. +The EMP can be increased beyond the Curzon-Ahlborn +limit by squeezing the cavity alone even if the baths are +unsqueezed. +We also show a linear dependence of the +EMP with respect to the reservoirs’ squeezing parameters +which we identify analytically with a slope proportional +to the dissipation into the cavity mode. The EMP with +respect to a system parameter, Ea doesn’t obey the uni- +versal slope of 1/2 for finite squeezing and is not equal to +a recently proposed general form of the EMP in presence +of squeezed reservoirs [11]. +ACKNOWLEDGMENTS +MJS and HPG acknowledge the support from Science +and Engineering Board, India for the start-up grant, +SERB/SRG/2021/001088. +[1] H.-T. Quan, Y.-x. Liu, C.-P. Sun, and F. Nori, Physical +Review E 76, 031105 (2007). +[2] R. Kosloff and A. Levy, Annual Review of Physical +Chemistry 65, 365 (2014). +[3] M. Campisi, J. Pekola, +and R. Fazio, New Journal of +Physics 17, 035012 (2015). +[4] H. Scovil and E. Schulz-DuBois, Phys. Rev. Lett. 2, 262 +(1959). +[5] J.-P. Brantut, +C. Grenier, +J. Meineke, +D. Stadler, +S. Krinner, C. Kollath, T. Esslinger, +and A. Georges, +Science 342, 713 (2013). +[6] J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, +K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walm- +sley, R. Uzdin, +and E. Poem, Phys. Rev. Lett. 122, +110601 (2019). +[7] M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, +and A. Svidzinsky, Proc. Natl. Acad. Sci. U.S.A. 108, +15097 (2011). +[8] M. +O. +Scully, +M. +S. +Zubairy, +G. +S. +Agar- +wal, +and +H. +Walther, +Science +299, +862 +(2003), +https://www.science.org/doi/pdf/10.1126/science.1078955. +[9] X. L. Huang, T. Wang, and X. X. Yi, Phys. Rev. E 86, +051105 (2012). +[10] G. Manzano, F. Galve, R. Zambrini, and J. M. R. Par- +rondo, Phys. Rev. E 93, 052120 (2016). +[11] J. Wang, J. He, and Y. Ma, Phys. Rev. E 100, 052126 +(2019). +[12] G. Manzano, Phys. Rev. E 98, 042123 (2018). +[13] D. F. Walls, nature 306, 141 (1983). +[14] R. Puri, pramana 48, 787 (1997). +[15] L. Dupays and A. Chenu, Quantum 5, 449 (2021). +[16] A. Kumar, T. Bagarti, S. Lahiri, and S. Banerjee, arXiv +preprint arXiv:2209.06433 (2022). +[17] J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Physical +Review X 7, 031044 (2017). + +9 +[18] J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Phys. +Rev. X 7, 031044 (2017). +[19] S. Pal, T. Mahesh, and B. K. Agarwalla, Physical Review +A 100, 042119 (2019). +[20] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and +E. Lutz, Physical review letters 112, 030602 (2014). +[21] Y. Zou, Y. Jiang, Y. Mei, X. Guo, and S. Du, Physical +Review Letters 119, 050602 (2017). +[22] F. V. Melo, N. S´a, I. Roditi, R. S. Sarthour, I. S. Oliveira, +and A. M. Souza, arXiv preprint arXiv:2203.13773 +(2022). +[23] W. Niedenzu, D. Gelbwaser-Klimovsky, A. G. Kofman, +and G. Kurizki, New Journal of Physics 18, 083012 +(2016). +[24] M. Lostaglio, D. Jennings, +and T. Rudolph, Nature +Communications 6 (2015), 10.1038/ncomms7383. +[25] M. +Lostaglio, +K. +Korzekwa, +D. +Jennings, +and +T. Rudolph, Phys. Rev. X 5, 021001 (2015). +[26] K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jen- +nings, New Journal of Physics 18, 023045 (2016). +[27] O. Abah and E. Lutz, EPL (Europhysics Letters) 106, +20001 (2014). +[28] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and +E. Lutz, Phys. Rev. Lett. 112, 030602 (2014). +[29] J. Um, K. E. Dorfman, +and H. Park, Physical Review +Research 4, L032034 (2022). +[30] H. P. Goswami and U. Harbola, Phys. Rev. A 88, 013842 +(2013). +[31] S. Rahav, U. Harbola, +and S. Mukamel, Phys. Rev. A +86, 043843 (2012). +[32] C. L. Latune, I. Sinayskiy, and F. Petruccione, The Eu- +ropean Physical Journal Special Topics 230, 841 (2021). +[33] G. Manzano, F. Galve, R. Zambrini, +and J. M. R. +Parrondo, Physical Review E 93 (2016), 10.1103/phys- +reve.93.052120. +[34] B. K. Agarwalla, J.-H. Jiang, +and D. Segal, +(2017), +10.48550/ARXIV.1706.06206. +[35] R. Long and W. Liu, Phys. Rev. E 91, 062137 (2015). +[36] G. Chen, D. A. Church, B.-G. Englert, C. Henkel, B. Ro- +hwedder, M. O. Scully, +and M. S. Zubairy, Quan- +tum computing devices: principles, designs, and analysis +(Chapman and Hall/CRC, 2006). +[37] M. Teich and B. Saleh, Quantum Optics Journal of the +European Optical Society Part B 1, 153 (1989). +[38] R. +R. +TUCCI, +International +Journal +of +Modern +Physics +B +05, +545 +(1991), +https://doi.org/10.1142/S021797929100033X. +[39] B. K. Agarwalla, J.-H. Jiang, and D. Segal, Phys. Rev. +B 96, 104304 (2017). +[40] D. Newman, F. Mintert, and A. Nazir, Phys. Rev. E 95, +032139 (2017). +[41] F. L. Curzon and B. Ahlborn, American Journal of +Physics 43, 22 (1975). +[42] C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005). +[43] S. H. Lee, J. Um, and H. Park, Phys. Rev. E 98, 052137 +(2018). +[44] Z. Ye and V. Holubec, Phys. Rev. E 103, 052125 (2021). +[45] T. Abebe, D. Jobir, C. Gashu, and E. Mosisa, Advances +in Mathematical Physics 2021 (2021). +[46] S.-W. Li et al., Physical Review E 96, 012139 (2017). +[47] M. J. Sarmah, A. Bansal, +and H. P. Goswami, arXiv +preprint arXiv:2206.07606 (2022). +[48] U. Harbola, S. Rahav, +and S. Mukamel, EPL (Euro- +physics Letters) 99, 50005 (2012). +[49] Q. Bouton, J. Nettersheim, S. Burgardt, D. Adam, +E. Lutz, +and A. Widera, Nature Communications 12, +2063 (2021). +[50] H. K. Yadalam, B. K. Agarwalla, and U. Harbola, Phys. +Rev. A 105, 062219 (2022). +[51] V. Dodonov, Journal of Optics B: Quantum and Semi- +classical Optics 4, R1 (2002). +[52] S. K. Giri and H. P. Goswami, Phys. Rev. E 99, 022104 +(2019). +[53] M. Esposito, K. Lindenberg, +and C. Van den Broeck, +Phys. Rev. Lett. 102, 130602 (2009). +[54] M. Esposito, R. Kawai, K. Lindenberg, and C. Van den +Broeck, Phys. Rev. Lett. 105, 150603 (2010). +[55] S. K. Giri and H. P. Goswami, Phys. Rev. E 106, 024131 +(2022). +[56] H. Liu, J. He, and J. Wang, Journal of Applied Physics +131, 214303 (2022). +[57] A. Dechant, N. Kiesel, and E. Lutz, EPL (Europhysics +Letters) 119, 50003 (2017). +[58] I. Iyyappan and R. S. Johal, EPL (Europhysics Letters) +128, 50004 (2020). + diff --git a/ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/load_file.txt b/ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c09e2fd9d03237f1a5d645eeaf05dc96fb6ff4a8 --- /dev/null +++ b/ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/load_file.txt @@ -0,0 +1,1029 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf,len=1028 +page_content='Work flux and efficiency at maximum power of a triply squeezed engine Manash Jyoti Sarmah and Himangshu Prabal Goswami∗ Department of Chemistry, Gauhati University, Jalukbari, Guwahati-781014, Assam, India (Dated: January 30, 2023) We explore the effects of quantum mechanical squeezing on the nonequilibrium thermodynamics of a coherent heat engine with squeezed reservoirs coupled to a squeezed cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We observe that the standard known phenomenon of flux- optimization beyond the classical limit with respect to quantum coherence is destroyed in presence of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under extreme nonequilibrium conditions, the flux is rendered independent of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The efficiency at maximum power (EMP) obtained by optimizing the cavity’s squeezing parameter is greater than what was predicted by Curzon and Ahlborn even in the absence of reservoir squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP with respect to the either of reservoirs’ squeezing parameters is surprisingly equal and linear in ηC with a slope unequal to the universally accepted slope, 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The slope is found to be proportional to the dissipation into the cavity mode and an intercept equal to a specific numerical value of the engine’s efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' INTRODUCTION One or more quantum systems that operate between two separate reservoirs make up a Quantum Heat En- gine (QHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' QHEs have the primary function of convert- ing heat into work [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Apart from traditional thermal reservoirs, the use of non-thermal baths, which are con- structed reservoirs with correlated characteristics, have provided a thorough setting for examining the relation- ship between quantum effects and thermodynamic quan- tities [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Squeezed states or non-canonical initial states [13–15] are such non-thermal baths which allow additional control over any quantum systems’ dynamics garnering tremendous interest off late in the context of open quantum systems [11, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Current technologies permit experimental realization of such states [17] and its effects on the thermodynamics are experimentally realizable through recently designed experimental quantum heat engines (QHE)[18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In- tense efforts have been made to interrogate QHEs on the role of coherence, correlations or entanglement on the underlying dynamics [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' It has already been demonstrated that certain quantum resources can be ex- ploited to bend the limits of classical thermodynamics [16, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Coherence enhanced power and efficiency and optimization of the flux via quantum coherences in QHEs are well studied and established phenomena [7, 29– 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Squeezed thermal baths too have proven crucial, especially in the light of a proof-of-concept experiment based on a nanobeam heat engine[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Efficiency greater than that of Carnot has also been predicted [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' On the theoretical front, quantum thermodynamic analysis of QHEs s are performed by combining principles from quantum optics and nonequilibrium statistical me- chanics [9, 33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In quantum optics, squeezing [36, 37] generally leads to less observation of quantum noise than thermal states [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Squeezing alters the entropy flow associated with the heat exchanged with the system and ∗ hpg@gauhati.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='in introduces an additional term proportional to the second- order coherences which determines the asymmetry in the second-order moments of the mode quadratures, which takes into account both the relative variance shape and the relative optical phase space displacements[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This manifests in an increased efficiency, even surpassing the Carnot bound [10, 18, 23, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' To account for a re- alistic performance of such QHEs, usually a finite time assessment is performed by evaluating the efficiency at maximum power (EMP), originally introduced in a clas- sical context [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' From a nonequilibrium quantum sta- tistical point of view, the near equilibrium EMP is univer- sally accepted to be ηC/2 [42], with ηC being the stan- dard Carnot efficiency of a classical heat engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Re- cently, this robust expression has been showed to be in- valid if the engine is locally optimized [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP has been shown to be modified into several forms as one keeps changing or introducing or optimizing addi- tional system parameters [39, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' One particularly in- teresting form of the EMP has been predicted recently which holds in the presence of squeezed reservoirs, be- ing equal to η2 m/(ηm − (1 − ηm) ln(1 − ηm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Here, ηm being a squeezing-dependent effective Carnot efficiency [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' However, the validity of such robust thermody- namic expressions remains questionable when engines op- erate in presence of both quantum coherences and quan- tum squeezing since the general framework on which such studies were based didn’t take such effects into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The current work is motivated on this latter aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In this work, we address how the thermodynamics of a QHE coupled to squeezed cavity respond to reservoir squeezing in presence of coherences using a quantum mas- ter equation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Such a technique is standard and has already been used in nonequilibrium quantum transport studies with squeezed reservoirs [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Un- squeezed dynamics of the engine that we cosider has also been well studied [7, 31, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (II), we in- troduce our triple squeezed QHE model and its dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (III), we explore the effects of squeezing on the flux into the cavity mode, which we call the work- flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (IV), we evaluate the EMP with respect to three squeezing parameters and a system parameter after arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='11607v1 [quant-ph] 27 Jan 2023 2 which we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' SQUEEZED ENGINE DYNAMICS The QHE model consists of four quantum levels cou- pled asymmetrically to two squeezed baths with the up- per two levels coupled to a squeezed unimodal cavity as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Experimentally, similar QHEs have been realized in cold Rb and Cs atoms us- ing magneto optical traps [21, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The squeezed density matrices of the QHE can be written as[46, 50], ¯ρℓ = 1 Zℓ exp{−βℓ ˆSℓ ˆHℓ ˆS† ℓ}, (1) ¯ρν = 1 Zν exp{−βν ˆSν ˆHν ˆS† ν}, ν = h, c, (2) with βz = (kBTz)−1, z = ℓ, h, c being the inverse temper- atures of the cavity, hot and cold reservoirs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' ˆS( ˆSν) is the squeezing operator on the squeezed cavity’s 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 xc Ρ12 ss 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 xh Ρ12 ss 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 x Ρ12 ss a b d c Th Tc xh xc x (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (Color online) a) Level scheme of the model quan- tum heat engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A pair of degenerate levels |1⟩ , |2⟩ is reso- nantly coupled to two excited levels |a⟩ and |b⟩ by two ther- mally populated squeezed field modes with hot (Th) and cold (Tc) temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Levels |a⟩ and |b⟩ are coupled through a squeezed cavity mode of frequency νℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Emission of pho- tons into this squeezed cavity is the work done by the QHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The engine parameters are fixed through out the manuscript at E1 = E2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, Eb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4, Ea = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, g = 1, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 in the unit of kB → 1 and ¯h → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' b) The solid (dot- ted) curves represent the steadystate coherence, ρss 12 (solved by setting the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8)=0) as a function of the b) cold bath squeezing parameter xc evaluated at different values of xh = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 1, 2, bottom to top with x = 1 (x = 0), c) hot squeezing parameter, xh with xc = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 1, 2, bottom to top and x = 1 (x = 0), d) cavity squeezing,x with the solid curves (bottom to top) evaluated at xh = 0, xc = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The dotted ones represent xc = 0, xh = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' mode (reservoirs’ modes) given by : ˆSℓ = e 1 2 (xˆa†2 ℓ −h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='c), (3) ˆSν = � k e 1 2 (λ∗ kνˆa†2 kν−h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='c), (4) λkν = xkνeiθkν, xkν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (5) θkν and xkν are the squeezing parameters of the reservoirs and x is the squeezing parameter [46, 47, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' ˆHℓ = ϵℓˆa† ℓˆaℓ is the Hamiltonian for the cavity mode and ˆHν = � k ϵkνˆa† kνˆakν is the Hamiltonian for the ν-th reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The total Hamiltonian of the four level QHE is ˆHT = � ν = 1,2,a,b Eν|ν⟩⟨ν|+ ˆHℓ+ ˆHν+ ˆVsb+ ˆVsc, with the system- reservoir and system-cavity coupling Hamiltonians given by, ˆVsb = � k ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='c � i = 1,2 � x = a,b rikˆak|x⟩⟨i| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='c (6) ˆVsc = gˆa† ℓ|b⟩⟨a| + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7) ϵk, ϵℓ and Eν denote the energy of the kth mode of the two thermal reservoirs, the unimodal cavity and system’s νth energy level respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The system-reservoir cou- pling of the ith state with the kth mode of the reservoirs is denoted by rik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' ˆa†(ˆa) are the bosonic creation (an- nihilation) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The radiative decay originating from the transition |a⟩ → |b⟩ is the work done by the engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Unsqueezed version of such a QHE has been thoroughly studied using a Markovian quantum mas- ter equation [7, 30, 31, 48, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Following such a stan- dard procedure to derive of a quantum master equation [46, 48] for the matrix elements of the reduced density matrix ρ (supplementary information) has four popula- tions, ρii, i = 1, 2, a, b coupled to the real part of a co- herence term, ρ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The coherence ρ12 between states |1⟩ and |2⟩ arise due to interactions with the hot and the cold baths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This thermally induced coherence couples to populations due to transition involving the states |1⟩ and |2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under the symmetric coupling regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' we can now write down five coupled first order differential equations describing the time-evolution of the four populations and the coherence (under symmetric coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' given by ˙ρ12 = −ry 2 ρ11 − ry 2 ρ22 + rph ˜Nhρaa + rpc ˜Ncρbb − r(n + τ)ρ12 (8) ˙ρii = −rnρii + r ˜Nhρaa + ˜Ncρbb − ryρ12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 2 (9) ˙ρbb = rNcρ11 + rNcρ22 + g2 ˜Nℓρaa − (g2Nℓ + 2r ˜Nc)ρbb + 2rpcNcρ12 (10) ˙ρaa = rNhρ11 + rNhρ22 − (g2 ˜Nℓ + 2r ˜Nh)ρaa + g2Nℓρbb + 2rphNhρ12 (11) with,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' � i ρii = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' b and n = Nc + Nh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' y = Ncpc + Nhph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' with the reorganized occupation factors 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='30 t Ρij �a� Ρ11,Ρ22�black� Ρaa�green� Ρbb�blue� Ρ12�brown� 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='15 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='07 x Ρbb ss �Ρaa ss x 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='03 ph j�jo ph �d� (b) (c) (a) (b) (d) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' a) The solid (dotted) curves represent time evo- lution of ρij with, x = 2 (without, x = 0) squeezing ob- tained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8-11) for Th = 2, Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, Tl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' b) Steady state values as a function of the squeezing pa- rameters for the same parameters as (a) c) Ratio of the steady state values between states |b⟩ and |a⟩ reaching unity highlighting the equipopulated nature under high squeezing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 from the top to the bottom curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (d) Optimization of the flux ratio as a function of hot coher- ence parameter, ph for different squeezing parameters under far from equilibrium conditions and pc = 1 (top to bottom: x = 0, π/6, π/π/2, 2π/3, 5π/6, π, 3π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Other parameters are same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' given by Nz = cosh(2xz)(nz + 1 2) − 1 2, z = h, c, (12) Nℓ = cosh(2x)(nℓ + 1 2) − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (13) Here, nc, nh andnl are the Bose-Einstein distributions for the cold reservoir, hot reservoir and the cavity re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' These factors are now squeezing dependent via the dimensionless parameters, xh, xc and x represent- ing the extent of squeezing in the hot, cold reservoirs and the cavity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' pν = | cos φν|, ν = h, c are two dimensionless parameters that governs the strength of coherences and whose values are dictated by the an- gles of relative orientation (φν) of the ν−th bath induced transition in the system [7, 48, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A phenomenological dimensionless rate τ has been added to take care of the dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Setting ˙ρ = 0, at the steady state, we can solve for the steady state values of ρaa, ρbb, ρ11, ρ22, and ρ12 and obtain these analytically (supplementary text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The steadystate value of the coherence term ρss 12 as a function of the squeezing parameters, xh, xc and x are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (1b,c,d)) for different engine parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The different curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (1b) represent ρss 12 evaluated for different xh and x values as a function of xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The solid (dotted) lines represent ρss 12 when xh ̸= 0(xh = 0) and x = 0(x ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' At high xc values, the coherence is re- duced and saturates to a lower value in comparison to ρss 12 values of lower xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' At high xh values (black curve), ρss 12 steadily increases and reaches a maximum value around some intermediate xc value and then sharply drops as 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='00 j� jo ph 0 1 1 ph j� jo ph FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Failure of coherence to optimize the flux beyond classical values (j/j0 > 1) under high squeezing as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Inset: Linear dependence of the flux ratio on p − j under high squeezing (x ≫ 0) and Tl ≫ 0 given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (18) evaluated at pc = 1, Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, Th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The square boxes represent linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' xc keeps increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This behavior is however absent for lower xh values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (1c) represent ρss 12 evaluated for dif- ferent xc and x values as a function of xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The solid (dotted) lines represent ρss 12 when xc ̸= 0(xc = 0) and x = 0(x ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' At high xh values, the steady state values of the coherence term increases and saturates to a higher value in comparison to coherence at lower xh values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We can rationalize that, xc(xh) tend to reduce (increase) the steadystate values of the coherences as we keep squeez- ing the baths more and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The same however cannot be said for ρss 12 vs x as seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The solid (dotted) lines represent the behavior at xh = 0(xh ̸= 0) for finite xc values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The time evolution of each of the equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8-11)) for various engine parameters for xh = xc = 0 and x = 2 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (2b), the steadystate values of the populations as a function of x is shown where solid (dotted) curves represent cavity-squeezed, x ̸= 0 (cavity- unsqueezed, x = 0) evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Note that under high squeezing of the cavity mode, the steady state values, ρss aa and ρss bb equipopulate giving, lim x→∞ ρss bb ρss aa = 1 (14) and is shown numerically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (2c) for different values of the hot coherence parameter, ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The analytical ex- pressions for the steadystate values are provided in the supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' WORK FLUX We interprete the emission of photons into the squeezed cavity as the work done by the engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This photon exchange process between the levels |a⟩, |b⟩ with the squeezed cavity is quantified by the rate of photon exchange with the cavity which we refer to as the work flux, j = d dt⟨a† ℓaℓ⟩, where the trace is with respect to 4 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 ph j�jo ph �a� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9 pc ph � Tc�Colour 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1�Pink 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2�Blue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3�Red 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4�Orange 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5�Blue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0�Black pc �b� 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 x ph � x �c� 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 x Ζ�Ζx�0 Tl�10 Tl�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 Tl�1 Tl�2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 x �d� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' a) Loss of optimization of flux as a function of ph for different squeezing parameters, near equilibrium (Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9, Th = 1, Tl = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (b) Loss of linear dependence of pc on the optimal value p∗ h as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The topmost curve represents Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (c) Plot showing breakdown of the coherent optimization of the flux as a function of squeezing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The shaded region is not allowed since the maxi- mum possible value of p∗ h is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under far from equilibrium condition p∗ h exists which saturates (bottom curve) at higher values of x given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The top curve shows the behav- ior of p∗ h near equilibrium which is nonexistent after a certain squeezing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (d) (d) Lowering of thermodynamic affinity as a function of squeezing evaluated at Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, Th = 2 and xc = xh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' the squeezed cavity density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Following a stan- dard procedure to second order in the coupling as devel- oped in[30, 48] we get, j = g2( ˜Nℓρss aa − Nℓρss bb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We can substitute the values of the steadystate populations to obtain an analytical expression for the flux (supplemen- tary information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' When, the hot and the cold coherence parameters individually go to zero (pc = ph = 0), the coherence vanishes (ρss 12=0) and we obtain a coherence - unaffected value of the flux, which we denote as jo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Note that, jo depends on the squeezing parameters x, xh and xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In the absence of squeezing (xh = xc = x = 0), jo shall be denoted by j0 o, which we refer to as the classical value of the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' There are no effects of coherence or squeezing on j0 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' It is a well known phenomena that, in absence of squeezing, j > jo can be achieved as a func- tion of coherence parameter, ph [7, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We plot the ratio j/jo in shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (2d) for different squeezing values of the cavity for xc = xh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As the cavity squeezing parameter is increased the optimal value of the flux grad- ually decreases and the ph value that optimizes the ratio (denoted as p∗ h) shifts towards larger ph values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We now attempt to explore the dependence of the flux in presence of squeezing on the coherences in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Since the ana- lytical expressions of j and j0 o are too lengthy we focus on some limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under high cavity squeezing, (x → ∞), we obtain ρaa aa = ρss bb as seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The expression for the flux in this case is simply given by, lim x→∞ j = g2( lim x→∞ ρss aa), (15) which under the condition pc = 0, ph = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (15) is, lim x→∞ jo = r(Nh − Nc) 2(n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (16) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (15), with pc = 1 can be expressed as, lim x→∞ j|pc=1 = r(Nh − Nc) � Nh � 1 − p2 h � + t � (1 − ph)fn + 2τ(n + 1) (17) with fn = 4NcNh + n(2Nh(ph + 1) + ph + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (16) is always greater than RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (17) as seen from the numerical result in Fig (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The physical in- terpretation is that the coherences are no longer able to increase the flux beyond the non coherence values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Un- der this condition, the ratio is bounded below unity as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We can analytically prove this by invok- ing a few conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (16) and (17), if τ = 0 and Nh = zNc, z being a positive integer), the ratio between the two fluxes becomes, lim x→∞ j|pc=1 lim x→∞ jo ���� Nh=zNc = 2z(ph+1)(Ncz+ Nc+ 1) 2Nc(ph+1)z2+z(4Nc+ ph+2)+1 (18) which is a rational fraction of two linear terms of ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (18) can be shown to have a linear dependence on ph for some appropriate conditions of the coefficients which is graphically shown as an inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (18), for z = 1 and Nh = Nc (no bias), we see a flux value that solely depends on only the coherence value, given by lim x→∞ j jo |Nh=Nc = 2(1 + ph) (3 + ph) (19) ≤ 1 (20) and is linear in ph for small values as seen in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (3) and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (4a), the flux ratio j/jo is plotted for different squeezing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The squeezing decreases from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' For smaller ph, the linearity is prominent, but for higher ph values, the linearity is gradually less apparent as the squeezing pa- rameter increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' It has been previously reported that p∗ h increases lin- early in pc under the unsqueezed case [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In the current case, we observe that under an extremely biased scenario (Nh ≫ 0) and high squeezing, x ≫ 0, the linear depen- dence is lost as shown graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (4b) and the dependence of p∗ h on the cold coherence parameter, pc is given by the nonlinear function, p∗ h| = � (1−p2c) (4N 2c (1−p2c)+4Nc+1)+2Nc � p2 c+1 � +1 4Ncpc + pc (21) which reduces to unity when pc = 1 as seen in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The nonlinear dependence takes a simplistic form when Tc → 0, where the above expression reduces to, p∗ h|Tc=0 = 1 − � 1 − p2c pc (22) 5 which is shown as the topmost curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (21) also has a strange dependence on the cav- ity squeezing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' p∗ h increases as a function of x and saturates at higher x values as shown in the bottom- most curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' However under extremely biased conditions, p∗ h sharply rises beyond unity and goes to the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The shaded region is not allowed as the maximum value of p∗ h is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Since an analytical expres- sion of p∗ h as a function of x is beyond the scope of sim- plistic analysis, the exact identification of this numerical fallout range is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We simply speculate that such a breakdown happens when the cavity temperature Tℓ is set to be very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Since nℓ is a function of Tℓ, the numerics blows when there is competition between x and Tℓ to dominate the behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The upper dashed curve in the shaded portion also corresponds to an unrealistic p∗ h evaluated at a high cavity temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (4d), we plot the thermodynamic force as a function of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The force can be identified from the analytical expression of the flux (supplementary text) and is given by, ζ = ˜Nc ˜ NℓNh Nc ˜NhNℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (23) When ζ > (<)1, j > (<)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (4d), we plot the ratio between the thermodynamic forces in presence and ab- sence of squeezing for different cavity temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As squeezing increases, the ratio decreases for a fixed set of engine parameters and then saturates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This leads to lower magnitude of the flux in comparison to the un- squeezed case and is more prominent when the cavity temperature is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (5a,b and c), we plot the ratio between the total flux j and the classical flux j0 o as a function of xc, xh and x respectively for the same parameters as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As a function of both the baths’ squeezing parameters, the increase of the total flux is quite large in comparison to the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' All of the curves show saturation be- havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Particularly interesting is the ratio’s dependence on xh where the saturation value of the ratio is always greater than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We now focus on an extreme biased case (Th ≫ Tc, a limit which we invoke by taking Th → ∞ and Tc → 0), a scenario when the temperature gradient is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This case is different from a standard extreme nonequilib- rium case where the thermodynamic force must be very high (ζ ≫ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under the high temperature gradient sce- nario, the steadystate populations of the upper two states are given by, lim Th≫Tc ρss aa = � p2 h + 1 � � g2Nℓ + 2r � g2 (4Nℓ + p2 h + 1) − 2 (p2 h − 3) r (24) lim Th≫Tc ρss bb = g2 ˜Nℓ � p2 h + 1 � g2 (4Nℓ + p2 h + 1) − 2 (p2 h − 3) r, (25) which no longer depends on the squeezing parameters of the two baths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Using these above values the flux can be (a) (b) (d) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Giant increase of the total flux (in presence of squeez- ing as well as coherence) in comparison to the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The solid (dotted) lines represent the ratio between the to- tal flux j and the classical flux j0 o as a function of a) xc evaluated at x = 0(1), xh = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 1, 2, b) xh evaluated at x = 0(1), xc = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' c) Solid (dotted) curves indicate the total flux ratio as a function of cavity squeezing x evaluated at Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1) with {xh, xc} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1}, {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5}, {0, 0} (top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (d) Change in the sign of the thermody- namic affinity, A = log ζ as function of cavity squeezing pa- rameter evaluated at{xh, xc} = {1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1} (upper curve) and {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, 1} (lower curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The sign change happens at x∗ given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' recast as, lim Th≫Tc j = 2g2r ˜Nℓ(1 + p2 h) g2(1 + 4Nl + p2 h) − 2r(p2 h − 3) (26) while the coherence-unaffected value of the flux is simply, lim Th≫Tc jo = 2g2 ˜Nℓr g2(1 + 4Nℓ) + 6r (27) It is interesting to note that, in this highly biased sce- nario, the flux expression (RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (26)) doesn’t de- pend on the cold coherence parameter any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In the above two expressions, if we invoke the high squeezing scenario (x → ∞), we can write down the ratio between the two fluxes as, lim x→∞ lim Th≫Tc j lim Th≫Tc jo = (1 + p2 h) (28) Note that, the above expression is bound, 1 ≤ 1 + p2 h ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In this limit with ph = 1(pc ̸= 1), coherences can double the value of the flux from its zero coherence value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Likewise, the ratio between the flux in this limit and the classical value of the flux can be written as, lim x→∞ lim Th≫Tc j lim Th≫Tc j0 o = (1 + p2 h)(1 + 6r − 3g2 4g2˜nℓ ) (29) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (30) As long as r > g2/2 and pc ̸= ph, within the high bias scenario and maximal cavity-squeezing, the flux is always greater than unity in comparison to the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 20 15 15 10 10 5 5 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Xc Xh 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 X X6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='30 x W�Wo Tl�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 Tl�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 Tl�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 Tl�10 x �a� 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 3 �1 0 1 2 3 x W�Wo Tc�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1 Tc�2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 Tc�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 Tc�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 x �b� 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='005 ph ΗEa � �Ηo � x�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 x�0 x�1 x�2Π �c� ph 0 1 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 x ΗEa � �Ηo � Tl�0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9 Tl�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 Tl�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 Tl�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 x �d� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (a) Squeezing induced increase of the work done beyond classical limits (Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, Th = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The increase is larger when the cavity temperature is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (b) Negative work done as a function of squeezing for different Tc( Th = 2, Tl = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (c) EMP with respect to Ea as a function of ph for different squeezing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (d) EMP with respect to Ea for the range of squeezing at different cavity temperatures (pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, ph = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' EFFICIENCY AT MAXIMUM POWER We now move to perform a thorough analysis on the efficiency at maximum power (EMP or η∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In a stan- dard context, the EMP is calculated by maximizing the efficiency with respect to a system parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In our QHE model, the efficiency is defined as η = W/Qh with Qh = (Ea −E1), and the useful work done (W) is defined as, W = Ea − Eb − WdissTc, (31) with Wdiss = kBln ˜ Nℓ Nℓ is the dissipation into the cavity mode [30, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' W doesn’t depend on the squeezing pa- rameters of the two squeezed reservoirs or the noise in- duced coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (6a), we show the variation of W/Wo (Wo being the useful work in absence of squeez- ing, x = 0) as a function of x for several values of the cavity temperature, Tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As can be seen, the work done increases as Tl is lowered and saturates at higher values of x and is always greater than unity as long as Tc > Tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' When Tc < Tℓ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (6)b), the work done is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In general, the work changes its sign at x = x∗, given by x∗ = 1 2ℜ � cosh−1 � ˜NcNh + Nc ˜Nh (2nℓ + 1)(Nc − Nh) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (32) Although W and η are independent of coherences and the reservoir squeezing parameters, the EMP however de- pends on these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP obtained by max- imizing P with respect to any system parameter puts an implicit dependence via the optimized value of the chosen parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We choose the three squeezing parameters xc, xh, x and Ea to optimize the EMP and denote these by η∗ xc, η∗ xh, η∗ x and η∗ Ea respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The squeezing unaf- fected values of the EMP are denoted by η∗ o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (6c), 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='010 ph Ηx ��Ηo � �a� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='60 Ηx � �b� r � g Η�2 Η��2�Η� ΗC Ηc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='60 Ηx � �c� r � g Η��2�Η� ΗC Ηc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='70 ΗEa � �d� ΗC Η��2�Η� (c) (a) (b) (b) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (Color online)(a) EMP with respect to squeezing as a function of ph fr various pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In a), b) and c), the black curves (overlayed with red color) represent the evaluated EMP of our QHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The green dashed curve is the upper bound on the EMP, η∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The brown dashed line represents ηCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The dotted line represent ηL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (b) and (c) EMP with respect to x as a function of ηC with r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7, g = 1) and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, g = 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' When r ≈ g, η∗ x > ηCA as seen in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (d) EMP with respect to Ea as a function of ηC with r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7, g = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Here, η∗ Ea > ηCA with x = 1(xc = xh = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' we show the dependence of the ratio η∗ Ea/η∗ o as a function of ph for several x-values evaluated at xc = xh = 0 and pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The dependence of this ratio on ph is extremely nonlinear and is unity at ph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 where effects of coher- ence vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' At lower (higher) squeezing values, the ratio decreases (increases) to unity and then sharply increases beyond unity as a function of ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We can theorize that, lower ph values (under the condition ph < pc), smaller values of cavity squeezing favor increasing the EMP be- yond classical values while for larger ph (ph > pc), high squeezing favor increase of the EMP beyond classical val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (6d), we plot the same ratio as a function of cavity squeezing parameter for different cavity tempera- tures, Tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' There is an optimization of the EMP at lower values of x and the hump keeps shifting leftward to even smaller values as Tl is increased and the EMP ratio keeps decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (6d), we can conclude that lower values of Tℓ yield very high values of EMP with respect to Ea under moderate squeezing conditions of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7a), we plot η∗ x as a function of ph for differ- ent combinations of xc and xh for a fixed pc value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Here, for a fixed set of engine parameters, when xc < xh leads to a larger optimized value (around ph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5) of the EMP with respect to x (blue curve in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' How- ever as ph approaches unity, there is a sharper fall in the EMP and goes below unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' For the case when xc = xh, the behavior is similar (dotted curve) but the increase is not as high as the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' When squeezed to the limits, xc → ∞, xh → ∞, the EMP with respect to x no longer depends on the coherence (dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This is due to the fact that, under this scenario, the power cannot be optimized with respect to x and the maximum value occurs at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In general, the EMP has a universally accepted for- mula, the Curzon-Ahlborn EMP, ηCA = 1 − √1 − ηC 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 Ηc Ηxc � Ηc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 Ηc Ηxh � Ηc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='75 Tc Η,Η� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='85 Ηc ΗEa � (a) (b) (d) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Linear dependence of η∗ xc(a) and η∗ xh(b) as a func- tion of ηC, governed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (33) evaluated at x = ∞, 1 and 0 (top to bottom ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Note that η∗ xh = η∗ xc with the upper (mid- dle) curves having a slope of m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='02(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='19) and intercept of c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='76(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' c) Solid line represents the EMP, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (34) while the dotted line is simply the normal efficiency, η = W/Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' d) Appearance of a quadratic term and an in- tercept for η∗ Ea as a function of ηC, evaluated at x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5(∞) denoted by lower (upper) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The fit parameters for the upper (lower) curves are a1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='85(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='76), a2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='7(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='75), a3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='05(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='28), a4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='07(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [41, 53] and is represented by the dashed curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7b,c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As a function of ηC, the EMP is bound between ηC/2 ≤ η∗ ≤ η∗∗, where the upper bound is η∗∗ = ηC 2−ηC [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7b,c and d), we show the be- havior of our engine’s EMP as a function of the Carnot efficiency, ηC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The solid (topmost green) curve repre- sent the upper bound η∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP of the QHE op- timized with respect to x for xc = xh = 0 is repre- sented by the solid line highlighted with red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7b,c), η∗ x ≥ (<)ηCA is observed under the condition r ≥ (<)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Values of EMP larger than ηCA has been previously reported with squeezed reservoirs [17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In our case, one can have EMP more than the predicted ηCA just by squeezing the cavity even in the absence of squeezed reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7d), for nonzero values of cavity-squeezing, η∗ Ea > ηCA is shown (solid black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This result is valid irrespective of r and g values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The upper bound is always obeyed in presence of squeezing as evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8b,c and d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP of the QHE is always lower than the upper dashed curve (η∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Note that the universal slope of 1/2 (any EMP = ηC/2 near equilibrium)[42] is maintained in all the curves for smaller values of ηC when maximized with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We now move to discuss a rather interesting finding observed when the EMP is maximized with respect to a reservoir squeezing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8a and b), both η∗ xh and η∗ xc are found to be linear in ηC with a slope which is not equal to the universally predicted value of 1/2[53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' By a linear curve fitting tech- nique, we infer that the EMP with respect to xc or xh is dictated by the equation, η∗ xh = η∗ xc = mηC + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (33) Our numerical results reveal that the slope, m is equal to the numerical value of Wdiss/Qh and the intercept, c being given by the numerical value of the quantity, (Eab− Wdiss)/Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This intercept is interestingly the efficiency of the engine albeit with Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Note that, η∗ xc = η∗ xh and is shown as two identical plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(8a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In these two figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The numerical plots reveal that the m ̸= 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Such a breakdown of the universality of the linear coefficient has also been observed in presence of geometric phaselike effects [52, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Since Wdiss > 1, the EMP increases as x is increased (for fixed Tℓ) to a maximum value of Eab/Qh at ηC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The efficiency of the QHE, η = W/Qh is always less than η∗ ν and is shown as a function of Tc in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This linear dependence doesn’t exist for η∗ Ea for finite x as seen from the numerical results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (8d) for x = 1 and x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' It has been previously reported that such a nonlinear dependence of the EMP on the squeezing parameter x takes the form η∗ ∗ = 1 − � sech(2x)√1 − ηC [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We assess the validity if this expression by defining two curve fitting equations, η∗ Ea ≈ a1 − � sech(a2x)√a3 − a4ηC (34) ≈ a5ηC + a6η2 C + c (35) that can best represent the EMP with respect to the sys- tem parameter Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Here, ai-s are fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We observe that a1 ̸= a3 ̸= a4 ̸= 1 and a3 ̸= 2 result- ing in η∗ Ea ̸= η∗ ∗ and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Further, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (35), a5 ̸= 1/2 and a6 ̸= 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' In this engine, it is al- ready known that the quadratic coefficient is not 1/8 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Both the above equations are good fits (solid curves) on the numerically evaluated η∗ Ea (dots) as function of ηC as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' It is interesting to note that the in- tercept of η∗ Ea as a function of ηC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (35) is the same numerical value of the engine’s efficiency of the engine, η = W/Qh similar to what was observed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' This lets us rationalize that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (35) is a better representation of η∗ Ea vs ηC than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='(34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' At ηC = 1, η∗ Ea again reaches a maximum value of Eab/Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' For x = 0, m = 1/2 is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Further for x = 0, the intercept in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (34) also vanishes by mixing with the quadratic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Since we cannot derive analytical expressions for these coeffi- cients, we demonstrated it this numerically shown as the bottom-most dotted line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP also has other interesting logarithmic expressions[43, 57, 58], one particularly claimed to be valid for squeezed states[11], η∗ L = η2 m/{1−(1−ηm) ln(1− ηm)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' ηm is a modified Carnot efficiency given by ηm = 1−Tc/T m h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' T m h is a modified but fictitious reservoir tem- perature and is directly proportional to the energy of the squeezed mode and inversely proportional to the logarith- mic ratio of the squeezed mode’s occupation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' By an analogy with this previous work [11], we can express the modified temperature in our QHE to be, T m h = Ea − E1 ln 1+Nh Nh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (36) 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='0 Ηc ΗEa � Ηc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Disagreement between the QHE’s EMP optimized with respect to Ea and the predicted EMP, η∗ L for the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' η∗ L is evaluated using the definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The dotted (dashed) curves represent η∗ Ea(η∗ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Parameters used are Th = 3, xc = xh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='6 (top dotted), Th = 4, xc = xh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2, x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='5 (middle dotted) and Th = 6, xc = xh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='2, x = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We numerically evaluate ηE∗ a for different squeezing pa- rameters and Th values and plot it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (9) along side the corresponding η∗ L values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' As can be seen, η∗ Ea ̸= η∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Further since η∗ xh and η∗ xc is found to be linear in ηC, these anyway don’t agree with the predicted value η∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Un- der extremely low squeezing conditions of the hot bath, ηm C → ηC in the expression for η∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under this condition, η∗ L has been high lighted as dotted curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' (7b,c and d) and is seen to be unequal to η∗ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' CONCLUSION By deriving a coherence-population coupled quantum master equation, we carried out a comprehensive study of the thermodynamics of quantum heat engine coupled to two squeezed reservoirs and a squeezed unimodal cav- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We showed that the steadystate value of the co- herence term of the density matrix vanishes (saturates) under maximal squeezing of the cold (hot) bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Under high squeezing conditions of the cavity, the two upper states of the engine equipopulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We showed that under high squeezing of the cavity, the quantum coherence can no longer optimize the flux beyond the classical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We also showed how the flux can be linearized with re- spect to coherences under high squeezing conditions and equal Bose-Einstein distributions for the hot and cold baths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We also showed that larger EMP favors lower val- ues of cavity temperatures and lower values of squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP can be increased beyond the Curzon-Ahlborn limit by squeezing the cavity alone even if the baths are unsqueezed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' We also show a linear dependence of the EMP with respect to the reservoirs’ squeezing parameters which we identify analytically with a slope proportional to the dissipation into the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' The EMP with respect to a system parameter, Ea doesn’t obey the uni- versal slope of 1/2 for finite squeezing and is not equal to a recently proposed general form of the EMP in presence of squeezed reservoirs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' ACKNOWLEDGMENTS MJS and HPG acknowledge the support from Science and Engineering Board, India for the start-up grant, SERB/SRG/2021/001088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Quan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Sun, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Nori, Physical Review E 76, 031105 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kosloff and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Levy, Annual Review of Physical Chemistry 65, 365 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Campisi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Pekola, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Fazio, New Journal of Physics 17, 035012 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Scovil and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Schulz-DuBois, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 2, 262 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Brantut, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Grenier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Meineke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Stadler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Krinner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kollath, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Esslinger, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Georges, Science 342, 713 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Klatzow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Becker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Ledingham, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Weinzetl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kaczmarek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Saunders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Nunn, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Walm- sley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Uzdin, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Poem, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 122, 110601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Scully, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Chapin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Dorfman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kim, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Svidzinsky, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 108, 15097 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Scully, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Zubairy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Agar- wal, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Walther, Science 299, 862 (2003), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1078955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Wang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Yi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 86, 051105 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Manzano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Galve, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Zambrini, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Par- rondo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 93, 052120 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' He, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Ma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 100, 052126 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Manzano, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 98, 042123 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Walls, nature 306, 141 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Puri, pramana 48, 787 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Dupays and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Chenu, Quantum 5, 449 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kumar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Bagarti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lahiri, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Banerjee, arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='06433 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Klaers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Faelt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Imamoglu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Togan, Physical Review X 7, 031044 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 9 [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Klaers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Faelt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Imamoglu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Togan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' X 7, 031044 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Pal, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Mahesh, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Agarwalla, Physical Review A 100, 042119 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Roßnagel, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Abah, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Schmidt-Kaler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Singer, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lutz, Physical review letters 112, 030602 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Zou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Mei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Guo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Du, Physical Review Letters 119, 050602 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Melo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S´a, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Roditi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Sarthour, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Oliveira, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Souza, arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='13773 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [23] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Niedenzu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Gelbwaser-Klimovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kofman, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kurizki, New Journal of Physics 18, 083012 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lostaglio, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jennings, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rudolph, Nature Communications 6 (2015), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1038/ncomms7383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lostaglio, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Korzekwa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jennings, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rudolph, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' X 5, 021001 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Korzekwa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lostaglio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Oppenheim, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jen- nings, New Journal of Physics 18, 023045 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [27] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Abah and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lutz, EPL (Europhysics Letters) 106, 20001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Roßnagel, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Abah, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Schmidt-Kaler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Singer, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lutz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 112, 030602 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Um, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Dorfman, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Park, Physical Review Research 4, L032034 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Goswami and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Harbola, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A 88, 013842 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rahav, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Harbola, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Mukamel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A 86, 043843 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Latune, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Sinayskiy, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Petruccione, The Eu- ropean Physical Journal Special Topics 230, 841 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [33] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Manzano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Galve, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Zambrini, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Parrondo, Physical Review E 93 (2016), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1103/phys- reve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='052120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Agarwalla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jiang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Segal, (2017), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='06206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Long and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Liu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 91, 062137 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [36] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Church, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Englert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Henkel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Ro- hwedder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Scully, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Zubairy, Quan- tum computing devices: principles, designs, and analysis (Chapman and Hall/CRC, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Teich and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Saleh, Quantum Optics Journal of the European Optical Society Part B 1, 153 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' TUCCI, International Journal of Modern Physics B 05, 545 (1991), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='1142/S021797929100033X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [39] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Agarwalla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jiang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Segal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' B 96, 104304 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [40] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Newman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Mintert, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Nazir, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 95, 032139 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [41] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Curzon and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Ahlborn, American Journal of Physics 43, 22 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [42] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Van den Broeck, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 95, 190602 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Um, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Park, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 98, 052137 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [44] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Ye and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Holubec, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 103, 052125 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [45] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Abebe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Jobir, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Gashu, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Mosisa, Advances in Mathematical Physics 2021 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=', Physical Review E 96, 012139 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Sarmah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Bansal, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Goswami, arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content='07606 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [48] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Harbola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rahav, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Mukamel, EPL (Euro- physics Letters) 99, 50005 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [49] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Bouton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Nettersheim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Burgardt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Adam, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lutz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Widera, Nature Communications 12, 2063 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [50] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Yadalam, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Agarwalla, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Harbola, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' A 105, 062219 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [51] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Dodonov, Journal of Optics B: Quantum and Semi- classical Optics 4, R1 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [52] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Giri and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Goswami, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 99, 022104 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [53] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Esposito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lindenberg, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Van den Broeck, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 102, 130602 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [54] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Esposito, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kawai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lindenberg, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Van den Broeck, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' 105, 150603 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [55] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Giri and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Goswami, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' E 106, 024131 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [56] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' He, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Wang, Journal of Applied Physics 131, 214303 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [57] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Dechant, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Kiesel, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Lutz, EPL (Europhysics Letters) 119, 50003 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' [58] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Iyyappan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} +page_content=' Johal, EPL (Europhysics Letters) 128, 50004 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFJT4oBgHgl3EQfrC3Z/content/2301.11607v1.pdf'} diff --git a/AtAyT4oBgHgl3EQfRvdA/vector_store/index.pkl b/AtAyT4oBgHgl3EQfRvdA/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ab9ad54d7376fad47db48bf4becdd6029422c6bd --- /dev/null +++ b/AtAyT4oBgHgl3EQfRvdA/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2463740c54760cb886095259b4160ca74a8a2fc22863f0d561a998a863ef0fc4 +size 113711 diff --git a/AtAzT4oBgHgl3EQfTPzA/content/tmp_files/2301.01247v1.pdf.txt b/AtAzT4oBgHgl3EQfTPzA/content/tmp_files/2301.01247v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..df2a7e0a1f0dd6894f8a0aa2ff377c71d818fe4d --- /dev/null +++ b/AtAzT4oBgHgl3EQfTPzA/content/tmp_files/2301.01247v1.pdf.txt @@ -0,0 +1,151 @@ +Rate Adaptive Autoencoder-based Geometric Constellation +Shaping + +Ognjen Jovanovic, Metodi P. Yankov, Francesco Da Ros and Darko Zibar +Department of Electrical and Photonics Engineering, Technical University of Denmark, Kgs. Lyngby, 2800, Denmark +ognjo@dtu.dk + +Abstract: An autoencoder is used to optimize bit-to-symbol mappings for geometric constellation +shaping. The mappings allow for net rate adaptivity without additional hardware complexity, while +achieving up to 300km of transmission distance compared to uniform QAM. © 2023 The Author(s) + +1. Introduction +State of the art coherent optical communications need to be deployed in dynamic network scenarios, which require +a certain degree of adaptivity to varying channel conditions [1]. Classically, this is handled by varying the modulation +format size, which 1) often produces a coarse granularity in the rate with steps of 2 bits/symbol; and 2) requires the +transceiver to support bit to symbol mapping and demapping to/from constellations of different size, increasing the +complexity. The probabilistic amplitude shaping (PAS) scheme [1] has emerged as an efficient architecture that +provides a solution to the first problem, at the cost of rate matcher and dematcher, which increases complexity. +Autoencoders (AEs) are becoming a popular tool to optimize the signaling constellation of digital communication +transceivers [2, 3]. In optical communications, AEs have been employed for geometric constellation shaping (GCS) +[4], bit labeling [5], mostly with the target of mitigating the impact of fiber nonlinearities, as well as transceiver +impairment mitigation [6, 7]. GCS is beneficial over PAS because it does not require explicit matcher and dematcher +blocks. However, rate adaptivity with GCS typically requires a rate-flexible forward error correction (FEC), which +may increase the complexity of the digital logic. Further, GCS does not allow for straight-forward Gray labeling to be +performed, which may lead to sub-optimality when combined with conventional bit-metric decoders as in standard +coherent optical communications [8]. +In this paper, an AE is used to 1) find optimal bit mappings; and 2) find optimized constellation points for a variety +of net rates while maintaining a fixed FEC and demapper logic (i.e. log-likelihood ratio (LLR) computation). The +system therefore allows for shaping gain to be achieved with a finer granularity without a complexity increase w.r.t. +conventional bit-interleaved coded modulation (BICM). +2. System description +The AE-based GCS training setup is given in Fig. 1 a). The +mapping function is learned using the AE architecture from [5] to +jointly optimize bit labeling and constellation position on the I/Q +plane. During the AE training, the transmitter and receiver employ +neural networks for mapping of bits and demapping to LLRs, +respectively. The transmitter of the proposed system is given in Fig +1. b). During testing, the mapping and demapping functionalities are +replaced by a look-up table (LUT) and conventional Gaussian bit- +metric receiver [8], respectively. +The AE requires that the modulation format size is known and fixed. For a fixed modulation format size, the +generalized mutual information (GMI) typically is penalized w.r.t. the mutual information, as the signal to noise ratio +(SNR) decreases. This is due to the penalty in the demapper function related to the inability to resolve constellation +points with similar likelihoods at the receiver. An AE implicitly addresses this problem by ‘merging’ such points +closer together and assigning (more than 2) labels with a very small Hamming distance to virtually the same point [7], +i.e. a many-to-one mapping (MOM) of bits-to-symbols is produced. Here, we exploit this fact to achieve rate adaptivity +in the following way. The GMI of the MOM is analyzed and the bit levels which are ambiguous are not used for data. +Instead, they are assigned dummy bits in order to maintain the bit flow and logic at the transmitter and receiver. For a +system with an FEC rate of ������������ = ������������/������������, a code length of N, information block length of K and a modulation format size +of M, the net data rate per dual polarization channel may be calculated as ������������′ = (2������������ − ������������������������)������������, where ������������ = ������������������������������������2 ������������ is +the number of bit carried by the constellation and ������������������������ is the number of dummy bits in the labeling. If ������������������������ is even, each +polarization gets the same number of dummy bits, whereas if it is odd, the allocation is done such that the GMI per + +a) +bits- +>.Channeli +LLRS +nd@ +b) +LUT +FEC +Channel +bits- +S/P +K/N +2m - nd +Fig. 1. a) Training setup; b) Testing setup with +dummy bit insertionpolarization is similar for both polarization. In this paper, the ������������������������ bit positions are selected by sorting their per-bit GMI +and choosing as many as required from the lowest ones. For example, when the SNR decreases, ������������������������ can be increased. +The performance degradation of the large-size constellation at low SNR is compensated for by the increased Euclidean +distance of the effectively smaller constellation achieved via MOM. The receiver architecture does not change since +K, N, and M are fixed. The only addition w.r.t. BICM is the additional LUTs for mapping depending on the channel +conditions. +3. Results +A wavelength division multiplexing system is optimized using the nonlinear interference noise model from [9] +with dual polarization, 5 channels, 100km spacing, FEC of rate 3/4 and modulation size ������������ = 256. Fig. 2 a) shows the +maximum distance for a given data rate at which the GMI is above the target rate for the AE-based MOM and uniform +QAM (red dotted line, ������������������������ = 0) for the central channel. The AE-optimized MOM achieve an extra span of transmission +distance with respect to both 256QAM and 128QAM. Here, due to the SNR reduction with the increase of distance, +the AE learns a MOM allowing the system to be rate adaptive through insertion of dummy bits without losing the +shaping gain or changing the modulation format. Fig. 2 b) shows the constellation learned for 8 spans transmission +which does not achieve MOM. In Fig. 2 c), the constellation learned for 20 spans that achieves MOM is shown. In the +latter, the AE “merged” some of the points together as shown in Fig. 2 d). The red box shows the bits that are not +shared by the 4 points. These 2 bits effectively carry no information and can be assigned dummy bits. The system then +exploits the resulting increased Euclidean distance of the constellation to improve the performance. + +Fig. 2. a) Net rate per dual polarization channel w.r.t. distance for uniform QAM and AE-based GCS with 3/4 FEC; +GCS learned at: b) 8 spans; c) 20 spans; d) Zoomed in point to show that 4 points collapsed to each other. +4. Conclusion +An autoencoder (AE) is used to optimize rate dependent bit-to-symbol mappings for QAM of fixed size and FEC +of fixed rate. Rate adaptivity is achieved through the AE-optimized mapping functions as a result of many-to-one +mapping. It achieves shaping gain for a variety of net rates without changing the receiver architecture, the modulation +format size or requiring a distribution matcher and dematcher, resulting in a hardware friendly flexible architecture. +Acknowledgement: This work was financially supported by the ERC-CoG FRECOM project (grant no. 771878), +the Villum Young Investigator OPTIC-AI project (grant no. 29334), and DNRF SPOC, DNRF123. +References +[1] G. Böcherer, et. al, “Bandwidth efficient and rate matched low-density parity-check coded modulation,” IEEE Trans. on Comm., 2015. +[2] T. O’Shea and J. Hoydis, “An Introduction to Deep Learning for the Physical Layer,” IEEE Trans. on Cognitive Comm. and Net., 2017. +[3] B. Karanov, et. al., “End-to-End Deep Learning of Optical Fiber Communications,” JLT, 2018. +[4] R. T. Jones, et. al., “Deep Learning of Geometric Constellation Shaping Including Fiber Nonlinearities,” in ECOC, 2018. +[5] R. T. Jones, et. al., “End-to-end learning for GMI optimized geometric constellation shape,” in ECOC, 2019. +[6] J. Song, et.al., “Model-Based End-to-End Learning for WDM Systems With Transceiver Hardware Impairments,” IEEE JOSTQE, 2022. +[7] O. Jovanovic, et. al.” Geometric Constellation Shaping for Fiber-Optic Channels via End-to-End Learning,” arXiv:2211.04311, 2022. +[8] G. Böcherer, et. al., "Probabilistic Shaping and Forward Error Correction for Fiber-Optic Communication Systems," JLT, 2019. +[9] R. Dar, et. al., “Accumulation of nonlinear interference noise in fiber-optic systems,” Optics Express, 2014. + +0.71 +256QAM +12 +Xn +=0 +10101100 +10111100 +Xn +3/4 FEC +11 +256GS +b) +0.7 +128QAM +-2 +0 +10.5 += 2 +2 +10 +Xn_=3 +10110 +00 +9.5 +0.69 +1 +10100100 + 64QAM +n +(e 6 +c) +d) +-2 +0 +5 +10 +15 +20 +-2 +0 +2 +1.125 +1.135 +1.145 +Number of spans \ No newline at end of file diff --git a/AtAzT4oBgHgl3EQfTPzA/content/tmp_files/load_file.txt b/AtAzT4oBgHgl3EQfTPzA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69edb22c87db0cf80267034da2f2eecf07f65a0c --- /dev/null +++ b/AtAzT4oBgHgl3EQfTPzA/content/tmp_files/load_file.txt @@ -0,0 +1,139 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf,len=138 +page_content='Rate Adaptive Autoencoder-based Geometric Constellation Shaping Ognjen Jovanovic, Metodi P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Yankov, Francesco Da Ros and Darko Zibar Department of Electrical and Photonics Engineering, Technical University of Denmark, Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Lyngby, 2800, Denmark ognjo@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='dk Abstract: An autoencoder is used to optimize bit-to-symbol mappings for geometric constellation shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The mappings allow for net rate adaptivity without additional hardware complexity, while achieving up to 300km of transmission distance compared to uniform QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' © 2023 The Author(s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Introduction State of the art coherent optical communications need to be deployed in dynamic network scenarios, which require a certain degree of adaptivity to varying channel conditions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Classically, this is handled by varying the modulation format size, which 1) often produces a coarse granularity in the rate with steps of 2 bits/symbol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' and 2) requires the transceiver to support bit to symbol mapping and demapping to/from constellations of different size, increasing the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The probabilistic amplitude shaping (PAS) scheme [1] has emerged as an efficient architecture that provides a solution to the first problem, at the cost of rate matcher and dematcher, which increases complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Autoencoders (AEs) are becoming a popular tool to optimize the signaling constellation of digital communication transceivers [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' In optical communications, AEs have been employed for geometric constellation shaping (GCS) [4], bit labeling [5], mostly with the target of mitigating the impact of fiber nonlinearities, as well as transceiver impairment mitigation [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' GCS is beneficial over PAS because it does not require explicit matcher and dematcher blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' However, rate adaptivity with GCS typically requires a rate-flexible forward error correction (FEC), which may increase the complexity of the digital logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Further, GCS does not allow for straight-forward Gray labeling to be performed, which may lead to sub-optimality when combined with conventional bit-metric decoders as in standard coherent optical communications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' In this paper, an AE is used to 1) find optimal bit mappings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' and 2) find optimized constellation points for a variety of net rates while maintaining a fixed FEC and demapper logic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' log-likelihood ratio (LLR) computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The system therefore allows for shaping gain to be achieved with a finer granularity without a complexity increase w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' conventional bit-interleaved coded modulation (BICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' System description The AE-based GCS training setup is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The mapping function is learned using the AE architecture from [5] to jointly optimize bit labeling and constellation position on the I/Q plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' During the AE training, the transmitter and receiver employ neural networks for mapping of bits and demapping to LLRs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The transmitter of the proposed system is given in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' During testing, the mapping and demapping functionalities are replaced by a look-up table (LUT) and conventional Gaussian bit- metric receiver [8], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The AE requires that the modulation format size is known and fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' For a fixed modulation format size, the generalized mutual information (GMI) typically is penalized w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' the mutual information, as the signal to noise ratio (SNR) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' This is due to the penalty in the demapper function related to the inability to resolve constellation points with similar likelihoods at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' An AE implicitly addresses this problem by ‘merging’ such points closer together and assigning (more than 2) labels with a very small Hamming distance to virtually the same point [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' a many-to-one mapping (MOM) of bits-to-symbols is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Here, we exploit this fact to achieve rate adaptivity in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The GMI of the MOM is analyzed and the bit levels which are ambiguous are not used for data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Instead, they are assigned dummy bits in order to maintain the bit flow and logic at the transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' For a system with an FEC rate of ������������ = ������������/������������, a code length of N, information block length of K and a modulation format size of M, the net data rate per dual polarization channel may be calculated as ������������′ = (2������������ − ������������������������)������������, where ������������ = ������������������������������������2 ������������ is the number of bit carried by the constellation and ������������������������ is the number of dummy bits in the labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' If ������������������������ is even, each polarization gets the same number of dummy bits, whereas if it is odd, the allocation is done such that the GMI per a) bits- >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='Channeli LLRS nd@ b) LUT FEC Channel bits- S/P K/N 2m - nd Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' a) Training setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' b) Testing setup with dummy bit insertionpolarization is similar for both polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' In this paper, the ������������������������ bit positions are selected by sorting their per-bit GMI and choosing as many as required from the lowest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' For example, when the SNR decreases, ������������������������ can be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The performance degradation of the large-size constellation at low SNR is compensated for by the increased Euclidean distance of the effectively smaller constellation achieved via MOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The receiver architecture does not change since K, N, and M are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The only addition w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' BICM is the additional LUTs for mapping depending on the channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Results A wavelength division multiplexing system is optimized using the nonlinear interference noise model from [9] with dual polarization, 5 channels, 100km spacing, FEC of rate 3/4 and modulation size ������������ = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 2 a) shows the maximum distance for a given data rate at which the GMI is above the target rate for the AE-based MOM and uniform QAM (red dotted line, ������������������������ = 0) for the central channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The AE-optimized MOM achieve an extra span of transmission distance with respect to both 256QAM and 128QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Here, due to the SNR reduction with the increase of distance, the AE learns a MOM allowing the system to be rate adaptive through insertion of dummy bits without losing the shaping gain or changing the modulation format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 2 b) shows the constellation learned for 8 spans transmission which does not achieve MOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 2 c), the constellation learned for 20 spans that achieves MOM is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' In the latter, the AE “merged” some of the points together as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 2 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The red box shows the bits that are not shared by the 4 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' These 2 bits effectively carry no information and can be assigned dummy bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' The system then exploits the resulting increased Euclidean distance of the constellation to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' a) Net rate per dual polarization channel w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' distance for uniform QAM and AE-based GCS with 3/4 FEC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' GCS learned at: b) 8 spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' c) 20 spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' d) Zoomed in point to show that 4 points collapsed to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Conclusion An autoencoder (AE) is used to optimize rate dependent bit-to-symbol mappings for QAM of fixed size and FEC of fixed rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Rate adaptivity is achieved through the AE-optimized mapping functions as a result of many-to-one mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' It achieves shaping gain for a variety of net rates without changing the receiver architecture, the modulation format size or requiring a distribution matcher and dematcher, resulting in a hardware friendly flexible architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Acknowledgement: This work was financially supported by the ERC-CoG FRECOM project (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 771878), the Villum Young Investigator OPTIC-AI project (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 29334), and DNRF SPOC, DNRF123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Böcherer, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al, “Bandwidth efficient and rate matched low-density parity-check coded modulation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' on Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' O’Shea and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Hoydis, “An Introduction to Deep Learning for the Physical Layer,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' on Cognitive Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' and Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Karanov, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', “End-to-End Deep Learning of Optical Fiber Communications,” JLT, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Jones, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', “Deep Learning of Geometric Constellation Shaping Including Fiber Nonlinearities,” in ECOC, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Jones, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', “End-to-end learning for GMI optimized geometric constellation shape,” in ECOC, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Song, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', “Model-Based End-to-End Learning for WDM Systems With Transceiver Hardware Impairments,” IEEE JOSTQE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [7] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Jovanovic, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al.” Geometric Constellation Shaping for Fiber-Optic Channels via End-to-End Learning,” arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='04311, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Böcherer, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', "Probabilistic Shaping and Forward Error Correction for Fiber-Optic Communication Systems," JLT, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' Dar, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=', “Accumulation of nonlinear interference noise in fiber-optic systems,” Optics Express, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='71 256QAM 12 Xn =0 10101100 10111100 Xn 3/4 FEC 11 256GS b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='7 128QAM 2 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='5 = 2 2 10 Xn_=3 10110 00 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='69 1 10100100 64QAM n (e 6 c) d) 2 0 5 10 15 20 2 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='135 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} +page_content='145 Number of spans' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'} diff --git a/BdE1T4oBgHgl3EQf9gYX/content/tmp_files/2301.03556v1.pdf.txt b/BdE1T4oBgHgl3EQf9gYX/content/tmp_files/2301.03556v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc0c3a2d0f663979ad32bb74de3a0c25ec2d1890 --- /dev/null +++ b/BdE1T4oBgHgl3EQf9gYX/content/tmp_files/2301.03556v1.pdf.txt @@ -0,0 +1,2331 @@ +Nuclear shape fluctuations in high-energy heavy ion collisions +Aman Dimri,1, ∗ Somadutta Bhatta,1 and Jiangyong Jia1, 2, † +1Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA +2Physics Department, Brookhaven National Laboratory, Upton, NY 11976, USA +(Dated: January 10, 2023) +Atomic nuclei often exhibit a quadrupole shape that fluctuates around some average profile. +We investigate the impact of nuclear shape fluctuation on the initial state geometry in heavy ion +collisions, particularly its eccentricity ε2 and inverse size d⊥, which can be related to the elliptic flow +and radial flow in the final state. The fluctuation in overall quadrupole deformation enhances the +variances and modifies the skewness and kurtosis of the ε2 and d⊥ in a controllable manner. The +fluctuation in triaxiality reduces the difference between prolate and oblate shape for any observable, +whose values, in the large fluctuation limit, approach those obtained in collisions of rigid triaxial +nuclei. The method to disentangle the mean and variance of the quadrupole deformation is discussed. +PACS numbers: 25.75.Gz, 25.75.Ld, 25.75.-1 +I. +INTRODUCTION +Ultra-relativistic heavy ion physics aims to understand the dynamics and properties of the Quark-Gluon Plasma +(QGP) created in collisions of atomic nuclei at very high energy [1]. +Achieving this goal is currently limited by +the lack of understanding of the initial condition, i.e. how the energy is deposited in the overlap region before the +formation of QGP [2]. The energy deposition process is not calculable from first principles and is often parameterized +via phenomenological approaches with multiple free parameters [3]. On the other hand, heavy atomic nuclei are +well-studied objects exhibiting a wide range of shapes and radial profiles [4], which are often characterized by a few +collective nuclear structure parameters such as quadrupole, triaxial, and octupole deformations, nuclear radius and +skin thickness. One can leverage species with similar mass numbers but different structures, such as isobars, to directly +probe the energy deposition mechanism and hence constrain the initial condition. The efficacy of this approach has +been investigated recently [5–7]. +One good example demonstrating this possibility is the 96Ru+96Ru and 96Zr+96Zr collisions, recently carried out +by the STAR Collaboration at the relativistic heavy ion collider [8, 9]. Ratios of many bulk observables between +the isobars, such as harmonic flow vn, charged particle multiplicity Nch, and average transverse momentum ⟨pT⟩, +have been measured, which show significant and observable- and centrality-dependent deviation from unity. Model +studies show that these ratios are insensitive to final-state effects and are controlled mainly by the differences of the +collective nuclear structure parameters between 96Ru and 96Zr [10]. Comparing the calculations with experimental +data, Refs. [5, 11] have estimated structure parameters that are broadly consistent with general knowledge from low +energy. However, these studies also suggest a sizable octupole collectivity for Zr, not predicted by mean field structure +models [12]. The rich and versatile information from isobar or isobar-like collisions provides a new constraint on the +heavy ion initial condition and a new way to probe nuclear structure at high energy [13]. +However, it is important to point out that atomic nuclei in the ground state often do not have a static shape, but +can fluctuate due to interplay between collective modes and single-particle states [14]. The potential energy surface of +such species usually has shallow minimums as a function of deformation parameters, such as quadruple deformation +β and triaxiality γ. The ground state nuclear wave function is often treated as a mixture of configurations with +different (β,γ) values [4, 15]. Then there are the phenomena of shape coexistence, which happens when the same +nuclei can have multiple low-lying states with widely different shapes but small energy differences [16]. From the +nuclear structure side, the quadrupole fluctuations can be estimated from the sum rules of matrix elements of various +moments of quadrupole operators that can be measured experimentally [17, 18]. From the heavy ion collision side, +the shape fluctuations can be accessed using multi-particle correlations, which probe moments of the nucleon position +in the initial condition [19]. For instance, the elliptic flow v2 in each event is proportional to the elliptic eccentricity +ε2, v2 = kε2 calculable from participating nucleons [20]. Therefore, the fluctuations of flow are related to fluctuations +of quadruple deformation via their respective moments: ⟨vm +2 ⟩ = km ⟨εm +2 ⟩ ∝ ⟨βm⟩,m = 2,4... In principle, one could +∗Electronic address: aman.dimri@stonybrook.edu +†Electronic address: jiangyong.jia@stonybrook.edu +arXiv:2301.03556v1 [nucl-th] 9 Jan 2023 + +2 +constrain the mean and variance of quadrupole fluctuations from the ⟨β2⟩ and ⟨β4⟩, which in terms can be determined +from ⟨v2 +2⟩ and ⟨v4 +2⟩. +This paper extends our previous study to investigate the influence of fluctuations of quadruple deformation param- +eters (β,γ) to several selected two-, three- and four-particle heavy-ion observables. We first derive simple analytical +relations between these observables and the means and variances of (β,γ). We then perform a more realistic Glauber +model simulation, assuming Gaussian fluctuations, to quantify the region of validity of these relations. We discuss +the sensitivity of these observables on the nuclear shape, as well as the prospect of separating the average shape from +shape fluctuations. +II. +EXPECTATION AND MODEL SETUP +We consider the eccentricity vector ϵ2 and inverse transverse size d⊥, which are estimators for elliptic flow V2 ≡ +v2e2iΨ2 and average transverse momentum ⟨pT⟩ or radial flow, calculated from the transverse position of nucleon +participants in each event, +ϵ2 = − +⟨r2 +⊥ei2φ⟩ +⟨r2⊥⟩ +, d⊥ = +√ +Npart/⟨r2⊥⟩, +(1) +where r⊥ is the transverse radius and Npart is the number of participating nucleons. Following the heuristic argument +from Ref. [21], for collisions of nuclei with small quadrupole deformation, the eccentricity vector and d⊥ in a given +event have the following leading-order form: +δd⊥ +d⊥ +≈ δd + p0(Ωp,γp)βp + p0(Ωt,γt)βt , ϵ2 ≈ ϵ0 + p2(Ωp,γp)βp + p2(Ωt,γt)βt, +(2) +where the scalar δd and vector ϵ0 are valued for spherical nuclei, and we are considering the general situation where +the projectile and target, denoted by subscripts “p” and “t”, have different deformation values. The p0 and p2 are +phase space factors, which depend on γ and the Euler angles Ω. +Since the fluctuations of δd (ϵ0) are uncorrelated with p0 (p2), an average over collisions with different Euler angles +is expected to give the following leading-order expressions for the variances, skewness, and kurtosis of the fluctuations +c2,ϵ{2} ≡ ⟨ε2 +2⟩ = ⟨ε2 +0⟩ + ⟨p2(γp)p∗ +2(γp)⟩β2 +p + ⟨p2(γt)p∗ +2(γt)⟩β2 +t +cd{2} ≡ ⟨(δd⊥ +d⊥ +) +2 +⟩ = ⟨δ2 +d⟩ + ⟨p0(γp)2⟩β2 +p + ⟨p0(γt)2⟩β2 +t +Cov ≡ ⟨ε2 +2 +δd⊥ +d⊥ +⟩ = ⟨ε2 +0δd⟩ + ⟨p0(γp)p2(γp)p2(γp)∗⟩β3 +p + ⟨p0(γt)p2(γt)p2(γt)∗⟩β3 +t +cd{3} ≡ ⟨(δd⊥ +d⊥ +) +3 +⟩ = ⟨δ3 +d⟩ + ⟨p0(γp)3⟩β3 +p + ⟨p0(γt)3⟩β3 +t +c2,ϵ{4} ≡ ⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2 = ⟨ε4 +0⟩ − 2⟨ε2 +0⟩ +2 + (⟨p2 +2p2∗ +2 ⟩⟨β4⟩ − 2⟨p2p∗ +2⟩2 ⟨β2⟩ +2) +p + (⟨p2 +2p2∗ +2 ⟩⟨β4⟩ − 2⟨p2p∗ +2⟩2 ⟨β2⟩ +2) +t . +(3) +These quantities relate directly to the final state observables, ⟨v2 +2⟩, ⟨(δpT/⟨pT⟩)2⟩, ⟨v2 +2 +δpT +⟨pT⟩⟩, ⟨(δpT/⟨pT⟩)3⟩ and +⟨v4 +2⟩ − 2⟨v2 +2⟩ +2, respectively. +Previous studies have demonstrated that the moments ⟨p2 +0⟩, ⟨p2p∗ +2⟩, and ⟨p2 +2p2∗ +2 ⟩ are independent of γ, while +⟨p0p2p∗ +2⟩ and ⟨p3 +0⟩ have leading order dependence on γ: c + bcos(3γ). Here, c ≪ b for ⟨p0p2p∗ +2⟩, whereas c ≲ b for +⟨p3 +0⟩ [21]. In the presence of quadrupole fluctuations, we also need to average these quantities over “independent” + +3 +fluctuations for projectile and target, giving, +⟨(δd⊥ +d⊥ +) +2 +⟩ = a0 + b0 +2 (⟨β2 +p⟩ + ⟨β2 +t ⟩) = a0 + b0 ⟨β2⟩ +⟨ε2 +2⟩ = a1 + b1 +2 (⟨β2 +p⟩ + ⟨β2 +t ⟩) = a1 + b1 ⟨β2⟩ +⟨ε2 +2 +δd⊥ +d⊥ +⟩ = a2 − 1 +2 (⟨(c2 + b2 cos(3γp))β3 +p⟩ + ⟨(c2 + b2 cos(3γt))β3 +t ⟩) = a2 − ⟨(c2 + b2 cos(3γ))β3⟩ +⟨(δd⊥ +d⊥ +) +3 +⟩ = a3 + 1 +2 (⟨(c3 + b3 cos(3γp))β3 +p⟩ + ⟨(c3 + b3 cos(3γt)β3 +t ⟩) = a3 + ⟨(c3 + b3 cos(3γ))β3⟩ +⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2 = a4 + b4 +2 (⟨β4 +p⟩ + ⟨β4 +t ⟩) − c4 +2 (⟨β2 +p⟩ +2 + ⟨β2 +t ⟩ +2) = a4 + b4 ⟨β4⟩ − c4 ⟨β2⟩ +2 , +(4) +where the averages are performed over fluctuations in β and γ, and the coefficients an, bn and cn are centrality- +dependent positive quantities satisfying c2 ≪ b2 and c3 ≲ b3 [21]. The second part of these equations is obtained by +assuming that the fluctuations of the projectile and target are sampled from the same probability density distributions. +For a more quantitative estimation, we consider the liquid-drop model where the nucleon density distribution has +a sharp surface. For head-on collisions with zero impact parameter, it predicts the following simple relations [21], +δd⊥ +d⊥ += +√ +5 +16π β2 (cosγD2 +0,0 + sinγ +√ +2 +[D2 +0,2 + D2 +0,−2]) , ϵ2 = − +√ +15 +2π β2 (cosγD2 +2,0 + sinγ +√ +2 +[D2 +2,2 + D2 +2,−2]) , +(5) +where the Dl +m,m′(Ω) are the Wigner matrices. The analytical results obtained for various cumulants are listed in +Table I. They provide approximate estimates for the values of bn in most central collisions. +⟨(δd⊥/d⊥)2⟩ +⟨(δd⊥/d⊥)3⟩ +⟨(δd⊥/d⊥)4⟩ − 3 ⟨(δd⊥/d⊥)2⟩ +2 +1 +32π ⟨β2⟩ +√ +5 +896π3/2 ⟨cos(3γ)β3⟩ +− +3 +14336π2 (7 ⟨β2⟩ +2 − 5 ⟨β4⟩) +⟨ε2 +2⟩ +⟨ε4 +2⟩ − 2 ⟨ε2 +2⟩ +2 +(⟨ε6 +2⟩ − 9 ⟨ε4 +2⟩ ⟨ε2 +2⟩ + 12 ⟨ε2 +2⟩ +3)/4 +3 +4π ⟨β2⟩ +− +9 +112π2 (7 ⟨β2⟩ +2 − 5 ⟨β4⟩) +81 +256π3 [⟨β2⟩ +3 − 45 +14 ⟨β4⟩ ⟨β2⟩ − 1175 +6006 ⟨β6⟩ + +25 +3003 ⟨cos(6γ)β6⟩] +⟨ε2 +2(δd⊥/d⊥)⟩ +⟨ε2 +2(δd⊥/d⊥)2⟩ − ⟨ε2 +2⟩ ⟨(δd⊥/d⊥)2⟩ +⟨ϵ2 +2ϵ∗ +4⟩ +− +3 +√ +5 +112π3/2 ⟨cos(3γ)β3⟩ +− +3 +1792π2 (7 ⟨β2⟩ +2 − 5 ⟨β4⟩) +45 +56π2 ⟨β4⟩ +TABLE I: The leading-order results of various cumulants calculated for the nucleus with a sharp surface via Eq. 5. The two +nuclei are placed with zero impact parameter and results are obtained by averaging over random orientations. +To make further progress, we consider the case where the fluctuations of β and γ are independent of each other. +The observables in Eq. 4 and Table I can be expressed in terms of central moments. Assuming Gaussian fluctuations +with means ¯β or ¯γ and variances σβ or σγ, Eq. 4 becomes +⟨ε2 +2⟩ = a1 + b1(¯β2 + σ2 +β) ,⟨(δd⊥ +d⊥ +) +2 +⟩ = a0 + b0(¯β2 + σ2 +β) +⟨ε2 +2 +δd⊥ +d⊥ +⟩ = a2 − (b2e− +9σ2 +γ +2 cos(3¯γ) + c2)¯β(¯β2 + 3σ2 +β) +⟨(δd⊥ +d⊥ +) +3 +⟩ = a3 + (b3e− +9σ2 +γ +2 cos(3¯γ) + c3)¯β(¯β2 + 3σ2 +β) +⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2 = a4 + b4(¯β4 + 6σ2 +β ¯β2 + 3σ4 +β) − c4(¯β2 + σ2 +β)2 . +(6) +where we have used the well-known expression for Gaussian smearing of an exponential function, ⟨einγ⟩ = e− +n2σ2 +γ +2 +ein¯γ. + +4 +If the fluctuations of β and γ are non-Gaussian, one should also consider the higher cumulants of β. For example, +⟨β3⟩ = ¯β(¯β + 3σ2 +β) + k3,β and ⟨β4⟩ = ¯β4 + 6σ2 +β ¯β2 + 3σ4 +β + 4¯βk3,β + k4,β, where k3,β = ⟨(β − ¯β)3⟩ and k4,β = ⟨(β − ¯β)4⟩ − +3⟨(β − ¯β)2⟩ +2 are the skewness and kurtosis of the β fluctuation. The expectation value of cos(nγ) can be expressed +via the cumulant generating function of γ. Keeping the cumulants km,γ up to leading order correction in skewness +and kurtosis, k3,γ = ⟨(γ − ¯γ)3⟩ and k4,γ = ⟨(γ − ¯γ)4⟩ − 3⟨(γ − ¯γ)2⟩ +2, we have, +⟨cos(nγ)⟩ = 1 +2 (⟨ein¯γ⟩ + ⟨e−in¯γ⟩) = 1 +2 (exp( +∞ +∑ +m=1 +κm,γ +(in)m +m! +) + exp( +∞ +∑ +m=1 +κm,γ +(−in)m +m! +)) += exp( +∞ +∑ +m=1 +κ2m,γ +(−1)m(n)2m +2m! +)[cos( +∞ +∑ +m=1 +κ2m+1,γ +(−1)m(n)2m+1 +(2m + 1)! ++ n¯γ)] +≈ e− +n2σ2 +γ +2 ++ +n4k4,γ +24 +cos(n¯γ + n3 +6 k3,γ) ≈ e− +n2σ2 +γ +2 +[cos(n¯γ) + sin(n¯γ)n3 +6 k3,γ](1 + n4 +24k4,γ). +(7) +Clearly, the net effect of skewness is a rotation of ¯γ by k3,γn2/6, and the net effect of kurtosis is to increase or decrease +the overall variation with γ depending on its sign. +For a more realistic estimation of the influences of shape fluctuations, we perform a Monte-Carlo Glauber model +simulation of 238U+238U collisions. The setup of the model and the data used in this analysis are the same as those +used in our previous work [19]. We simulate ultra-central collisions with zero impact parameter, where the impact of +nuclear deformation reaches maximum. The nucleon distribution is described by a deformed Woods-Saxon function +ρ(r,θ,φ) = +ρ0 +1 + e[r−R(θ,φ)/a] , R(θ,φ) = R0 (1 + β[cosγY2,0(θ,φ) + sinγY2,2(θ,φ)]), +(8) +where the nuclear surface R(θ,φ) is expanded into spherical harmonics Y2,m in the intrinsic frame. Each nucleus is +assigned a random (β,γ) value, sampled from Gaussian distributions with means (¯β, ¯γ) and variances (σβ,σγ). The +nucleus is then rotated by random Euler angles before they are set on a straight line trajectory towards each other +along the z direction. Furthermore, three quark constituents are generated for each nucleon according to the quark +Glauber model from Ref. [22]. From this, the nucleons or the constituent quarks in the overlap region are identified, +which are used to calculate the ε2 and d⊥ defined in Eqs. (1), and the results are presented as a function of deformation +parameters. +For the study of the β fluctuation, we fix the γ = 0 (prolate nucleus) and choose 11 values each for ¯β2 and +σ2 +β with 0, 0.01,...,0.09, 0.1. So a total of 11 × 11 = 121 simulations have been performed. For the study of the +γ fluctuation, we fix β = 0.28 and choose seven ¯γ and seven σγ values: cos(3¯γ) = 1,0.87,0.5,0,−0.5,0.87,−1 and +σγ = 0,π/18,2π/18,...,6π/18, so a total of 7 × 7 = 49 simulation have been performed. For each case, about 50 Million +events were generated and all the observables were calculated. Our discussion is mainly based on the nucleon Glauber +model, and the results from the quark Glauber model are included in the Appendix. +III. +IMPACT OF TRIAXIALITY FLUCTUATION +Due to the three-fold symmetry of nuclei shape in triaxiality, the γ dependence of a given observable can be +generally expressed as a0 +∑∞ +n=1 [an cos(3n¯γ) + bn sin(3n¯γ)]e− +n2σ2 +γ +2 +. If we further impose the condition that a random +fluctuation for a triaxial nucleus does not impact the value of the observable, which is found to be true in our analysis. +This leads to the γ dependence of the form a0 + ∑∞ +n=1 [an(cos(3n¯γ) − cos(3n π +6 )) + bn(sin(3n¯γ) − sin(3n π +6 ))]e− +n2σ2 +γ +2 +. +We first discuss the impact of triaxiality fluctuation on three-particle observables ⟨ε2 +2 +δd⊥ +d⊥ ⟩ and ⟨(δd⊥/d⊥)3⟩. We first +subtract them by the values for the undeformed case, to isolate the second term in Eq. 4 containing the triaxiality. +Figure 1 show the results obtained for different values of cos3¯γ as a function of σγ. The values for triaxial nucleus +cos(3¯γ) = 0 are indeed independent of σγ. The fluctuation of γ reduces the difference between the prolate ¯γ = 0 and +the oblate ¯γ = π/3 shape. This reduction is largely described by e− +9σ2 +γ +2 cos(3¯γ), except for a small asymmetry between +¯γ = 0 and ¯γ = π/3, clearly visible for ⟨(δd⊥/d⊥)3⟩. +We account for this small asymmetry by including higher-order terms in the fit function allowed by symmetry. + +5 +0 +0.2 +0.4 +0.6 +0.8 +1 +γ +σ +0.2 +− +0.15 +− +0.1 +− +0.05 +− +0 +0.05 +0.1 +0.15 +0.2 +3 +− +10 +× + +=0 +β + Cov - Cov +1.00 +0.50 +-0.50 +-1.00 + + + + + + + + +-5 +10 +× + = -0.85 +0 +a +-5 +10 +× +) = (-19.98,-0.03) +2 +,a +1 +(a +U+U Glauber Model +-5 +10 +× +) = (0.28,0.08) +2 +,b +1 +(b + = 0.28 +β +b = 0 fm, +) Data Fit +γ +Cos(3 +0 +0.2 +0.4 +0.6 +0.8 +1 +γ +σ +2 +− +0 +2 +4 +6 +8 +6 +− +10 +× + +=0 +β +{3} +d + - C +{3} +d +C +0.87 +0.00 +-0.87 + + + + + + +-6 +10 +× + = 3.19 +0 +a +-6 +10 +× +) = (5.18,-0.18) +2 +,a +1 +(a +U+U Glauber Model +-6 +10 +× +) = (0.27,0.04) +2 +,b +1 +(b + = 0.28 +β +b = 0 fm, +) Data Fit +γ +Cos(3 +FIG. 1: +The dependence of ⟨ε2 +2 +δd⊥ +d⊥ ⟩ (left) and ⟨(δd⊥/d⊥)3⟩ (right) on smearing in triaxiality σγ for different values of ¯γ. The +lines indicate a simultaneous fit to Eq. 10 with the parameter values displayed on the plot. +Keeping leading and subleading terms, we have, +⟨ε2 +2 +δd⊥ +d⊥ +⟩ − ⟨ε2 +2 +δd⊥ +d⊥ +⟩ +β=0 += [a′ +0 + (a′ +1 cos(3¯γ) + b′ +1 [sin(3¯γ) − 1])e− +9σ2 +γ +2 ++ (a′ +2 [cos(6¯γ) + 1] + b′ +2 sin(6¯γ))e− +36σ2 +γ +2 ] ¯β3 +(9) += a0 + (a1 cos(3¯γ) + b1 [sin(3¯γ) − 1])e− +9σ2 +γ +2 ++ (a2 [cos(6¯γ) + 1] + b2 sin(6¯γ))e− +36σ2 +γ +2 +. +(10) +The same fit function is also used to describe ⟨(δd⊥/d⊥)3⟩. The parameters in the first line and those in the second +line differ by a scale factor ¯β3 = 0.283 = 0.021. From the values of parameters displayed in Fig. 1, we concluded that +the magnitude of the high-order order terms is less than 2% of the magnitude of a1 for ⟨ε2 +2 +δd⊥ +d⊥ ⟩ but reaches up to 5% +for ⟨(δd⊥/d⊥)3⟩. +Figure 1 shows that the signature of triaxiality in heavy ion collisions is greatly reduced for large value of σγ, often +found in γ-soft nuclei. A twenty-degree fluctuation in triaxiality, for example, reduces the signal by nearly 50%. It +would be difficult to distinguish between static rigid triaxial nuclei and nuclei with large fluctuations around ¯γ = π/6 +using heavy ion collisions. In particular, nuclei that fluctuate uniformly between prolate and oblate shapes would give +the same three-particle correlation signal as rigid triaxial nuclei! Such strong smearing also degrades the prospects of +using higher-order cumulants of ε2 to infer the value of σγ. +For the other three observables, ⟨ε2 +2⟩, ⟨(δd⊥/d⊥)2⟩ and ⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2, γ dependence are known to be very weak [19]. +Nevertheless, up to a few percent dependence is observed, which can also be parameterized by Eq. 9, except that +we should change ¯β3 to ¯β2 for the variances and to ¯β4 for ⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2. However, since ¯β is fixed at 0.28, all these +observables can be parameterized by Eq. 10. The data and the results of the fits are shown in Fig. 2. First, we observe +that the parameter a0, representing the baseline contribution associated with ¯β is by far the largest, and the other +terms only cause a few percent of modulation. Secondly, while the ⟨ε2 +2⟩ and ⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2 can be largely described +by including the cos(3¯γ) term, the description of ⟨(δd⊥/d⊥)2⟩ requires the inclusion of sin(3¯γ), cos(6¯γ) and sin(6¯γ) +terms with comparable magnitudes. Lastly, all three observables have no sensitivity to ¯γ at large σγ. +IV. +IMPACT OF QUADRUPLE DEFORMATION FLUCTUATION +Next, we consider the impact of β fluctuations. +For this purpose, we shall fix the γ to be prolate shape, e.g +cos(3γ) = 1. Figure 3 displays the finding for two-particle observables ⟨ε2 +2⟩ and ⟨(δd⊥/d⊥)2⟩, again they are corrected +by the undeformed baseline. Although approximately-linear dependencies on ¯β2 are observed for both observables, +the slopes of the data points also vary with σβ. To describe this feature, we include two higher-order terms, + +6 +0 +0.2 +0.4 +0.6 +0.8 +1 +γ +σ +15.8 +16 +16.2 +16.4 +16.6 +16.8 +3 +− +10 +× +=0 +β〉 +2 +2 +ε 〈 + - +〉 +2 +2 +ε 〈 +1.00 +0.50 +-0.50 +-1.00 + + + + + + + + +-5 +10 +× + = 1621.87 +0 +a +-5 +10 +× +) = (52.46,-0.29) +2 +,a +1 +(a +U+U Glauber Model +-5 +10 +× +) = (-0.79,-0.53) +2 +,b +1 +(b + = 0.28 +β +b = 0 fm, +) Data Fit +γ +Cos(3 +0 +0.2 +0.4 +0.6 +0.8 +1 +γ +σ +0.6 +0.62 +0.64 +0.66 +0.68 +3 +− +10 +× +=0 +β +{2} +d + - C +{2} +d +C +0.87 +0.00 +-0.87 + + + + + + +-5 +10 +× + = 65.83 +0 +a +-5 +10 +× +) = (-1.89,-1.31) +2 +,a +1 +(a +U+U Glauber Model +-5 +10 +× +) = (1.48,0.08) +2 +,b +1 +(b + = 0.28 +β +b = 0 fm, +) Data Fit +γ +Cos(3 +0 +0.2 +0.4 +0.6 +0.8 +1 +γ +σ +0.105 +− +0.1 +− +0.095 +− +0.09 +− +0.085 +− +3 +− +10 +× +=0 +β +{4} +ε +2, + - c +{4} +ε +2, +c +-5 +10 +× + = -9.76 +0 +a +-5 +10 +× +) = (1.03,0.10) +2 +,a +1 +(a +U+U Glauber Model +-5 +10 +× +) = (-0.13,-0.01) +2 +,b +1 +(b + = 0.28 +β +b = 0 fm, +FIG. 2: +The dependence of ⟨ε2 +2⟩ (left), ⟨(δd⊥/d⊥)2⟩ (middle), and ⟨ε4 +2⟩ − 2 ⟨ε2 +2⟩ +2 (right) on σγ for different values of ¯γ. The +dashed lines indicate a simultaneous fit to Eq. 10, with fit results are displayed on the plot. +⟨ε2 +2⟩ − ⟨ε2 +2⟩β=0 or ⟨(δd⊥ +d⊥ +) +2 +⟩ − ⟨(δd⊥ +d⊥ +) +2 +⟩ +β=0 += c1 ⟨β2⟩ + c2 ⟨β3⟩ + c3 ⟨β4⟩ += c1(¯β2 + σ2 +β) + c2 ¯β(¯β2 + 3σ2 +β) + c3(¯β4 + 6σ2 +β ¯β2 + 3σ4 +β) +(11) +The fits including only the leading term and all three terms are shown in the first row and the last row of Fig. 3, +respectively. The fits in the middle row include the c1 and c3 terms for ⟨ε2 +2⟩, while they include c1 and c2 terms for +⟨(δd⊥/d⊥)2⟩. Clearly, the behavior of ⟨(δd⊥/d⊥)2⟩ at large ¯β or σβ requires the presence of the ⟨β3⟩ term in Eq. 11 +with a negative coefficient c2 < 0. In general, a large fluctuation σβ tends to reduce the slope of the dependence on +¯β2. +For the three-particle correlators, we include three terms in the fitting function as +⟨ε2 +2 +δd⊥ +d⊥ +⟩ − ⟨ε2 +2 +δd⊥ +d⊥ +⟩ +β=0 +or ⟨(δd⊥ +d⊥ +) +3 +⟩ − ⟨(δd⊥ +d⊥ +) +3 +⟩ +β=0 += c1 ⟨β3 cos(3γ)⟩ + c2 ⟨β4 cos(3γ)⟩ + c3 ⟨β5 cos(3γ)⟩ += [c1 ¯β(¯β2 + 3σ2 +β) + c2(¯β4 + 6σ2 +β ¯β2 + 3σ4 +β) + c3(¯β5 + 10σ3 +β ¯β2 + 15¯βσ4 +β)]cos(3γ) +(12) +The fitting results are shown in Fig. 4 as a function of ¯β3 for the prolate case cos(3γ) = 1. Inclusion of the high-order +terms, mostly contribution from the ⟨β5⟩ component, improves the description of ⟨ε2 +2 +δd⊥ +d⊥ ⟩ in the region of large σβ. +However, they are not sufficient to describe the ⟨(δd⊥/d⊥)3⟩ in the region of large ¯β and σβ. In particular, the fit also +misses all points at ¯β = 0. We checked that the fit can be systematically improved by including more higher moment +terms, albeit only very slowly. +Lastly, we consider the four-particle observable c2,ε{4} = ⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2. According to findings in Fig. 3, the Taylor +expansion of ⟨ε2 +2⟩ should give the first two terms as c1 ⟨β2⟩ + c2 ⟨β4⟩, similarly the first few terms of ⟨ε4 +2⟩ has the form +of a0 ⟨β2⟩ +2 + a1 ⟨β4⟩ + a2 ⟨β6⟩. Therefore, the natural expression for c2,ε{4} up to second order correction should be +c2,ε{4} − c2,ε{4}β=0 = a0 ⟨β2⟩ +2 + a1 ⟨β4⟩ + a2 ⟨β6⟩ − (c1 ⟨β2⟩ + c2 ⟨β4⟩)2 ≈ a1 ⟨β4⟩ − b1 ⟨β2⟩ +2 + a2 ⟨β6⟩ − b2 ⟨β2⟩⟨β4⟩ += a1(¯β4 + 6¯β2σ2 +β + 3σ4 +β) − b1(¯β2 + σ2 +β)2 + a2(¯β6 + 15¯β4σ2 +β + 45¯β2σ4 +β + 15σ6 +β) − b2(¯β2 + σ2 +β)(¯β4 + 6¯β2σ2 +β + 3σ4 +β) +(13) +with b1 = c2 +1 − a0 and b2 = 2c1c2. The leading order correction includes the first two terms with a1 and b1, while the +remaining two terms are the subleading-order corrections. +The results from the Glauber model and the fit to Eq. 13 are shown in the left panel of Fig. 5. The strong variation +of c2,ε{4} with both ¯β and σβ is captured nicely by the fit. For small values of σβ, the deformation has a negative +contribution to c2,ε{4} that is proportional to ¯β4. +For large values of σβ, c2,ε{4} becomes positive. +A previous +study shows that the centrality fluctuation also tends to give a positive value of c2,ε{4} [23]. Therefore, a negative +c2,ε{4} which decreases further in central collisions would be an unambiguous indication for a large static quadrupole +deformation of the colliding nuclei. +The values of the fit parameters show some interesting relations, i.e. b1 ≈ 3/2a1 and b2 ≈ 2a2. This means that the + +7 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +5 +10 +15 +20 +25 +30 +35 +40 +3 +− +10 +× +0,0 +〉 +2 +2 +ε 〈 + - +〉 +2 +2 +ε 〈 +0.00 +0.01 +0.02 +0.03 + + + + + + + + +U+U Glauber Model +-1 +10 +× + = 1.93 +1 +c +b = 0 fm + Data Fit +2 +β +σ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +0.2 +0.4 +0.6 +0.8 +1 +3 +− +10 +× +0,0 +{2} +d + - C +{2} +d +C +U+U Glauber Model +-3 +10 +× + = 5.94 +1 +c +b = 0 fm +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +5 +10 +15 +20 +25 +30 +35 +40 +3 +− +10 +× +0,0 +〉 +2 +2 +ε 〈 + - +〉 +2 +2 +ε 〈 +0.04 +0.05 +0.06 +0.07 + + + + + + + + +U+U Glauber Model +-1 +10 +× +) = (2.15,-0.68) +3 +,c +1 +(c +b = 0 fm + Data Fit +2 +β +σ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +0.2 +0.4 +0.6 +0.8 +1 +3 +− +10 +× +0,0 +{2} +d + - C +{2} +d +C +U+U Glauber Model +-3 +10 +× +) = (8.89,-5.98) +2 +,c +1 +(c +b = 0 fm +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +5 +10 +15 +20 +25 +30 +35 +40 +3 +− +10 +× +0,0 +〉 +2 +2 +ε 〈 + - +〉 +2 +2 +ε 〈 +0.08 +0.09 +0.10 + + + + + + +U+U Glauber Model +-1 +10 +× +) = (2.10,0.27,-0.91) +3 +,c +2 +,c +1 +(c +b = 0 fm + Data Fit +2 +β +σ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +0.2 +0.4 +0.6 +0.8 +1 +3 +− +10 +× +0,0 +{2} +d + - C +{2} +d +C +U+U Glauber Model +-3 +10 +× +) = (8.74,-3.62,-3.01) +3 +,c +2 +,c +1 +(c +b = 0 fm +FIG. 3: +The simultaneous fit of the ⟨ε2 +2⟩ (¯β, σβ) (left column) and ⟨(δd⊥/d⊥)2⟩ (¯β, σβ) (right column) calculated in U+U +collisions with zero impact parameter. The top row shows the fits to Eq. 11 with only the leading term and the last row +shows the fits with all three terms. The middle row show the fits including c1 and c3 terms for ⟨ε2 +2⟩ and c1 and c2 terms for +⟨(δd⊥/d⊥)2⟩. +distribution can also be described by the following alternative form, +c2,ε{4} − c2,ε{4}β=0 ≈ a1 +2 (6¯β2σ2 +β + 3σ4 +β − ¯β4) + a2(¯β4σ2 +β + 27¯β2σ4 +β + 9σ6 +β − ¯β6) +(14) + +8 +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +3 +β +0.8 +− +0.7 +− +0.6 +− +0.5 +− +0.4 +− +0.3 +− +0.2 +− +0.1 +− +0 +0.1 +3 +− +10 +× +0,0 +Cov - Cov +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 + + + + + + + + + + + + +U+U Glauber Model +b = 0 fm +-3 +10 +× + = -6.63 +1 +c + Data Fit +2 +β +σ +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +3 +β +4 +− +2 +− +0 +2 +4 +6 +8 +10 +12 +14 +6 +− +10 +× +0,0 +{3} +d + - C +{3} +d +C +U+U Glauber Model +b = 0 fm +-4 +10 +× + = 1.36 +1 +c +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +3 +β +0.8 +− +0.7 +− +0.6 +− +0.5 +− +0.4 +− +0.3 +− +0.2 +− +0.1 +− +0 +0.1 +0.2 +3 +− +10 +× +0,0 +Cov - Cov +0.06 +0.07 +0.08 +0.09 +0.10 + + + + + + + + + + +U+U Glauber Model +b = 0 fm +-3 +) = (-8.96,-0.47,5.95)*10 +3 +,c +2 +,c +1 +(c + Data Fit +2 +β +σ +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +3 +β +4 +− +2 +− +0 +2 +4 +6 +8 +10 +12 +14 +6 +− +10 +× +0,0 +{3} +d + - C +{3} +d +C +U+U Glauber Model +b = 0 fm +-4 +) = (3.11,-0.69,-2.74)*10 +3 +,c +2 +,c +1 +(c +FIG. 4: +The simultaneous fit of the ⟨ε2 +2 +δd⊥ +d⊥ ⟩ (¯β, σβ) (left column) and ⟨(δd⊥/d⊥)3⟩ (¯β, σβ) (right column) calculated in U+U +collisions with zero impact parameter. The top row shows the results of the fit to Eq. 12 with only the leading term and the +second row shows the fits with all three terms. The fit results imply that the contribution from ⟨β4⟩ is negligible, though. +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0.4 +− +0.2 +− +0 +0.2 +0.4 +0.6 +3 +− +10 +× +0,0 +{4} +ε +2, + - c +{4} +ε +2, +c +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 + + + + + + + + + + + + +U+U Glauber Model +b = 0 fm +-2 +10 +× +) = (2.78,4.20,-1.55,-2.92) +2 +,b +2 +,a +1 +,b +1 +(a + Data Fit +2 +β +σ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0.4 +− +0.2 +− +0 +0.2 +0.4 +0.6 +3 +− +10 +× +0,0 +{4} +ε +2, + - c +{4} +ε +2, +c +0.06 +0.07 +0.08 +0.09 +0.10 + + + + + + + + + + +U+U Glauber Model +b = 0 fm +-2 +10 +× +) = (2.74,-1.64) +2 +,a +1 +(a +Modified Fit Function + Data Fit +2 +β +σ +FIG. 5: +The fit of the c2,ε{4}(¯β, σβ) data calculated in U+U collisions with zero impact parameter to Eq. 13 (left) and Eq. 14 +(right). +The contribution of residual terms is only a few percent. Indeed, a fit of this form describes the data very well as +shown in the right panel of Fig. 5. This behavior provides clear intuition on how the fluctuation terms containing σβ +compete with the terms containing only ¯β. For example, assuming ¯β = σβ, the contribution from fluctuation-related +terms is a factor of 9 (37) times the ¯β4 (¯β6) in the leading-order (subleading order). Thus, even a relatively small + +9 +0 +0.005 +0.01 +4 +β +0.5 +− +0.4 +− +0.3 +− +0.2 +− +0.1 +− +0 +3 +− +10 +× +0,0 +2〉 +2 +2 +ε〈 + - k +〉 +2 +4 +ε〈 +- +2〉 +2 +2 +ε〈 + - k +〉 +2 +4 +ε〈 +2 +β +σ +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +k=2.541 +Nucleon Glauber +k +2.4 +2.5 +2.6 +2.7 +5 +10 + (ab. units) +2 +χ +0 +0.005 +0.01 +4 +β +0.5 +− +0.4 +− +0.3 +− +0.2 +− +0.1 +− +0 +3 +− +10 +× +0,0 +2〉 +2 +2 +ε〈 + - k +〉 +2 +4 +ε〈 +- +2〉 +2 +2 +ε〈 + - k +〉 +2 +4 +ε〈 +2 +β +σ +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +=2.530 +0 +k +Quark Glauber +k +2.4 +2.5 +2.6 +2.7 +5 +10 + (ab. units) +2 +χ +FIG. 6: +The values of ⟨ε4 +2⟩−K ⟨ε2 +2⟩ +2 for the value of K that minimize the dependence on σβ in the nucleon Glauber model (left) +and quark Glauber model (right). The inserts shows the K dependence of χ2, which is calculated as χ2 = ∑i ∑j(fij − ¯fi)2/σ2 +i.j, +where fij = f(¯βi, σβ,j, ¯fi = ∑j fij/ ∑j and σij is the statistical error bar on the i, j-th data point. +fluctuation could have a stronger impact on c2,ε{4} than the modestly large ¯β. Note that the liquid-drop model +results in Table I predict b1 = 7/5a1, slightly smaller than the Glauber model expectation. +Experimentally, we can measure ⟨v2 +2⟩ and ⟨v2 +4⟩, which are linearly related to ⟨ε2 +2⟩ and ⟨ε4 +2⟩, respectively. Thus, it is +natural to ask whether one could constrain the ¯β and σβ from these two quantities. So far we have learned that the +combination in the cumulant definition c2,ε{4} = ⟨ε4 +2⟩ − 2⟨ε2 +2⟩ +2 is not sufficient to achieve such separation. Motivated +by this fact, we tried a more general combination f(¯β,σβ;k) = ⟨ε4 +2⟩ − k ⟨ε2 +2⟩ +2, and identify the k value for which the +f(¯β,σβ;k) have the least variation in σβ. The best value found is k = k0 = 2.541, for which the data points follow an +approximately-linear dependence ¯β4 as shown in the left panel of Fig. 6. The right panel of Fig. 6 shows a similar +exercise in the quark Glauber model which gives a nearly identical k0 value. The data points yet do not collapse on a +single curve, implying a small σβ dependence remaining. The amount of spread is estimated to be about 25% relative +for a given ¯β, corresponding to a variation of ¯β of about 1 − +4√ +0.75 = 7%. This 7% value is the best precision for +determining the ¯β in the Glauber model using this method. The determined ¯β value can then be plugged into Eq. 11 +(considering only the leading order is sufficient for ⟨ε2 +2⟩ as shown in Fig. 3) to determine the σβ. +In the study of flow fluctuations in heavy ion collisions, it is often desirable to calculate the normalized quantities +between high-order cumulant and low-order cumulants, which have the advantage of canceling the final state effects. +Here we study the following three quantities following the convention from Ref. [21], +ρ = +⟨ε2 +2 +δd⊥ +d⊥ ⟩ − ⟨ε2 +2 +δd⊥ +d⊥ ⟩ +β=0 +(⟨ε2 +2⟩ − ⟨ε2⟩β=0) +√ +⟨( δd⊥ +d⊥ ) +2 +⟩ − ⟨( δd⊥ +d⊥ ) +2 +⟩ +β=0 +, ncd{3} = +⟨( δd⊥ +d⊥ )3⟩ − ⟨( δd⊥ +d⊥ )3⟩ +β=0 +(⟨( δd⊥ +d⊥ ) +2 +⟩ − ⟨( δd⊥ +d⊥ ) +2 +⟩ +β=0 +) +3/2 , ncε{4} = c2,ε{4} − c2,ε{4}β=0 +(⟨ε2 +2⟩ − ⟨ε2 +2⟩β=0) +2 +(15) +Since 96Zr has little quadruple deformation βZr ≈ 0, these quantities can be constructed from measurements in +96Ru+96Ru and 96Zr+96Zr collisions. +The results from our Glauber model calculation are shown in Fig. 7 for prolate nuclei cos(3γ) = 0. The values of ρ +are nearly independent of ¯β and have a weak dependence on σβ. In the large ¯β region, ρ quickly converges to a value +around −0.62 nearly independent of σβ. In the moderate ¯β region say ¯β ∼ 0.2, the ρ first decreases quickly to a value +around -0.6, but then increases gradually with σβ. The values of ncd{3} have similar convergence trends towards large +¯β around 0.4, but much more slowly compare to ρ. The ncε{4} has a negative and nearly constant value when σβ = 0, +while it increases rather quickly with σβ. Even for a value of σ2 +β = 0.01, the ncε{4} stays positive until ¯β2 > 0.06. For +larger values of σ2 +β, the ncε{4} decreases with increasing ¯β2, but always remains positive over the range of ¯β studied. +V. +SUMMARY +We studied the impact of the fluctuations of nuclear quadrupole deformation on the heavy ion observables in a Monte +Carlo Glauber model. In particular, we focus on eccentricity ε2 and inverse size d⊥ in each event, which can be related + +10 +2 +β +0 +0.05 +0.1 +ρ +0.6 +− +0.5 +− +0.4 +− +0.3 +− + +2 +β +σ +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +2 +β +0 +0.05 +0.1 +{3} +d +nc +0 +0.2 +0.4 +0.6 +0.8 +) = 0 +γ +cos(3 +2 +β +0 +0.05 +0.1 +{4} +ε +nc +0 +0.5 +1 +FIG. 7: +The normalized three-particle correlators, ρ (left) and ncd{3} (middle) and normalized four-particle correlator ncε{4} +(right) defined in Eq. 11 as a function of ¯β2 for different values of σ2 +β. +to the event-wise elliptic flow and mean transverse momentum in the final state. The triaxiality γ has a strong impact +on three-particle correlators, but the impact diminishes for larger σγ. In particular, when σγ is large, the observables +do not distinguish between prolate deformation and oblate deformation, i.e. the values of all observables approach +those obtained in collisions of rigid triaxial nuclei with the same β. The mean and variance of quadrupole fluctuations, +¯β and σ2 +β, have a strong influence on all observables. The influence on two-particle observables ⟨ε2 +2⟩ and ⟨(δd⊥/d⊥)2⟩ is +proportional to ⟨β2⟩ = ¯β2 +σ2 +β, however, the ⟨(δd⊥/d⊥)2⟩ also has a sizable subleading order term proportional to ⟨β3⟩. +The three-particle observables to the leading order are proportional to ⟨cos(3γ)β3⟩ = cos(3γ)¯β(¯β + 3σ2 +β), whereas the +four-particle observables to the leading order are proportional to ⟨β4⟩ = ¯β4 + 6σ2 +β ¯β2 + 3σ4 +β. Hence, the variance of β +fluctuation has a stronger impact than ¯β for these higher-order observables. +By combining two and four-particle cumulant of ε2, we have constructed a simple formula to constrain parameters ¯β +and σβ simultaneously. Such separation becomes less effective when σβ is comparable or larger than ¯β. In the future, +it would be interesting to carry out a full hydrodynamic model simulation to quantify the efficacy of this method on +the final state flow observables. +This research is supported by DOE DE-FG02-87ER40331. +Appendix +The default results in this paper are obtained with the nucleon Glauber model. We have repeated the analysis for +the quark Glauber model and compared it with the nucleon Glauber model results in Figs. 8 and 9 for the impact of +γ fluctuation and β fluctuation, respectively. The trends are mostly very similar. A few exceptions are observed. In +particular, the results of the two models are shifted vertically from each other in Fig. 8. In the case of β fluctuation +in Fig. 9, the variance cd{2} and skewness cd{3} are systematically different between the two models in the high ¯β +region. + +11 +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +γ +σ +0.2 +− +0.15 +− +0.1 +− +0.05 +− +0 +0.05 +0.1 +0.15 +0.2 +3 +− +10 +× +=0 +β +Cov - Cov +Nucleon Glauber +Quark Glauber +U+U Glauber Model + = 0.28 +β +b = 0 fm, +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +γ +σ +2 +− +0 +2 +4 +6 +8 +6 +− +10 +× +=0 +β +{3} +d + - C +{3} +d +C +Nucleon Glauber +Quark Glauber +U+U Glauber Model + = 0.28 +β +b = 0 fm, +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +γ +σ +0.12 +− +0.11 +− +0.1 +− +0.09 +− +0.08 +− +3 +− +10 +× +=0 +β +{4} +ε +2, + - c +{4} +ε +2, +c +Nucleon Glauber +Quark Glauber +U+U Glauber Model + = 0.28 +β +b = 0 fm, +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +γ +σ +15.8 +16 +16.2 +16.4 +16.6 +16.8 +3 +− +10 +× +=0 +β〉 +2 +2 +ε 〈 + - +〉 +2 +2 +ε 〈 +) +γ +Cos(3 +1.00 +0.87 +0.50 +0.00 +-0.50 +-0.87 +-1.00 +Nucleon Glauber +Quark Glauber +U+U Glauber Model + = 0.28 +β +b = 0 fm, +0 +0.2 +0.4 +0.6 +0.8 +1 +2 +γ +σ +0.54 +0.56 +0.58 +0.6 +0.62 +0.64 +0.66 +3 +− +10 +× +=0 +β +{2} +d + - C +{2} +d +C +Nucleon Glauber +Quark Glauber +U+U Glauber Model + = 0.28 +β +b = 0 fm, +FIG. 8: +Comparison of the five observables between nucleon Glauber model (symbols) and quark Glauber model (lines with +matching colors) as a function of σγ for different values of ¯γ. +! +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0.8 +− +0.7 +− +0.6 +− +0.5 +− +0.4 +− +0.3 +− +0.2 +− +0.1 +− +0 +3 +− +10 +× +0,0 +Cov - Cov +Nucleon Glauber +Quark Glauber +U+U Glauber Model +b = 0 fm +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +2 +− +0 +2 +4 +6 +8 +10 +12 +6 +− +10 +× +0,0 +{3} +d + - C +{3} +d +C +Nucleon Glauber +Quark Glauber +U+U Glauber Model +b = 0 fm +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0.1 +− +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +3 +− +10 +× +0,0 +{4} +ε +2, + - c +{4} +ε +2, +c +Nucleon Glauber +Quark Glauber +U+U Glauber Model +b = 0 fm +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +5 +10 +15 +20 +25 +30 +35 +40 +3 +− +10 +× +0,0 +〉 +2 +2 +ε 〈 + - +〉 +2 +2 +ε 〈 +2 +β +σ +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +Nucleon Glauber +Quark Glauber +U+U Glauber Model +b = 0 fm +0 +0.02 +0.04 +0.06 +0.08 +0.1 +2 +β +0 +0.2 +0.4 +0.6 +0.8 +1 +3 +− +10 +× +0,0 +{2} +d + - C +{2} +d +C +Nucleon Glauber +Quark Glauber +U+U Glauber Model +b = 0 fm +FIG. 9: +Comparison of the five observables between nucleon Glauber model (symbols) and quark Glauber model (lines with +matching colors) as a function of ¯β2 for different values of σβ. + +12 +[1] W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph] . +[2] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu, +and U. Heinz, Phys. Rev. C 94, 024907 (2016), arXiv:1605.03954 +[nucl-th] . +[3] G. Giacalone, (2022), arXiv:2208.06839 [nucl-th] . +[4] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003). +[5] J. Jia and C.-J. Zhang, (2021), arXiv:2111.15559 [nucl-th] . +[6] G. Nijs and W. van der Schee, (2021), arXiv:2112.13771 [nucl-th] . +[7] J. Jia, G. Giacalone, and C. Zhang, (2022), arXiv:2206.10449 [nucl-th] . +[8] M. Abdallah et al. (STAR), Phys. Rev. C 105, 014901 (2022), arXiv:2109.00131 [nucl-ex] . +[9] Haojie Xu and Chunjian Zhang (STAR Collabration), Constraints on neutron skin thickness and nuclear deformations using +relativistic heavy-ion collisions from STAR, “https://indico.cern.ch/event/895086/contributions/4724887/,https: +//indico.cern.ch/event/895086/contributions/4749420/,” (2022). +[10] C. Zhang, S. Bhatta, and J. Jia, Phys. Rev. C 106, L031901 (2022), arXiv:2206.01943 [nucl-th] . +[11] C. Zhang and J. Jia, Phys. Rev. Lett. 128, 022301 (2022), arXiv:2109.01631 [nucl-th] . +[12] Y. Cao, S. E. Agbemava, A. V. Afanasjev, W. Nazarewicz, +and E. Olsen, Phys. Rev. C 102, 024311 (2020), +arXiv:2004.01319 [nucl-th] . +[13] B. Bally et al., (2022), arXiv:2209.11042 [nucl-ex] . +[14] T. Otsuka, Y. Tsunoda, T. Togashi, N. Shimizu, and T. Abe, in European Physical Journal Web of Conferences, European +Physical Journal Web of Conferences, Vol. 178 (2018) p. 02003. +[15] M. Bender and P.-H. Heenen, Phys. Rev. C 78, 024309 (2008), arXiv:0805.4383 [nucl-th] . +[16] K. Heyde and J. L. Wood, Rev. Mod. Phys. 83, 1467 (2011). +[17] K. Kumar, Phys. Rev. Lett. 28, 249 (1972). +[18] A. Poves, F. Nowacki, and Y. Alhassid, Phys. Rev. C 101, 054307 (2020), arXiv:1906.07542 [nucl-th] . +[19] J. Jia, Phys. Rev. C 105, 014905 (2022), arXiv:2106.08768 [nucl-th] . +[20] D. Teaney and L. Yan, Phys. Rev. C 83, 064904 (2011), arXiv:1010.1876 [nucl-th] . +[21] J. Jia, Phys. Rev. C 105, 044905 (2022), arXiv:2109.00604 [nucl-th] . +[22] C. Loizides, Phys. Rev. C94, 024914 (2016), arXiv:1603.07375 [nucl-ex] . +[23] M. Zhou and J. Jia, Phys. Rev. C 98, 044903 (2018), arXiv:1803.01812 [nucl-th] . + diff --git a/BdE1T4oBgHgl3EQf9gYX/content/tmp_files/load_file.txt b/BdE1T4oBgHgl3EQf9gYX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3da09863e4d62c1aa0471649d011814898a410b2 --- /dev/null +++ b/BdE1T4oBgHgl3EQf9gYX/content/tmp_files/load_file.txt @@ -0,0 +1,932 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf,len=931 +page_content='Nuclear shape fluctuations in high-energy heavy ion collisions Aman Dimri,1, ∗ Somadutta Bhatta,1 and Jiangyong Jia1, 2, † 1Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA 2Physics Department, Brookhaven National Laboratory, Upton, NY 11976, USA (Dated: January 10, 2023) Atomic nuclei often exhibit a quadrupole shape that fluctuates around some average profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We investigate the impact of nuclear shape fluctuation on the initial state geometry in heavy ion collisions, particularly its eccentricity ε2 and inverse size d⊥, which can be related to the elliptic flow and radial flow in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The fluctuation in overall quadrupole deformation enhances the variances and modifies the skewness and kurtosis of the ε2 and d⊥ in a controllable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The fluctuation in triaxiality reduces the difference between prolate and oblate shape for any observable, whose values, in the large fluctuation limit, approach those obtained in collisions of rigid triaxial nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The method to disentangle the mean and variance of the quadrupole deformation is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' PACS numbers: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='Gz, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='Ld, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='-1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' INTRODUCTION Ultra-relativistic heavy ion physics aims to understand the dynamics and properties of the Quark-Gluon Plasma (QGP) created in collisions of atomic nuclei at very high energy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Achieving this goal is currently limited by the lack of understanding of the initial condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' how the energy is deposited in the overlap region before the formation of QGP [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The energy deposition process is not calculable from first principles and is often parameterized via phenomenological approaches with multiple free parameters [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' On the other hand, heavy atomic nuclei are well-studied objects exhibiting a wide range of shapes and radial profiles [4], which are often characterized by a few collective nuclear structure parameters such as quadrupole, triaxial, and octupole deformations, nuclear radius and skin thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' One can leverage species with similar mass numbers but different structures, such as isobars, to directly probe the energy deposition mechanism and hence constrain the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The efficacy of this approach has been investigated recently [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' One good example demonstrating this possibility is the 96Ru+96Ru and 96Zr+96Zr collisions, recently carried out by the STAR Collaboration at the relativistic heavy ion collider [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Ratios of many bulk observables between the isobars, such as harmonic flow vn, charged particle multiplicity Nch, and average transverse momentum ⟨pT⟩, have been measured, which show significant and observable- and centrality-dependent deviation from unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Model studies show that these ratios are insensitive to final-state effects and are controlled mainly by the differences of the collective nuclear structure parameters between 96Ru and 96Zr [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Comparing the calculations with experimental data, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [5, 11] have estimated structure parameters that are broadly consistent with general knowledge from low energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' However, these studies also suggest a sizable octupole collectivity for Zr, not predicted by mean field structure models [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The rich and versatile information from isobar or isobar-like collisions provides a new constraint on the heavy ion initial condition and a new way to probe nuclear structure at high energy [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' However, it is important to point out that atomic nuclei in the ground state often do not have a static shape, but can fluctuate due to interplay between collective modes and single-particle states [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The potential energy surface of such species usually has shallow minimums as a function of deformation parameters, such as quadruple deformation β and triaxiality γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The ground state nuclear wave function is often treated as a mixture of configurations with different (β,γ) values [4, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Then there are the phenomena of shape coexistence, which happens when the same nuclei can have multiple low-lying states with widely different shapes but small energy differences [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' From the nuclear structure side, the quadrupole fluctuations can be estimated from the sum rules of matrix elements of various moments of quadrupole operators that can be measured experimentally [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' From the heavy ion collision side, the shape fluctuations can be accessed using multi-particle correlations, which probe moments of the nucleon position in the initial condition [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For instance, the elliptic flow v2 in each event is proportional to the elliptic eccentricity ε2, v2 = kε2 calculable from participating nucleons [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Therefore, the fluctuations of flow are related to fluctuations of quadruple deformation via their respective moments: ⟨vm 2 ⟩ = km ⟨εm 2 ⟩ ∝ ⟨βm⟩,m = 2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In principle, one could ∗Electronic address: aman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='dimri@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='edu †Electronic address: jiangyong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='jia@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03556v1 [nucl-th] 9 Jan 2023 2 constrain the mean and variance of quadrupole fluctuations from the ⟨β2⟩ and ⟨β4⟩, which in terms can be determined from ⟨v2 2⟩ and ⟨v4 2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This paper extends our previous study to investigate the influence of fluctuations of quadruple deformation param- eters (β,γ) to several selected two-, three- and four-particle heavy-ion observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We first derive simple analytical relations between these observables and the means and variances of (β,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We then perform a more realistic Glauber model simulation, assuming Gaussian fluctuations, to quantify the region of validity of these relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We discuss the sensitivity of these observables on the nuclear shape, as well as the prospect of separating the average shape from shape fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' EXPECTATION AND MODEL SETUP We consider the eccentricity vector ϵ2 and inverse transverse size d⊥, which are estimators for elliptic flow V2 ≡ v2e2iΨ2 and average transverse momentum ⟨pT⟩ or radial flow, calculated from the transverse position of nucleon participants in each event, ϵ2 = − ⟨r2 ⊥ei2φ⟩ ⟨r2⊥⟩ , d⊥ = √ Npart/⟨r2⊥⟩, (1) where r⊥ is the transverse radius and Npart is the number of participating nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Following the heuristic argument from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [21], for collisions of nuclei with small quadrupole deformation, the eccentricity vector and d⊥ in a given event have the following leading-order form: δd⊥ d⊥ ≈ δd + p0(Ωp,γp)βp + p0(Ωt,γt)βt , ϵ2 ≈ ϵ0 + p2(Ωp,γp)βp + p2(Ωt,γt)βt, (2) where the scalar δd and vector ϵ0 are valued for spherical nuclei, and we are considering the general situation where the projectile and target, denoted by subscripts “p” and “t”, have different deformation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The p0 and p2 are phase space factors, which depend on γ and the Euler angles Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Since the fluctuations of δd (ϵ0) are uncorrelated with p0 (p2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' an average over collisions with different Euler angles is expected to give the following leading-order expressions for the variances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' skewness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' and kurtosis of the fluctuations c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='ϵ{2} ≡ ⟨ε2 2⟩ = ⟨ε2 0⟩ + ⟨p2(γp)p∗ 2(γp)⟩β2 p + ⟨p2(γt)p∗ 2(γt)⟩β2 t cd{2} ≡ ⟨(δd⊥ d⊥ ) 2 ⟩ = ⟨δ2 d⟩ + ⟨p0(γp)2⟩β2 p + ⟨p0(γt)2⟩β2 t Cov ≡ ⟨ε2 2 δd⊥ d⊥ ⟩ = ⟨ε2 0δd⟩ + ⟨p0(γp)p2(γp)p2(γp)∗⟩β3 p + ⟨p0(γt)p2(γt)p2(γt)∗⟩β3 t cd{3} ≡ ⟨(δd⊥ d⊥ ) 3 ⟩ = ⟨δ3 d⟩ + ⟨p0(γp)3⟩β3 p + ⟨p0(γt)3⟩β3 t c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='ϵ{4} ≡ ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2 = ⟨ε4 0⟩ − 2⟨ε2 0⟩ 2 + (⟨p2 2p2∗ 2 ⟩⟨β4⟩ − 2⟨p2p∗ 2⟩2 ⟨β2⟩ 2) p + (⟨p2 2p2∗ 2 ⟩⟨β4⟩ − 2⟨p2p∗ 2⟩2 ⟨β2⟩ 2) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (3) These quantities relate directly to the final state observables, ⟨v2 2⟩, ⟨(δpT/⟨pT⟩)2⟩, ⟨v2 2 δpT ⟨pT⟩⟩, ⟨(δpT/⟨pT⟩)3⟩ and ⟨v4 2⟩ − 2⟨v2 2⟩ 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Previous studies have demonstrated that the moments ⟨p2 0⟩, ⟨p2p∗ 2⟩, and ⟨p2 2p2∗ 2 ⟩ are independent of γ, while ⟨p0p2p∗ 2⟩ and ⟨p3 0⟩ have leading order dependence on γ: c + bcos(3γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Here, c ≪ b for ⟨p0p2p∗ 2⟩, whereas c ≲ b for ⟨p3 0⟩ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In the presence of quadrupole fluctuations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' we also need to average these quantities over “independent” 3 fluctuations for projectile and target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' giving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' ⟨(δd⊥ d⊥ ) 2 ⟩ = a0 + b0 2 (⟨β2 p⟩ + ⟨β2 t ⟩) = a0 + b0 ⟨β2⟩ ⟨ε2 2⟩ = a1 + b1 2 (⟨β2 p⟩ + ⟨β2 t ⟩) = a1 + b1 ⟨β2⟩ ⟨ε2 2 δd⊥ d⊥ ⟩ = a2 − 1 2 (⟨(c2 + b2 cos(3γp))β3 p⟩ + ⟨(c2 + b2 cos(3γt))β3 t ⟩) = a2 − ⟨(c2 + b2 cos(3γ))β3⟩ ⟨(δd⊥ d⊥ ) 3 ⟩ = a3 + 1 2 (⟨(c3 + b3 cos(3γp))β3 p⟩ + ⟨(c3 + b3 cos(3γt)β3 t ⟩) = a3 + ⟨(c3 + b3 cos(3γ))β3⟩ ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2 = a4 + b4 2 (⟨β4 p⟩ + ⟨β4 t ⟩) − c4 2 (⟨β2 p⟩ 2 + ⟨β2 t ⟩ 2) = a4 + b4 ⟨β4⟩ − c4 ⟨β2⟩ 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (4) where the averages are performed over fluctuations in β and γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' and the coefficients an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' bn and cn are centrality- dependent positive quantities satisfying c2 ≪ b2 and c3 ≲ b3 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The second part of these equations is obtained by assuming that the fluctuations of the projectile and target are sampled from the same probability density distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For a more quantitative estimation, we consider the liquid-drop model where the nucleon density distribution has a sharp surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For head-on collisions with zero impact parameter, it predicts the following simple relations [21], δd⊥ d⊥ = √ 5 16π β2 (cosγD2 0,0 + sinγ √ 2 [D2 0,2 + D2 0,−2]) , ϵ2 = − √ 15 2π β2 (cosγD2 2,0 + sinγ √ 2 [D2 2,2 + D2 2,−2]) , (5) where the Dl m,m′(Ω) are the Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The analytical results obtained for various cumulants are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' They provide approximate estimates for the values of bn in most central collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨(δd⊥/d⊥)2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨(δd⊥/d⊥)3⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨(δd⊥/d⊥)4⟩ − 3 ⟨(δd⊥/d⊥)2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='32π ⟨β2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='896π3/2 ⟨cos(3γ)β3⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='14336π2 (7 ⟨β2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 5 ⟨β4⟩) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨ε4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ − 2 ⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='(⟨ε6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ − 9 ⟨ε4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ ⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ + 12 ⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3)/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4π ⟨β2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='112π2 (7 ⟨β2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 5 ⟨β4⟩) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='256π3 [⟨β2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='14 ⟨β4⟩ ⟨β2⟩ − 1175 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6006 ⟨β6⟩ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3003 ⟨cos(6γ)β6⟩] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2(δd⊥/d⊥)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2(δd⊥/d⊥)2⟩ − ⟨ε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2⟩ ⟨(δd⊥/d⊥)2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='⟨ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2ϵ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='112π3/2 ⟨cos(3γ)β3⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1792π2 (7 ⟨β2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 5 ⟨β4⟩) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='56π2 ⟨β4⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='TABLE I: The leading-order results of various cumulants calculated for the nucleus with a sharp surface via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The two nuclei are placed with zero impact parameter and results are obtained by averaging over random orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' To make further progress, we consider the case where the fluctuations of β and γ are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The observables in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 4 and Table I can be expressed in terms of central moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Assuming Gaussian fluctuations with means ¯β or ¯γ and variances σβ or σγ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 4 becomes ⟨ε2 2⟩ = a1 + b1(¯β2 + σ2 β) ,⟨(δd⊥ d⊥ ) 2 ⟩ = a0 + b0(¯β2 + σ2 β) ⟨ε2 2 δd⊥ d⊥ ⟩ = a2 − (b2e− 9σ2 γ 2 cos(3¯γ) + c2)¯β(¯β2 + 3σ2 β) ⟨(δd⊥ d⊥ ) 3 ⟩ = a3 + (b3e− 9σ2 γ 2 cos(3¯γ) + c3)¯β(¯β2 + 3σ2 β) ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2 = a4 + b4(¯β4 + 6σ2 β ¯β2 + 3σ4 β) − c4(¯β2 + σ2 β)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (6) where we have used the well-known expression for Gaussian smearing of an exponential function, ⟨einγ⟩ = e− n2σ2 γ 2 ein¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 4 If the fluctuations of β and γ are non-Gaussian, one should also consider the higher cumulants of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For example, ⟨β3⟩ = ¯β(¯β + 3σ2 β) + k3,β and ⟨β4⟩ = ¯β4 + 6σ2 β ¯β2 + 3σ4 β + 4¯βk3,β + k4,β, where k3,β = ⟨(β − ¯β)3⟩ and k4,β = ⟨(β − ¯β)4⟩ − 3⟨(β − ¯β)2⟩ 2 are the skewness and kurtosis of the β fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The expectation value of cos(nγ) can be expressed via the cumulant generating function of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Keeping the cumulants km,γ up to leading order correction in skewness and kurtosis, k3,γ = ⟨(γ − ¯γ)3⟩ and k4,γ = ⟨(γ − ¯γ)4⟩ − 3⟨(γ − ¯γ)2⟩ 2, we have, ⟨cos(nγ)⟩ = 1 2 (⟨ein¯γ⟩ + ⟨e−in¯γ⟩) = 1 2 (exp( ∞ ∑ m=1 κm,γ (in)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' ) + exp( ∞ ∑ m=1 κm,γ (−in)m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' )) = exp( ∞ ∑ m=1 κ2m,γ (−1)m(n)2m 2m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' )[cos( ∞ ∑ m=1 κ2m+1,γ (−1)m(n)2m+1 (2m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' + n¯γ)] ≈ e− n2σ2 γ 2 + n4k4,γ 24 cos(n¯γ + n3 6 k3,γ) ≈ e− n2σ2 γ 2 [cos(n¯γ) + sin(n¯γ)n3 6 k3,γ](1 + n4 24k4,γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (7) Clearly, the net effect of skewness is a rotation of ¯γ by k3,γn2/6, and the net effect of kurtosis is to increase or decrease the overall variation with γ depending on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For a more realistic estimation of the influences of shape fluctuations, we perform a Monte-Carlo Glauber model simulation of 238U+238U collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The setup of the model and the data used in this analysis are the same as those used in our previous work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We simulate ultra-central collisions with zero impact parameter, where the impact of nuclear deformation reaches maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The nucleon distribution is described by a deformed Woods-Saxon function ρ(r,θ,φ) = ρ0 1 + e[r−R(θ,φ)/a] , R(θ,φ) = R0 (1 + β[cosγY2,0(θ,φ) + sinγY2,2(θ,φ)]), (8) where the nuclear surface R(θ,φ) is expanded into spherical harmonics Y2,m in the intrinsic frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Each nucleus is assigned a random (β,γ) value, sampled from Gaussian distributions with means (¯β, ¯γ) and variances (σβ,σγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The nucleus is then rotated by random Euler angles before they are set on a straight line trajectory towards each other along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Furthermore, three quark constituents are generated for each nucleon according to the quark Glauber model from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' From this, the nucleons or the constituent quarks in the overlap region are identified, which are used to calculate the ε2 and d⊥ defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (1), and the results are presented as a function of deformation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For the study of the β fluctuation, we fix the γ = 0 (prolate nucleus) and choose 11 values each for ¯β2 and σ2 β with 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=',0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' So a total of 11 × 11 = 121 simulations have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For the study of the γ fluctuation, we fix β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 and choose seven ¯γ and seven σγ values: cos(3¯γ) = 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5,0,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87,−1 and σγ = 0,π/18,2π/18,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=',6π/18, so a total of 7 × 7 = 49 simulation have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For each case, about 50 Million events were generated and all the observables were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Our discussion is mainly based on the nucleon Glauber model, and the results from the quark Glauber model are included in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' IMPACT OF TRIAXIALITY FLUCTUATION Due to the three-fold symmetry of nuclei shape in triaxiality, the γ dependence of a given observable can be generally expressed as a0 +∑∞ n=1 [an cos(3n¯γ) + bn sin(3n¯γ)]e− n2σ2 γ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' If we further impose the condition that a random fluctuation for a triaxial nucleus does not impact the value of the observable, which is found to be true in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This leads to the γ dependence of the form a0 + ∑∞ n=1 [an(cos(3n¯γ) − cos(3n π 6 )) + bn(sin(3n¯γ) − sin(3n π 6 ))]e− n2σ2 γ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We first discuss the impact of triaxiality fluctuation on three-particle observables ⟨ε2 2 δd⊥ d⊥ ⟩ and ⟨(δd⊥/d⊥)3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We first subtract them by the values for the undeformed case, to isolate the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 4 containing the triaxiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Figure 1 show the results obtained for different values of cos3¯γ as a function of σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The values for triaxial nucleus cos(3¯γ) = 0 are indeed independent of σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The fluctuation of γ reduces the difference between the prolate ¯γ = 0 and the oblate ¯γ = π/3 shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This reduction is largely described by e− 9σ2 γ 2 cos(3¯γ), except for a small asymmetry between ¯γ = 0 and ¯γ = π/3, clearly visible for ⟨(δd⊥/d⊥)3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We account for this small asymmetry by including higher-order terms in the fit function allowed by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 γ σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 3 − 10 × =0 β Cov - Cov 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 5 10 × = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='85 0 a 5 10 × ) = (-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='98,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03) 2 ,a 1 (a U+U Glauber Model 5 10 × ) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08) 2 ,b 1 (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, ) Data Fit γ Cos(3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 γ σ 2 − 0 2 4 6 8 6 − 10 × =0 β {3} d C {3} d C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 6 10 × = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='19 0 a 6 10 × ) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='18,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='18) 2 ,a 1 (a U+U Glauber Model 6 10 × ) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='27,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04) 2 ,b 1 (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, ) Data Fit γ Cos(3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 1: The dependence of ⟨ε2 2 δd⊥ d⊥ ⟩ (left) and ⟨(δd⊥/d⊥)3⟩ (right) on smearing in triaxiality σγ for different values of ¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The lines indicate a simultaneous fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 10 with the parameter values displayed on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Keeping leading and subleading terms, we have, ⟨ε2 2 δd⊥ d⊥ ⟩ − ⟨ε2 2 δd⊥ d⊥ ⟩ β=0 = [a′ 0 + (a′ 1 cos(3¯γ) + b′ 1 [sin(3¯γ) − 1])e− 9σ2 γ 2 + (a′ 2 [cos(6¯γ) + 1] + b′ 2 sin(6¯γ))e− 36σ2 γ 2 ] ¯β3 (9) = a0 + (a1 cos(3¯γ) + b1 [sin(3¯γ) − 1])e− 9σ2 γ 2 + (a2 [cos(6¯γ) + 1] + b2 sin(6¯γ))e− 36σ2 γ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (10) The same fit function is also used to describe ⟨(δd⊥/d⊥)3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The parameters in the first line and those in the second line differ by a scale factor ¯β3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='283 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' From the values of parameters displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 1, we concluded that the magnitude of the high-order order terms is less than 2% of the magnitude of a1 for ⟨ε2 2 δd⊥ d⊥ ⟩ but reaches up to 5% for ⟨(δd⊥/d⊥)3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Figure 1 shows that the signature of triaxiality in heavy ion collisions is greatly reduced for large value of σγ, often found in γ-soft nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' A twenty-degree fluctuation in triaxiality, for example, reduces the signal by nearly 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' It would be difficult to distinguish between static rigid triaxial nuclei and nuclei with large fluctuations around ¯γ = π/6 using heavy ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In particular, nuclei that fluctuate uniformly between prolate and oblate shapes would give the same three-particle correlation signal as rigid triaxial nuclei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Such strong smearing also degrades the prospects of using higher-order cumulants of ε2 to infer the value of σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For the other three observables, ⟨ε2 2⟩, ⟨(δd⊥/d⊥)2⟩ and ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2, γ dependence are known to be very weak [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Nevertheless, up to a few percent dependence is observed, which can also be parameterized by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 9, except that we should change ¯β3 to ¯β2 for the variances and to ¯β4 for ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' However, since ¯β is fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28, all these observables can be parameterized by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The data and the results of the fits are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' First, we observe that the parameter a0, representing the baseline contribution associated with ¯β is by far the largest, and the other terms only cause a few percent of modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Secondly, while the ⟨ε2 2⟩ and ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2 can be largely described by including the cos(3¯γ) term, the description of ⟨(δd⊥/d⊥)2⟩ requires the inclusion of sin(3¯γ), cos(6¯γ) and sin(6¯γ) terms with comparable magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Lastly, all three observables have no sensitivity to ¯γ at large σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' IMPACT OF QUADRUPLE DEFORMATION FLUCTUATION Next, we consider the impact of β fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For this purpose, we shall fix the γ to be prolate shape, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='g cos(3γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Figure 3 displays the finding for two-particle observables ⟨ε2 2⟩ and ⟨(δd⊥/d⊥)2⟩, again they are corrected by the undeformed baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Although approximately-linear dependencies on ¯β2 are observed for both observables, the slopes of the data points also vary with σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' To describe this feature, we include two higher-order terms, 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 γ σ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 16 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 3 − 10 × =0 β〉 2 2 ε 〈 〉 2 2 ε 〈 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 5 10 × = 1621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 0 a 5 10 × ) = (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='46,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='29) 2 ,a 1 (a U+U Glauber Model 5 10 × ) = (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='79,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='53) 2 ,b 1 (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, ) Data Fit γ Cos(3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 γ σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='68 3 − 10 × =0 β {2} d C {2} d C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 5 10 × = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='83 0 a 5 10 × ) = (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='89,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='31) 2 ,a 1 (a U+U Glauber Model 5 10 × ) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='48,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08) 2 ,b 1 (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, ) Data Fit γ Cos(3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 γ σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='105 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='095 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='085 − 3 − 10 × =0 β {4} ε 2, c {4} ε 2, c 5 10 × = -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='76 0 a 5 10 × ) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10) 2 ,a 1 (a U+U Glauber Model 5 10 × ) = (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='13,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01) 2 ,b 1 (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 2: The dependence of ⟨ε2 2⟩ (left), ⟨(δd⊥/d⊥)2⟩ (middle), and ⟨ε4 2⟩ − 2 ⟨ε2 2⟩ 2 (right) on σγ for different values of ¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The dashed lines indicate a simultaneous fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 10, with fit results are displayed on the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' ⟨ε2 2⟩ − ⟨ε2 2⟩β=0 or ⟨(δd⊥ d⊥ ) 2 ⟩ − ⟨(δd⊥ d⊥ ) 2 ⟩ β=0 = c1 ⟨β2⟩ + c2 ⟨β3⟩ + c3 ⟨β4⟩ = c1(¯β2 + σ2 β) + c2 ¯β(¯β2 + 3σ2 β) + c3(¯β4 + 6σ2 β ¯β2 + 3σ4 β) (11) The fits including only the leading term and all three terms are shown in the first row and the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The fits in the middle row include the c1 and c3 terms for ⟨ε2 2⟩, while they include c1 and c2 terms for ⟨(δd⊥/d⊥)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Clearly, the behavior of ⟨(δd⊥/d⊥)2⟩ at large ¯β or σβ requires the presence of the ⟨β3⟩ term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 11 with a negative coefficient c2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In general, a large fluctuation σβ tends to reduce the slope of the dependence on ¯β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For the three-particle correlators, we include three terms in the fitting function as ⟨ε2 2 δd⊥ d⊥ ⟩ − ⟨ε2 2 δd⊥ d⊥ ⟩ β=0 or ⟨(δd⊥ d⊥ ) 3 ⟩ − ⟨(δd⊥ d⊥ ) 3 ⟩ β=0 = c1 ⟨β3 cos(3γ)⟩ + c2 ⟨β4 cos(3γ)⟩ + c3 ⟨β5 cos(3γ)⟩ = [c1 ¯β(¯β2 + 3σ2 β) + c2(¯β4 + 6σ2 β ¯β2 + 3σ4 β) + c3(¯β5 + 10σ3 β ¯β2 + 15¯βσ4 β)]cos(3γ) (12) The fitting results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 4 as a function of ¯β3 for the prolate case cos(3γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Inclusion of the high-order terms, mostly contribution from the ⟨β5⟩ component, improves the description of ⟨ε2 2 δd⊥ d⊥ ⟩ in the region of large σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' However, they are not sufficient to describe the ⟨(δd⊥/d⊥)3⟩ in the region of large ¯β and σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In particular, the fit also misses all points at ¯β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We checked that the fit can be systematically improved by including more higher moment terms, albeit only very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Lastly, we consider the four-particle observable c2,ε{4} = ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' According to findings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 3, the Taylor expansion of ⟨ε2 2⟩ should give the first two terms as c1 ⟨β2⟩ + c2 ⟨β4⟩, similarly the first few terms of ⟨ε4 2⟩ has the form of a0 ⟨β2⟩ 2 + a1 ⟨β4⟩ + a2 ⟨β6⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Therefore, the natural expression for c2,ε{4} up to second order correction should be c2,ε{4} − c2,ε{4}β=0 = a0 ⟨β2⟩ 2 + a1 ⟨β4⟩ + a2 ⟨β6⟩ − (c1 ⟨β2⟩ + c2 ⟨β4⟩)2 ≈ a1 ⟨β4⟩ − b1 ⟨β2⟩ 2 + a2 ⟨β6⟩ − b2 ⟨β2⟩⟨β4⟩ = a1(¯β4 + 6¯β2σ2 β + 3σ4 β) − b1(¯β2 + σ2 β)2 + a2(¯β6 + 15¯β4σ2 β + 45¯β2σ4 β + 15σ6 β) − b2(¯β2 + σ2 β)(¯β4 + 6¯β2σ2 β + 3σ4 β) (13) with b1 = c2 1 − a0 and b2 = 2c1c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The leading order correction includes the first two terms with a1 and b1, while the remaining two terms are the subleading-order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The results from the Glauber model and the fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 13 are shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The strong variation of c2,ε{4} with both ¯β and σβ is captured nicely by the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For small values of σβ, the deformation has a negative contribution to c2,ε{4} that is proportional to ¯β4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For large values of σβ, c2,ε{4} becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' A previous study shows that the centrality fluctuation also tends to give a positive value of c2,ε{4} [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Therefore, a negative c2,ε{4} which decreases further in central collisions would be an unambiguous indication for a large static quadrupole deformation of the colliding nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The values of the fit parameters show some interesting relations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' b1 ≈ 3/2a1 and b2 ≈ 2a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This means that the 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 5 10 15 20 25 30 35 40 3 − 10 × 0,0 〉 2 2 ε 〈 〉 2 2 ε 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 U+U Glauber Model 1 10 × = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='93 1 c b = 0 fm Data Fit 2 β σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 3 − 10 × 0,0 {2} d C {2} d C U+U Glauber Model 3 10 × = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='94 1 c b = 0 fm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 5 10 15 20 25 30 35 40 3 − 10 × 0,0 〉 2 2 ε 〈 〉 2 2 ε 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 U+U Glauber Model 1 10 × ) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='15,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='68) 3 ,c 1 (c b = 0 fm Data Fit 2 β σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 3 − 10 × 0,0 {2} d C {2} d C U+U Glauber Model 3 10 × ) = (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='89,-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='98) 2 ,c 1 (c b = 0 fm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 5 10 15 20 25 30 35 40 3 − 10 × 0,0 〉 2 2 ε 〈 〉 2 2 ε 〈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 U+U Glauber Model 1 10 × ) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='27,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='91) 3 ,c 2 ,c 1 (c b = 0 fm Data Fit 2 β σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 3 − 10 × 0,0 {2} d C {2} d C U+U Glauber Model 3 10 × ) = (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='74,-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='62,-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01) 3 ,c 2 ,c 1 (c b = 0 fm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 3: The simultaneous fit of the ⟨ε2 2⟩ (¯β, σβ) (left column) and ⟨(δd⊥/d⊥)2⟩ (¯β, σβ) (right column) calculated in U+U collisions with zero impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The top row shows the fits to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 11 with only the leading term and the last row shows the fits with all three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The middle row show the fits including c1 and c3 terms for ⟨ε2 2⟩ and c1 and c2 terms for ⟨(δd⊥/d⊥)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' distribution can also be described by the following alternative form, c2,ε{4} − c2,ε{4}β=0 ≈ a1 2 (6¯β2σ2 β + 3σ4 β − ¯β4) + a2(¯β4σ2 β + 27¯β2σ4 β + 9σ6 β − ¯β6) (14) 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 3 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='7 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 3 − 10 × 0,0 Cov - Cov 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 U+U Glauber Model b = 0 fm 3 10 × = -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='63 1 c Data Fit 2 β σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 3 β 4 − 2 − 0 2 4 6 8 10 12 14 6 − 10 × 0,0 {3} d C {3} d C U+U Glauber Model b = 0 fm 4 10 × = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='36 1 c 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 3 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='7 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 3 − 10 × 0,0 Cov - Cov 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 U+U Glauber Model b = 0 fm 3 ) = (-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='96,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='47,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='95)*10 3 ,c 2 ,c 1 (c Data Fit 2 β σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 3 β 4 − 2 − 0 2 4 6 8 10 12 14 6 − 10 × 0,0 {3} d C {3} d C U+U Glauber Model b = 0 fm 4 ) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='11,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='69,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='74)*10 3 ,c 2 ,c 1 (c FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 4: The simultaneous fit of the ⟨ε2 2 δd⊥ d⊥ ⟩ (¯β, σβ) (left column) and ⟨(δd⊥/d⊥)3⟩ (¯β, σβ) (right column) calculated in U+U collisions with zero impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The top row shows the results of the fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 12 with only the leading term and the second row shows the fits with all three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The fit results imply that the contribution from ⟨β4⟩ is negligible, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 3 − 10 × 0,0 {4} ε 2, c {4} ε 2, c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 U+U Glauber Model b = 0 fm 2 10 × ) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='78,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='20,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='55,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='92) 2 ,b 2 ,a 1 ,b 1 (a Data Fit 2 β σ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 3 − 10 × 0,0 {4} ε 2, c {4} ε 2, c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 U+U Glauber Model b = 0 fm 2 10 × ) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='74,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='64) 2 ,a 1 (a Modified Fit Function Data Fit 2 β σ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 5: The fit of the c2,ε{4}(¯β, σβ) data calculated in U+U collisions with zero impact parameter to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 13 (left) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 14 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The contribution of residual terms is only a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Indeed, a fit of this form describes the data very well as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This behavior provides clear intuition on how the fluctuation terms containing σβ compete with the terms containing only ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For example, assuming ¯β = σβ, the contribution from fluctuation-related terms is a factor of 9 (37) times the ¯β4 (¯β6) in the leading-order (subleading order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Thus, even a relatively small 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 4 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0 3 − 10 × 0,0 2〉 2 2 ε〈 k 〉 2 4 ε〈 2〉 2 2 ε〈 k 〉 2 4 ε〈 2 β σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 k=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='541 Nucleon Glauber k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='7 5 10 (ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' units) 2 χ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 4 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0 3 − 10 × 0,0 2〉 2 2 ε〈 k 〉 2 4 ε〈 2〉 2 2 ε〈 k 〉 2 4 ε〈 2 β σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='530 0 k Quark Glauber k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='7 5 10 (ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' units) 2 χ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 6: The values of ⟨ε4 2⟩−K ⟨ε2 2⟩ 2 for the value of K that minimize the dependence on σβ in the nucleon Glauber model (left) and quark Glauber model (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The inserts shows the K dependence of χ2, which is calculated as χ2 = ∑i ∑j(fij − ¯fi)2/σ2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='j, where fij = f(¯βi, σβ,j, ¯fi = ∑j fij/ ∑j and σij is the statistical error bar on the i, j-th data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' fluctuation could have a stronger impact on c2,ε{4} than the modestly large ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Note that the liquid-drop model results in Table I predict b1 = 7/5a1, slightly smaller than the Glauber model expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Experimentally, we can measure ⟨v2 2⟩ and ⟨v2 4⟩, which are linearly related to ⟨ε2 2⟩ and ⟨ε4 2⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Thus, it is natural to ask whether one could constrain the ¯β and σβ from these two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' So far we have learned that the combination in the cumulant definition c2,ε{4} = ⟨ε4 2⟩ − 2⟨ε2 2⟩ 2 is not sufficient to achieve such separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Motivated by this fact, we tried a more general combination f(¯β,σβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='k) = ⟨ε4 2⟩ − k ⟨ε2 2⟩ 2, and identify the k value for which the f(¯β,σβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='k) have the least variation in σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The best value found is k = k0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='541, for which the data points follow an approximately-linear dependence ¯β4 as shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 6 shows a similar exercise in the quark Glauber model which gives a nearly identical k0 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The data points yet do not collapse on a single curve, implying a small σβ dependence remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The amount of spread is estimated to be about 25% relative for a given ¯β, corresponding to a variation of ¯β of about 1 − 4√ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='75 = 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This 7% value is the best precision for determining the ¯β in the Glauber model using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The determined ¯β value can then be plugged into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 11 (considering only the leading order is sufficient for ⟨ε2 2⟩ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 3) to determine the σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In the study of flow fluctuations in heavy ion collisions, it is often desirable to calculate the normalized quantities between high-order cumulant and low-order cumulants, which have the advantage of canceling the final state effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Here we study the following three quantities following the convention from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [21], ρ = ⟨ε2 2 δd⊥ d⊥ ⟩ − ⟨ε2 2 δd⊥ d⊥ ⟩ β=0 (⟨ε2 2⟩ − ⟨ε2⟩β=0) √ ⟨( δd⊥ d⊥ ) 2 ⟩ − ⟨( δd⊥ d⊥ ) 2 ⟩ β=0 , ncd{3} = ⟨( δd⊥ d⊥ )3⟩ − ⟨( δd⊥ d⊥ )3⟩ β=0 (⟨( δd⊥ d⊥ ) 2 ⟩ − ⟨( δd⊥ d⊥ ) 2 ⟩ β=0 ) 3/2 , ncε{4} = c2,ε{4} − c2,ε{4}β=0 (⟨ε2 2⟩ − ⟨ε2 2⟩β=0) 2 (15) Since 96Zr has little quadruple deformation βZr ≈ 0, these quantities can be constructed from measurements in 96Ru+96Ru and 96Zr+96Zr collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The results from our Glauber model calculation are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 7 for prolate nuclei cos(3γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The values of ρ are nearly independent of ¯β and have a weak dependence on σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In the large ¯β region, ρ quickly converges to a value around −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='62 nearly independent of σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In the moderate ¯β region say ¯β ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2, the ρ first decreases quickly to a value around -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6, but then increases gradually with σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The values of ncd{3} have similar convergence trends towards large ¯β around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4, but much more slowly compare to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The ncε{4} has a negative and nearly constant value when σβ = 0, while it increases rather quickly with σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Even for a value of σ2 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01, the ncε{4} stays positive until ¯β2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' For larger values of σ2 β, the ncε{4} decreases with increasing ¯β2, but always remains positive over the range of ¯β studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' SUMMARY We studied the impact of the fluctuations of nuclear quadrupole deformation on the heavy ion observables in a Monte Carlo Glauber model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In particular, we focus on eccentricity ε2 and inverse size d⊥ in each event, which can be related 10 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 ρ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 2 β σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 {3} d nc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 ) = 0 γ cos(3 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 {4} ε nc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 7: The normalized three-particle correlators, ρ (left) and ncd{3} (middle) and normalized four-particle correlator ncε{4} (right) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 11 as a function of ¯β2 for different values of σ2 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' to the event-wise elliptic flow and mean transverse momentum in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The triaxiality γ has a strong impact on three-particle correlators, but the impact diminishes for larger σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In particular, when σγ is large, the observables do not distinguish between prolate deformation and oblate deformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' the values of all observables approach those obtained in collisions of rigid triaxial nuclei with the same β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The mean and variance of quadrupole fluctuations, ¯β and σ2 β, have a strong influence on all observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The influence on two-particle observables ⟨ε2 2⟩ and ⟨(δd⊥/d⊥)2⟩ is proportional to ⟨β2⟩ = ¯β2 +σ2 β, however, the ⟨(δd⊥/d⊥)2⟩ also has a sizable subleading order term proportional to ⟨β3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The three-particle observables to the leading order are proportional to ⟨cos(3γ)β3⟩ = cos(3γ)¯β(¯β + 3σ2 β), whereas the four-particle observables to the leading order are proportional to ⟨β4⟩ = ¯β4 + 6σ2 β ¯β2 + 3σ4 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Hence, the variance of β fluctuation has a stronger impact than ¯β for these higher-order observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' By combining two and four-particle cumulant of ε2, we have constructed a simple formula to constrain parameters ¯β and σβ simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Such separation becomes less effective when σβ is comparable or larger than ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In the future, it would be interesting to carry out a full hydrodynamic model simulation to quantify the efficacy of this method on the final state flow observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' This research is supported by DOE DE-FG02-87ER40331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Appendix The default results in this paper are obtained with the nucleon Glauber model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' We have repeated the analysis for the quark Glauber model and compared it with the nucleon Glauber model results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 8 and 9 for the impact of γ fluctuation and β fluctuation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' The trends are mostly very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' A few exceptions are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In particular, the results of the two models are shifted vertically from each other in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' In the case of β fluctuation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 9, the variance cd{2} and skewness cd{3} are systematically different between the two models in the high ¯β region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 2 γ σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 3 − 10 × =0 β Cov - Cov Nucleon Glauber Quark Glauber U+U Glauber Model = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 2 γ σ 2 − 0 2 4 6 8 6 − 10 × =0 β {3} d C {3} d C Nucleon Glauber Quark Glauber U+U Glauber Model = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 2 γ σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='12 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='11 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 − 3 − 10 × =0 β {4} ε 2, c {4} ε 2, c Nucleon Glauber Quark Glauber U+U Glauber Model = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 2 γ σ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 16 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 3 − 10 × =0 β〉 2 2 ε 〈 〉 2 2 ε 〈 ) γ Cos(3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00 Nucleon Glauber Quark Glauber U+U Glauber Model = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 2 γ σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='66 3 − 10 × =0 β {2} d C {2} d C Nucleon Glauber Quark Glauber U+U Glauber Model = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='28 β b = 0 fm, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 8: Comparison of the five observables between nucleon Glauber model (symbols) and quark Glauber model (lines with matching colors) as a function of σγ for different values of ¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='7 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 − 0 3 − 10 × 0,0 Cov - Cov Nucleon Glauber Quark Glauber U+U Glauber Model b = 0 fm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 2 − 0 2 4 6 8 10 12 6 − 10 × 0,0 {3} d C {3} d C Nucleon Glauber Quark Glauber U+U Glauber Model b = 0 fm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10 Nucleon Glauber Quark Glauber U+U Glauber Model b = 0 fm 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1 2 β 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='8 1 3 − 10 × 0,0 {2} d C {2} d C Nucleon Glauber Quark Glauber U+U Glauber Model b = 0 fm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 9: Comparison of the five observables between nucleon Glauber model (symbols) and quark Glauber model (lines with matching colors) as a function of ¯β2 for different values of σβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 12 [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Busza, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rajagopal, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' van der Schee, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 68, 339 (2018), arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='04801 [hep-ph] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Bernhard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Moreland, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Bass, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Liu, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Heinz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 94, 024907 (2016), arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='03954 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Giacalone, (2022), arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='06839 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Bender, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Heenen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Reinhard, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 75, 121 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Zhang, (2021), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='15559 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Nijs and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' van der Schee, (2021), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='13771 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Giacalone, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Zhang, (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='10449 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Abdallah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' (STAR), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 105, 014901 (2022), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00131 [nucl-ex] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [9] Haojie Xu and Chunjian Zhang (STAR Collabration), Constraints on neutron skin thickness and nuclear deformations using relativistic heavy-ion collisions from STAR, “https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='ch/event/895086/contributions/4724887/,https: //indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='ch/event/895086/contributions/4749420/,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Bhatta, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 106, L031901 (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01943 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Zhang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 128, 022301 (2022), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01631 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Agbemava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Afanasjev, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Nazarewicz, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Olsen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 102, 024311 (2020), arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01319 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Bally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=', (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='11042 [nucl-ex] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Otsuka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Tsunoda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Togashi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Shimizu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Abe, in European Physical Journal Web of Conferences, European Physical Journal Web of Conferences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 178 (2018) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 02003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Bender and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Heenen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 78, 024309 (2008), arXiv:0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='4383 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Heyde and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Wood, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 83, 1467 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Kumar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' 28, 249 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Poves, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Nowacki, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Alhassid, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 101, 054307 (2020), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07542 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 105, 014905 (2022), arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='08768 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Teaney and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 83, 064904 (2011), arXiv:1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='1876 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 105, 044905 (2022), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='00604 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Loizides, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C94, 024914 (2016), arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='07375 [nucl-ex] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Zhou and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Jia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content=' C 98, 044903 (2018), arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} +page_content='01812 [nucl-th] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQf9gYX/content/2301.03556v1.pdf'} diff --git a/FtAzT4oBgHgl3EQfxP5f/content/tmp_files/2301.01734v1.pdf.txt b/FtAzT4oBgHgl3EQfxP5f/content/tmp_files/2301.01734v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2744b940c1383a4d6c09b31ce35705b65fa74d83 --- /dev/null +++ b/FtAzT4oBgHgl3EQfxP5f/content/tmp_files/2301.01734v1.pdf.txt @@ -0,0 +1,2484 @@ +1 +Super-resolution with Sparse Arrays: A Non- +Asymptotic Analysis of Spatio-temporal Trade-offs +Pulak Sarangi, Mehmet Can H¨uc¨umeno˘glu, Robin Rajam¨aki, and Piya Pal +Abstract—Sparse arrays have emerged as a popular alternative to +the conventional uniform linear array (ULA) due to the enhanced +degrees of freedom (DOF) and superior resolution offered by +them. In the passive setting, these advantages are realized by +leveraging correlation between the received signals at different +sensors. This has led to the belief that sparse arrays require +a large number of temporal measurements to reliably estimate +parameters of interest from these correlations, and therefore +they may not be preferred in the sample-starved regime. In +this paper, we debunk this myth by performing a rigorous non- +asymptotic analysis of the Coarray ESPRIT algorithm. This +seemingly counter-intuitive result is a consequence of the scaling +of the singular value of the coarray manifold, which compensates +for the potentially large covariance estimation error in the limited +snapshot regime. Specifically, we show that for a nested array +operating in the regime of fewer sources than sensors (S = O(1)), +it is possible to bound the matching distance error between the +estimated and true directions of arrival (DOAs) by an arbitrarily +small quantity (ϵ) with high probability, provided (i) the number +of temporal snapshots (L) scales only logarithmically with the +number of sensors (P), i.e. L = Ω(ln(P)/ϵ2), and (ii) a suitable +separation condition is satisfied. Our results also formally prove +the well-known empirical resolution benefits of sparse arrays, +by establishing that the minimum separation between sources +can be Ω(1/P 2), as opposed to separation Ω(1/P) required by +a ULA with the same number of sensors. In addition to the +array geometry, our sample complexity expression reveals the +dependence on other key model parameters such as Signal to +Noise Ratio (SNR) and the dynamic range of the source powers. +This enables us to establish the superior noise-resilience of nested +arrays both theoretically and empirically. 1 +Index Terms—Sparse Arrays, Nested Sampling, Super-resolution, +Toeplitz Covariance Matrix, Non-Asymptotic Guarantees. +I. INTRODUCTION +The problem of source localization arises in different contexts +ranging from target detection in sonar and radar, hybrid +mmWave channel estimation, and DOA estimation in array +signal processing [1]–[3]. Traditionally, these applications +consider ULAs, which are known to resolve up to S = O(P) +sources with P sensors. However, deterministic sparse array +geometries, such as nested and coprime arrays [1], [2], have +recently gained significant attention primarily due to two +attractive properties. Firstly, sparse arrays are able to identify +up to S = O(P 2) uncorrelated sources using only P sensors. +Secondly, sparse arrays enjoy a performance gain showcased +by lower Cram´er-Rao bound and higher angular resolution [4]– +[8]. Both of these properties can be attributed to the enhanced +spatial DOF enabled by the so-called difference coarray, which +can be as large as Θ(P 2). +1This work was supported by Grants ONR N00014-19-1-2256, ONR +N00014-19-1-2227, NSF 2124929, and NSF CAREER ECCS 1700506. +The enhanced DOF of the coarray are realized by comput- +ing temporal correlations between the spatial measurements +and constructing an augmented covariance matrix called the +“coarray covariance matrix”, whose size is determined by the +size of the difference coarray. Following the construction of +the coarray covariance matrix, it is possible to fully harness +the power of the difference coarray and identify the unknown +source directions using classical subspace techniques, such +as MUSIC, ESPRIT or the matrix pencil method [9]–[11]. +Despite the success of coarray-based algorithms, a common +belief is that they require a large number of temporal snapshots +to fully utilize the number of DOFs provided by the coarray. +The root of this belief mainly lies in the inadequacy of +existing performance analyses, which are primarily based on +characterizing the asymptotic Mean Squared Error (MSE) of +the Coarray MUSIC [5] and Coarray ESPRIT algorithms [12]. +In particular, such asymptotic results primarily rely on the +first-order perturbation analysis framework proposed in [13], +which leaves two key questions unanswered regarding the +performance of coarray algorithms. Firstly, the perturbation +framework fails to theoretically explain the improvement in +resolution offered by sparse arrays over the ULA—a phe- +nomenon that has been extensively observed in numerical ex- +periments [5], [14]. Secondly, the analysis does not adequately +reveal the dependence of temporal snapshots on key model +parameters such as the array geometry, number of sensors, +SNR and dynamic range of the source powers. +The aforementioned shortcomings are partially addressed in +[15], which adapts recent advances in the theory of super- +resolution [16], [17] to the coarray setting. The analysis, +which is based on Total-Variational norm minimization, is +indeed non-asymptotic. However, it is possible to show that +the snapshot requirement in this setting scales quadratically +(rather than linearly) with the number of sensors P, which +is undesirable. In a parallel line of work using a grid-based +model, we recently showed that Ω(P 2) snapshots are sufficient +for ensuring exact support recovery with high probability +even for closely-spaced sources, where the smallest source +separation scales as Ω(1/P 2) [18]. Although the analysis is +applicable for scenarios where S > P (more sources than +sensors), the sample complexity Ω(P 2) is still conservative +when S ≤ P. In [19], an atomic norm formulation is adopted +to exploit the Toeplitz structure of the coarray covariance +matrix. The analysis provides a characterization of the covari- +ance matrix estimation error, but not of the sample complexity +required to achieve a desired DOA estimation error, which is +often the main quantity of interest. Indeed, common folklore +arXiv:2301.01734v1 [eess.SP] 4 Jan 2023 + +2 +suggests that the benefits of sparse arrays necessarily come +at the cost of a large number of snapshots, since the coarray +covariance matrix, which typically needs to be estimated, is of +size Θ(P 2). Hence, one might be tempted to falsely conclude +that sparse arrays are at a disadvantage compared to ULAs. +In this paper, our goal is to dispel this belief by providing +new non-asymptotic results on the performance of Coarray +ESPRIT with a focus on nested arrays in the regime S ≤ P. +Our analysis is motivated by contemporary applications such +as autonomous sensing and mmWave channel estimation [3], +[14], where identifying more sources than sensors may not be +necessary, and the number snapshots may be restricted either +due to coherent multipaths or a rapidly varying environment. +While subspace-based algorithms have been around for several +decades and actively used in practice, performance guarantees +characterizing their precise resolution limit were obtained only +recently [20]–[24]. This analysis has also been extended to +multi-snapshot setting in [25]. The key factor enabling these +guarantees is the characterization of the smallest singular value +of Vandermonde matrices [24]. However, all the aforemen- +tioned results are only applicable to the ULA. Furthermore, +no statistical assumptions are made on the source signals, and +hence, the coarray perspective is missing. The key difference +between deterministic and random sources is that in the latter +case, the perturbation to the subspace of interest is a con- +sequence of both noise as well as finite-snapshot covariance +estimation error. Therefore, extending the analysis in [20], [25] +to the stochastic case requires non-trivial modifications. +Contributions: Our first main contribution is to probabilis- +tically characterize the coarray covariance matrix estimation +error due to finite snapshots. Our second main contribution is a +non-asymptotic performance analysis for the Coarray ESPRIT +algorithm in terms of the matching distance error metric. +Specifically, we characterize the number of temporal snapshots +(sample complexity) required to bound the matching distance +error by a specified parameter. To the best of our knowledge, +our sample complexity expression (in terms of snapshots) is +the first to explicitly bring out the dependence on key model +parameters such as the array geometry, SNR and dynamic +range of the source powers. Furthermore, we establish that it +is possible to bound the matching distance error with an arbi- +trarily small quantity for both the nested array and ULA, using +the (order-wise) same number of snapshots L = Ω(ln P). +However, a nested array can achieve this in a much smaller +separation regime ∆min = Ω(1/P 2) compared to the ULA, for +which ∆min = Ω(1/P). Our analysis dispels the widely-held +belief that sparse arrays require significantly more snapshots +compared to ULAs when the number of sources is less than +the number of sensors, and at the same time establishes the +superior resolution capabilities of nested arrays. In addition +to advancing the theoretical understanding, this analysis could +also serve as a guiding principle for practitioners to determine +suitable operating conditions. Notations: Symbol ⊙ represents +the Khatri-Rao (columnwise Kronecker) product, whereas ∥·∥2 +and ∥·∥F denote the spectral and Frobenius norm of a matrix. +Moreover, σi(A) is the i-th largest singular value of A. For +a set real numbers {p1, p2, . . . , pK}, pmin and pmax denote +the minimum and maximum numbers in the set, respectively. +The symbol T := [0, 1) denotes the torus. For a sub-Gaussian +random variable X, ∥X∥ψ2 denotes its sub-Gaussian norm +defined as ∥X∥ψ2 := inf{t > 0 | E[exp X2/t2] ≤ 2}. +II. BACKGROUND ON SPARSE ARRAYS +Consider a sparse linear array (SLA) with P sensors located +at {dpλ/2}P +p=1, where λ is the wavelength of the incoming +far-field narrow-band source signals and dp belongs to an +integer set S (|S| = P). Suppose S sources with distinct +DOAs θ = {θ1, θ2, · · · , θS} impinge on the array where +θi ∈ (−π/2, π/2] for i = 1, . . . , S. The signal received at +the P sensors at time instance t is given by: +y(t) = AS(θ)x(t) + n(t), +t = 1, . . . , L. +(1) +The +matrix +AS(θ) += +[aS(θ1), aS(θ2), . . . , aS(θS)] +∈ +CP ×S +is the array manifold matrix where: aS(θi) += +[ejπd1 sin(θi), ejπd2 sin(θi) +. . . ejπdP sin(θi)]⊤, represents the +steering vector corresponding to the direction θi, L denotes +the total number of temporal snapshots, x(t) ∈ CS is the tth +temporal snapshot of the source signal vector and n(t) ∈ CP +is an additive noise term. We define the normalized spatial +frequencies (which we refer to as normalized DOAs) as +ωi = sin(θi)/2. Throughout this paper, we make the following +statistical assumptions on the source signals and noise: +[A1] Uncorrelated Gaussian Sources: The source signals +x(t) are assumed to be uncorrelated white circularly sym- +metric Gaussian CN(0, P) where P = diag(p1, p2, . . . , pS) +represents a diagonal covariance matrix of source powers. +[A2] Gaussian Noise: The noise n(t) follows a zero-mean +circularly symmetric complex Gaussian distribution n(t) ∼ +CN(0, σ2I), and is uncorrelated with x(t). +Under assumptions [A1-A2], the measurements follow y(t) ∼ +CN(0, Ry), where Ry is given by: +Ry = AS(θ)PAH +S (θ) + σ2IP ∈ CP ×P . +(2) +By vectorizing Ry, we obtain the “virtual measurements”: +ry = (AS(θ)∗ ⊙ AS(θ))p + σ2i, where i = vec(IP ) and +p = [p1, . . . , pS]T . The matrix AS(θ)∗ ⊙ AS(θ) can be +viewed as a “virtual array” with sensor locations given by +the difference set of the SLA. +Definition +II.1 +(Difference +Set). +Given +a +SLA +S += +{d1, d2, · · · , dP }, +its +difference +set +DS +is +defined +as: +DS = {dm − dn|dm, dn ∈ S}. +The difference set DS of S is also called its virtual difference +coarray. Let Mca > 0 be the largest integer such that the +set US := {0, 1, . . . , Mca} satisfies US ⊆ DS. This set +US denotes the largest contiguous non-negative segment of +the difference set and is essentially a ULA with Mca + 1 +sensors. By harnessing the structure of US, sparse arrays enjoy +enhanced degrees of freedom over the physical SLA. An +array is called hole-free if its difference set is a ULA, i.e., +DS = {−Mca, · · · , Mca}. We now introduce the notation for +a “generalized nested array”, which is a special hole-free array. + +3 +Definition II.2 (Nested array). A generalized nested array +S(N1,N2) with N1 ≥ N2 > 0, is defined as: S(N1,N2) = +{n}N1 +n=1 ∪ {m(N1 + 1)}N2 +m=1. +It can be shown that any nested array S(N1,N2) is hole-free, +i.e., US = {0, 1, · · · , Mca} with Mca = N2(N1 + 1) − 1. +Furthermore, Sula = S(P −1,1), i.e., choosing N1 = P − 1 and +N2 = 1, yields a ULA with P sensors. For a given P, if +N1 = ⌈ P +2 ⌉, N2 = ⌊ P +2 ⌋, then Mca + 1 = ⌊ P +2 ⌋(⌈ P +2 ⌉ + 1). It can +be verified that for P ≥ 3, we have: +P 2/5 ≤ Mca + 1 ≤ P 2. +(3) +Therefore, Mca = Θ(P 2) is indeed achievable. Next, we +introduce an important quantity that is essential for describing +correlation-based processing. +Definition II.3 (Weight Function). Consider a hole-free array +S. For every i ∈ DS, its weight function is defined as |Ωi|: +Ωi = {(m, n)|dm − dn = i, 1 ≤ m, n ≤ P} where the set Ωi +essentially captures all pairs (dm, dn) of sensor locations that +generate the difference of i = dm − dn. +Due to symmetry, it can be verified that |Ωi| = |Ω−i|. Next, +we review the widely-used “redundancy averaging” technique +used for correlation-domain processing. Following [1], [5], +[26], the virtual ULA measurements are given by: +t = Favry, +(4) +where t = [t−Mca, · · · , t−1, t0, t1, · · · , tMca]⊤ and Fav is the +redundancy averaging matrix given by: +[Fav]i+Mca+1,m+P (n−1) = +� +1 +|Ωi| +If dm − dn = i +0 +Otherwise, +(5) +with −Mca ≤ i ≤ Mca and 1 ≤ m, n ≤ P. The element +ti is obtained by averaging all entries [Ry]m,n whose indices +(m, n) generate a difference of i, i.e., dm − dn = i. Define +a Toeplitz operator TMca : C2Mca+1 → CMca+1×Mca+1 as: +[TMca(z)]m,n = zMca+1+m−n, 1 ≤ m, n ≤ Mca + 1. +If +the +vector z ∈ C2Mca+1 is conjugate symmetric, i.e., zMca+1+i = +z∗ +Mca+1−i, i = 0, 1, . . . , Mca, then TMca(z) is a Hermitian +matrix. Using the virtual measurement t, an augmented vir- +tual co-array covariance matrix Tca ∈ C(Mca+1)×(Mca+1) is +constructed as follows: +Tca := TMca(t) = AUS(θ)PAUS(θ)H + σ2IMca+1. +(6) +Once this virtual coarray covariance matrix has been obtained, +any subspace-based algorithm [9], [10] applied to Tca can +exactly recover the source DOAs provided Mca ≥ S. Hence, +this also reveals that by efficiently designing sparse arrays, +we can resolve up to Θ(P 2) sources with only P sensors. In +the next section, we describe how the correlation processing +is modified in the finite snapshot setting. +A. Finite-Snapshot Coarray Covariance Estimation +Let �Ry be the sample covariance matrix given by: +�Ry := 1 +L +L +� +t=1 +y(t)y(t)H. +(7) +With a finite L, all the operations on the true covariance matrix +are replaced by operations on the sample covariance matrix. +First, we apply the redundancy averaging on ˆry: +ˆt := Favˆry, where ˆry := vec( �Ry). +(8) +Here ˆt += +[ˆt−Mca, · · · , ˆt−1, ˆt0, ˆt1, · · · , ˆtMca]⊤ with ˆti += +1 +|Ωi| +� +dm−dn=i[ �Ry]m,n. Next, the estimated coarray covari- +ance matrix is obtained by constructing a Toeplitz Hermitian +matrix from ˆt as follows: +�Tca = TMca(ˆt). +(9) +For a hole-free sparse array S, from (4), the elements of the +matrix Ry are given by: +[Ry]m,n = tdm−dn 1 ≤ m, n ≤ P. +(10) +Similarly, using the estimated coarray covariance matrix �Tca, +we define matrix Rav ∈ CP ×P as +[Rav]m,n := �tdm−dn, 1 ≤ m, n ≤ P. +(11) +This essentially maps the entries ˆti into a P × P matrix with +the assignments specified by the difference set of the array +S. Since the sample covariance matrix �Ry is imperfect, the +estimate �Tca also incurs an error due to a finite number of +snapshots. We denote the covariance estimation error as: +EL = Tca − �Tca. +(12) +The error in estimating the coarray covariance matrix naturally +causes errors in DOA estimation as well. Since subspace based +algorithms are typically applied to this estimated covariance +matrix �Tca, it becomes crucial to probabilistically characterize +the estimation error EL and how it affects the DOA estimation +error. This paper provides such a rigorous theoretical charac- +terization of the DOA estimation error with limited snapshots. +B. Review of Existing Performance Analysis of Coarray-Based +Angle Estimation +The existing performance analyses for coarray-based algo- +rithms are largely asymptotic in nature. In particular, they rely +on the first-order perturbation analysis framework proposed +in [13], which has been used to obtain expressions for the +mean square error (MSE) of coarray MUSIC [5], and coarray +ESPRIT [12]. Consider the eigen decomposition Tca += +UΓsU + U⊥ΓnUH +⊥, where U ∈ CMca+1×S and U⊥ ∈ +CMca+1×Mca+1−S denote the eigenvectors corresponding to +the signal and noise subspaces, respectively. The correspond- +ing perturbed matrices are denoted as �Tca = Tca + ∆Tca, +�U⊥ = U⊥ + ∆U⊥ and �Γn = Γn + ∆Γn. The perturbed +matrices satisfy: (Tca + ∆Tca)(U⊥ + ∆U⊥) = (U⊥ + +∆U⊥)(Γn+∆Γn). The perturbation analysis in [5] hinges on +(i) the perturbations being “small enough” and (ii) ignoring +the higher order perturbation terms such as ∆Tca∆U⊥ etc. +One of the key drawbacks of this analysis is that a rigorous +characterization of an upper bound on the “small enough +perturbation” ∥∆Tca∥2 ≤ ϵ1 has not been provided explicitly. +Secondly, [5, Theorem 1] makes a critical assumption that “the +signal subspace and the noise subspace are well-separated”. + +4 +This assumption leaves open the possibility of problematic +(unidentifiable) source configurations, which have not been +explicitly addressed in their analysis. We address both of the +aforementioned issues by adopting a non-asymptotic analysis +framework that is free from any approximations. Our analysis +also explicitly characterizes source configurations that ensure +separation between the so-called signal and noise subspaces. In +[18], the first rigorous non-asymptotic probabilistic guarantees +were provided for support recovery using a grid-based model. +Although their analysis is valid for S > P, the sample +complexity L = Ω(P 2) is conservative when S < P as our +analysis in Section IV will show. +III. PERFORMANCE ANALYSIS OF COARRAY ESPRIT +WITH FINITE SNAPSHOTS +The Coarray ESPRIT algorithm, an adaptation of ESPRIT in +the coarray domain, was introduced in [12]. It applies ESPRIT +on the estimated coarray covariance matrix �Tca as opposed to +covariance matrix �Ry of the physical measurements. For a +self-contained exposition, we review the Coarray ESPRIT al- +gorithm and point out certain invariance properties of Coarray +ESPRIT. We describe Coarray ESPRIT for the ideal coarray +covariance matrix Tca. The extension to the sample covariance +estimate is straightforward. +A. The Coarray ESPRIT Algorithm +The coarray signal subspace is defined as the span of +the steering vectors: Sca := R (AUS(θ)) . Matrix T0 := +AUS(θ)PAUS(θ)H is positive semi-definite and permits the +following eigendecompostion: T0 = BΓBH, where the di- +agonal of Γ comprises of the eigenvalues ordered in non- +increasing fashion and B is a unitary matrix. We can partition +B as B = [U, U⊥], where the columns of U ∈ C(Mca+1)×S +denote the eigenvectors of T0 corresponding to its non-zero +eigenvalues. Following this decomposition, we write Tca as: +Tca = T0 + σ2IMca+1 = B(Γ + σ2IMca+1)BH. +(13) +If Mca ≥ S, the Vandermonde structure of AUS(θ) allows us +to argue that rank(AUS(θ)PAUS(θ)H) = S, and hence: +Sca = R(AUS(θ)) = R(AUS(θ)PAUS(θ)H) = R(U). (14) +As a result of (14), ∃ an invertible Q ∈ CS×S such that +U = AUS(θ)Q. +(15) +Let U0 ∈ CMca×S and U1 ∈ CMca×S denote the submatrices +corresponding to the first and last Mca rows of U. Similarly, +let V0, V1 ∈ CMca×S be the submatrices corresponding to the +first and last Mca rows of AUS(θ). Due to the Vandermonde +structure of AUS(θ), the following holds: V1 = V0D, where +D = diag(ejπ sin(θ1), ejπ sin(θ2), . . . , ejπ sin(θS)). By (15), ma- +trices U0 and U1 satisfy: +U0 = V0Q, +U1 = V0DQ. +(16) +Now, consider the matrix +Ψ = U† +0U1 ∈ CS×S. +(17) +Since U0 has full column rank (17) implies U† +0 = Q−1V† +0. +Plugging this in (17) and combining with (16), we have: +Ψ = Q−1DQ. Hence, the DOAs can be inferred from the +eigenvalues of Ψ. Since L is finite, we do not have access to +Tca and Coarray ESPRIT is instead applied on its estimate +�Tca defined in (9). If we can ensure that the error EL is +small enough (which we will rigorously specify using Weyl’s +inequality), �Tca will be at least rank-S. Let ˆU be the matrix +of eigenvectors corresponding to the largest S eigenvalues of +�Tca (which is well-defined). We can consider �U as a basis of +the perturbed coarray signal space ˆSca. From �U, we compute +the matrices �U0, �U1, �Ψ following the same construction as +U0, U1 and Ψ. Let ˆλi = riej �φi be the polar representation +of the eigenvalues of the matrix ˆΨ. The estimated normalized +frequencies ˆΩ = {�ωi}S +i=1 are then given by �ωi = +�φi +2π. +B. Basis Invariance Property of ESPRIT +In the previous section, ESPRIT is performed using the basis +given by the singular vectors �U (U) of �Tca (Tca). However, +the following Lemma shows that the output of ESPRIT is +invariant to the choice of the basis for the subspace. +Lemma 1. Let �U ∈ C(Mca+1)×S be another basis for R( �U). +Then, the matrix �Ψ := �U† +0 �U1 is similar to the matrix �Ψ, i.e., +�Ψ and �Ψ share the same eigenvalues. +Proof. Since R( �U) = R( �U), there exists an invertible matrix +W ∈ CS×S such that �U := +�UW. Thus, the following +holds: �U0 = �U0W, �U1 = �U1W. Since W is an invertible +matrix, �U† +0 = W−1 �U† +0 and �Ψ = �U† +0 �U1 = W−1 �U† +0 �U1W = +W−1 �ΨW. This completes the proof. +C. Covariance Estimation Error +In this section, we obtain tail bounds on ∥EL∥2 in terms of +array parameters in a finite snapshot setting. Such a bound +brings out the effect of the array geometry on the estimation +error. Our analysis leverages recent results derived in [27] +which we specialize for complex Toeplitz Hermitian matrices. +Some of our intermediate steps depart from [27] by invoking +a result on the bounding the supremum of a certain spectral +function from [28]. We first introduce the key quantities +and intermediate results on bounding the spectral norm of a +Toeplitz Hermitian matrix from [28], [29]. +Let M +∈ +CN×N be any Hermitian symmetric Toeplitz +matrix. Such a matrix can be completely described by only +its first column. Consider the “spectral function” associated +with m = [m−(N−1), . . . , m−1, m0, m1, . . . , mN−1]⊤ [29]: +fm(θ) = �N−1 +k=−(N−1) mk exp(−jkθ), where mk = Mk+1,1 +and m−k = m∗ +k as a result of the Hermitian Toeplitz structure. +Evidently, the spectral function is a trigonometric polynomial +of order N − 1 [28] whose coefficients are determined by the +vector m. This spectral function fm(θ) can be used to bound +∥M∥2 as indicated by the following lemma from [27]–[29]: +Lemma 2. Let M∈CN×N be a Hermitian symmetric Toeplitz +matrix and fm be the associated spectral function. Then, +∥M∥2 ≤ supθ∈[−π,π] |fm(θ)|. + +5 +Lemma 2 indicates that the spectral norm of a Hermitian +symmetric Toeplitz matrix can be bounded by the supre- +mum of its associated spectral function. Note that the co- +variance estimation error EL = Tca − �Tca is a Toeplitz +Hermitian matrix, satisfying EL += +T (e), where e += +[e−Mca, . . . , e−1, e0, e1, . . . , eMca]T is conjugate symmetric +and ei = ti − ˆti. Therefore, to bound ∥EL∥2 using Lemma 2, +we need to investigate the spectral function fe(θ): +fe(θ) := +Mca +� +k=−Mca +ek exp(−jθk). +(18) +Towards this purpose, define Λ(θ), for 1 ≤ m, n ≤ P, +[Λ(θ)]m,n = +1 +|Ωdm−dn| exp(j(dm − dn)θ), +(19) +and Ey := Ry − Rav, where Ry and Rav are defined in (10) +and (11), respectively. The elements of Ey are given by: +[Ey]m,n = tdm−dn − �tdm−dn = edm−dn, 1 ≤ m, n ≤ P. (20) +Proposition 1 provides a compact representation of fe(θ). +Proposition 1. Let fe(θ) be the spectral function defined in +(18). Then, the following equality holds: fe(θ) = tr (EyΛ(θ)) +where Λ(θ) and Ey are defined in (19) and (20), respectively. +Proof. +tr (EyΛ(θ)) = +P +� +m,n=1 +[Ey]m,n[Λ(θ)]n,m = +P +� +m,n=1 +edm−dn +e−j(dm−dn)θ +|Ωdm−dn| += +Mca +� +s=−Mca +� +m,n +dm−dn=s +es +exp(−jsθ) +|Ωs| += +Mca +� +s=−Mca +es exp(−jsθ). +We introduce a quantity referred to as “Redundancy coeffi- +cient” that will play an important role in bounding ∥EL∥2. +Definition III.1 (Redundancy Coefficient). Given a hole-free +sparse array S, let Mca be the largest element in its differ- +ence set DS. The redundancy coefficient ∆(S) is defined as: +∆(S) := �Mca +i=0 +1 +|Ωi|, where set Ωi is defined in Definition II.3. +The quantity ∆(S) is controlled by the redundancy pattern of +the sparse array S, i.e., the number of times an element repeats +in the difference set. We provide an illustrative example to +show how the quantity ∆(S) grows as a function of P. +Lemma 3. Given a generalized nested array S(N1,N2) +nest +with +P := N1 + N2 ≥ 3 sensors, the following holds: ln(P) ≤ +∆(S(N1,N2) +nest +) +≤ +2 ln(P), if N2 += +1, and P 2/16 +≤ +∆(S(N1,N2) +nest +) ≤ P 2, if N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋ ≥ 2. +Proof. Case I (N2 = 1): The choice N2 = 1 corresponds to +a ULA, with P = N1 + 1 sensors and |Ωi| = P − i, i ≥ 0. +Therefore, ∆(S(P −1,1) +nest +) = �P −1 +i=0 +1 +P −i. Such a harmonic sum +can be bounded as ln(P) ≤ �P −1 +i=0 +1 +P −i ≤ 1+ln(P) [30]. For +P ≥ 3, we get the desired bound since 1 + ln(P) ≤ 2 ln(P). +Case II (N2 = ⌊P/2⌋ ≥ 2): The differences between the +elements of the outer and inner ULA which are of the form +k = i(⌈P/2⌉ + 1) − j, 2 ≤ i ≤ ⌊P/2⌋ and 1 ≤ j ≤ +⌈P/2⌉, satisfy |Ωk| = 1. Therefore, we have ∆(S(N1,N2) +nest +) ≥ +⌈P/2⌉⌊P/2⌋/2 ≥ (P 2/8 − P/8) ≥ P 2/16, where the first +inequality follows from ⌊P/2⌋ − 1 ≥ ⌊P/2⌋/2 and the last +inequality uses P ≤ P 2/2 for P ≥ 2. Since S(N1,N2) +nest +is hole +free, it implies 1/|Ωi| ≤ 1 for all 0 ≤ i ≤ Mca. Therefore, we +can bound ∆(S(N1,N2) +nest +) ≤ Mca + 1 ≤ P 2. +As the following Theorem will show, ∆(S) determines the +sample complexity for controlling the covariance estimation +error. Therefore, with the same number of sensors, two differ- +ent array geometries could require drastically different sample +complexity for ensuring that the covariance estimation error is +bounded by the same quantity with high probability. +Theorem 1. Consider the measurement model (1) obeying +assumptions [A1-A2], where S is a hole-free sparse array with +redundancy coefficient ∆(S). Let Tca ∈ CMca+1×Mca+1 be +the coarray covariance matrix defined in (6) and �Tca be its +estimate given by (9). For any ϵ ≥ 0, we have +P +� +∥Tca − �Tca∥2 ≥ ϵ +� +≤ 8Mca exp +� +−c1L min +� +c2ϵ2 +∥Ry∥2 +2∆(S) , +ϵ +∥Ry∥2 +� +∆(S) +�� +, +where c1 and c2 are a positive universal constants. +Proof. The proof is in Appendix A-B. +D. Frequency/Angle Estimation Error of Coarray ESPRIT +We next bound the DOA estimation error in terms of the +covariance estimation error EL. Finally, we will combine this +bound with the probabilistic bounds on ∥EL∥2 in Theorem 1 +to obtain the main sample complexity result (in Theorem 3). +We will use the matching distance metric, defined as follows +[20]: +md(θ, ˆθ) := min +Π∈P max +j +min +k∈Z |ˆωΠ(j) − ωj + k| +(21) +where ωi (ˆωi) are the normalized DOAs and P denotes the +set of all possible permutations on {1, 2, · · · , S}. +For our analysis, we will use an additional assumption that +will be invoked whenever suitable: +[A3] The number of sources S = O(1), i.e., S is held +constant and does not grow with P. +Eigen Gap condition: Define: +β := pminσ2 +S(AUS(θ)) − σ2. +(22) +Henceforth, we will refer the condition β > 0 as the “eigen +gap condition” and it will play an important role in our analy- +sis. Recall, from the definition of Tca = AUS(θ)PAUS(θ)H + +σ2, β > 0 ensures that there is a margin between the smallest +singular value of AUS(θ)PAUS(θ)H and the (S+1)th singular +value of Tca (determined by the noise σ) as pminσ2 +S(AUS(θ)) +is a lower bound on σS(Tca). The following theorem relates +the DOA estimation error in terms of matching distance to the +covariance estimation error EL, provided the latter is upper +bounded by a suitable quantity. +Theorem 2. Let S be a hole-free sparse linear array with P +sensors. Let Tca ∈ CMca+1×Mca+1 be the coarray covariance + +6 +matrix defined in (6) and �Tca be its estimate given by (9). +If assumption [A3] holds and the following conditions are +satisfied: +β > 0 +and ∥EL∥2 ≤ CSβ +(23) +then the matching distance error of ESPRIT algorithm satisfies +md(θ, ˆθ) ≤ q∥EL∥2 +(24) +where +EL, +β +are +defined +in +(12), +(22), +q += +(C′ +S +√Mca + 1)/(βσS(AUS(θ))). Quantities CS, C′ +S +are +dependent only on S which is assumed to be O(1). +Proof. See Appendix B-A. +The following Lemma obtains both lower and upper bounds +on the spectral norm ∥Ry∥2 that are valid regardless of the +array geometry. +Lemma 4. Consider the covariance matrix Ry given by (2), +where S is any (sparse) array. Given a fixed S, signal powers +p and noise power σ2, for all θ the following holds: +pminP ≤ ∥Ry∥2 ≤ pmaxPS + σ2. +(25) +Proof. For any S, we can bound the spectral norm ∥Ry∥2 as: +∥Ry∥2 = σ1(AS(θ)PAS(θ)H) + σ2 ≤ pmaxσ1(AS(θ))2 + σ2 +≤ pmaxPS + σ2 +where +the +last +inequality +follows +from +the +fact +that +σ1(AS(θ))2 +≤ +∥AS(θ)∥2 +F += +PS. Similarly, we can +lower bound the norm ∥Ry∥2 ≥ σ1(AS(θ)PAS(θ)H) ≥ +pminσ2 +1(AS(θ)) ≥ pmin∥AS(θ)∥2 +F /S = pminP. +Combining Theorem 1 and 2, we next present a sufficient +condition on the number (L) of snapshots in terms of the +model parameters (array geometry, SNR and source config- +uration) that allows us to bound the matching distance error +by a prescribed ϵ with probability at least 1 − δ. +Theorem 3. Consider the measurement model (1), where S +is a hole-free sparse array. Suppose β > 0 and the statistical +assumptions [A1-A3] hold. Then for any 0 < δ < 1 and ϵ > 0, +the matching distance error satisfies md(θ, ˆθ) ≤ min(ϵ, CSβq) +with probability at least 1 − δ, provided +L≥c3 ln +�8Mca +δ +� +max +� +q2 +1∆(S) +c2ϵ2 +,q1 +� +∆(S) +ϵ +,L2 +0 +c2 +,L0 +� +. (26) +Here q1 = q∥Ry∥2, L0 = ∥Ry∥2 +� +∆(S)/(CSβ) and c2, c3 +are universal constants. +Proof. See Appendix B-C +Corollary 1. Consider the measurement model (1), where +S is a hole-free sparse array. Suppose β +> 0 and the +statistical assumptions [A1-A3] hold. Then for any 0 < δ < 1 +and 0 < ϵ ≤ q min(CSβ, pminP +� +∆(S)/c2), the matching +distance error satisfies md(θ, ˆθ) ≤ ϵ with probability at least +1 − δ provided +L ≥ c3 ln (8Mca/δ) q2 +1∆(S)/(c2ϵ2), +(27) +where q1, L0,c2, c3 are given in Theorem 3. +Proof. Using the lower bound on ∥Ry∥2 from Lemma 4, +we can see ϵ +≤ +min(CSβq, q1 +� +∆(S)/c2). Since β +≥ +ϵ/(CSq), +this +implies +L0 +≤ +q1 +� +∆(S)/ϵ. +This +in- +equality +also +implies +L2 +0/c2 +≤ +q2 +1∆(S)/(c2ϵ2). +Us- +ing ϵ +≤ +q1 +� +∆(S)/c2, we can conclude that L0 +≤ +(q1 +� +∆(S)/ϵ2)(q1 +� +∆(S)/c2) = q2 +1∆(S) +c2ϵ2 . Therefore, (27) im- +plies (26) since max( q2 +1∆(S) +c2ϵ2 , +q1√ +∆(S) +ϵ +, L2 +0 +c2 , L0) = q2 +1∆(S) +c2ϵ2 , and +the proof is completed. +Role of redundancy coefficient in determining Temporal +Sample Complexity: Corollary 1 indicates that if the number +of snapshots grows proportional to the redundancy coefficient +∆(S), then it is possible to bound the matching distance error +by an arbitrarily small ϵ. Recall that ∆(S) is a function of +the redundancy pattern of S and from Lemma 3 we have +∆(Sula) = Θ(ln(P)) and ∆(Snest) = Θ(P 2). Based on this, at +a cursory glance, one may be tempted to conclude from (27) +that for the same number of sensors, the snapshot requirement +for the nested array is significantly larger than for the ULA. +This is also consistent with an existing misconception that co- +array based processing requires a large number of snapshots. +However, in reality the sample complexity is also controlled +by the interaction of ∆(S) with other geometry dependent +terms in (27) such as q1 = q∥Ry∥2, which in turn depend on +both the physical array and coarray size. In the next section, +we clarify this misconception regarding the seemingly higher +snapshot requirement of nested arrays in the setting S = O(1). +Spatiotemporal trade-offs: The snapshot requirement in +Corollary 1 is inversely proportional to β (since q ∝ +1 +β ). +If the array geometry and source configuration are kept +fixed and we increase the SNR (either by increasing pmin +or decreasing noise power σ), Corollary 1 suggests that it +is possible to achieve the same probability of error with +fewer snapshots. Our simulations also are consistent with +this theoretical prediction. This SNR and geometry dependent +snapshot characterization is another novel contribution of our +work. +IV. A CLOSER LOOK AT THE SEPARATION CONDITION FOR +SUPER-RESOLUTION WITH SPARSE ARRAYS +In order to understand the behavior of the smallest non- +zero singular value σS(AUS(θ)), we consider the notion of +minimum separation [20]: +∆min(θ) = min +i,j∈Ω +i̸=j +min +k∈Z +���ωi − ωj + k +��� +(28) +where ωi is the normalized spatial frequency corresponding to +direction θi. By definition, for all θ we have 0 ≤ ∆min(θ) ≤ +1/2. Instead of analyzing an arbitrary source configuration θ, +one can obtain a more interpretable condition by representing + +7 +(23) as a function of the minimum separation. The source +configurations where ∆min(θ) is larger than some threshold +inversely proportional to Mca +1 (i.e. ∆min(θ) > +γ +Mca+1, γ > +1) will be referred to as the “well-separated” regime. We will +inspect what this means for specific array geometries such as +the ULA and nested array, and obtain tight bounds on L. +A. The “Well-Separated” Case +In this section, we turn our attention to how the eigen gap +condition can be utilized to obtain sufficient conditions on +SNR for different array geometries in the “well-separated” +regime. Let V ∈ CK×S be a Vandermonde matrix, with +[V]m,n = zm−1 +n +where {zn}S +n=1 are the so called “nodes” +of the matrix. We begin by summarizing results from [21], +[24], [31], [32] which characterize the minimum singular value +of a Vandermonde matrix in the well-separated regime. The +following Lemma follows from [32, Eq. (32)] which is an +intermediate result from [32, Theorem 1]. +Lemma 5. Let V(α) ∈ CK×S be a Vandermonde matrix with +zn = ej2παn for 1 ≤ n ≤ S and S ≤ K. If αi ∈ [0, 1) are all +distinct and satisfy: +min +i,j∈Ω +i̸=j +min +k∈Z +���αi − αj + k +��� ≥ γ +K +(29) +for some constant γ > 1, then the following holds: +σS(V(α))2 ≥ K/C′, where C′ := γ/(γ − 1). +(30) +From Lemma 5, for S = Sula if the source configurations θ +satisfies ∆min(θ) ≥ γ +P for some γ > 1 and S ≤ P then we +have the following lower bound: +σS(AUS(θ))2 ≥ P/C′ +(31) +In the following Proposition, we apply Lemma 5 to character- +ize lower bounds on σS(AUS) for the nested array. +Proposition 2 (Well-Separated). Let S = S(N1,N2) +nest +be a nested +array with N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋ with P ≥ 3. +Suppose ∆min(θ) ≥ +5γ +P 2 for some γ > 1 and S ≤ P 2/5. +Then, the following lower bound holds: +σS(AUS(θ))2 ≥ P 2/C′ +n, where C′ +n = 5γ/(γ − 1). +(32) +Proof. For the nested array with N1 = ⌈P/2⌉ and N2 = +⌊P/2⌋, from (3) we have Mca + 1 ≥ P 2 +5 . Hence, ∆min(θ) ≥ +5γ +P 2 implies ∆min(θ) ≥ +γ +Mca+1. Therefore, the condition on +∆min(θ) in Lemma 5 holds and we have the desired lower +bound: σS(AUS(θ))2 ≥ Mca+1 +C′ +≥ ( γ−1 +γ ) P 2 +5 = P 2 +C′n . +Proposition 2 shows that for a nested array, the sources +are well-separated if ∆min(θ) ≥ 5γ/P 2 and in this case, +σS(AUS(θ)) grows as Ω(P), owing to the the larger difference +coarray of a nested array. +In order to highlight the dependence of sample complexity +only on key model parameters, we define quantities to combine +parameters that are held fixed (such as S, pmin, pmax, σ): +Cula(S, σ, pmax) := 8C +′2 +S C +′3 c3 +c2 +(S + +σ2 +pmax +)2 +(33) +Cnest(S, σ, pmax) := 4C +′2 +S C +′3 +n +c3 +c2 +(S + +σ2 +pmax +)2 +(34) +where C′, C′ +n are universal constants and CS defined in +Theorem 2 is dependent only on S. Using Proposition 2, we +now specialize Corollary 1 for the ULA and nested array. +Theorem 4. Let S = Sula be a ULA with P sensors. Suppose +the minimum angular separation between the sources, and the +SNR satisfy the following conditions for some γ > 1: +∆min(θ) ≥ γ/P, +pmin/σ2 > 2C′/P, where C′ = +γ +γ − 1. +Under assumptions [A1-A3], for any 0 < δ < 1 and 0 < ϵ ≤ +C1(S) := CSC′ +S, md(θ, �θ) ≤ ϵ is satisfied with probability at +least 1 − δ, provided P ≥ 3 and +L ≥ Cula(S, σ, pmax) +ϵ2 +�pmax +pmin +�2 � +ln +�8P +δ +��2 +. +(35) +Proof. From Lemma 5, if ∆min(θ) +≥ +γ/P, we have +σ2 +S(AUS(θ)) ≥ +P +C′ . Under the assumption on the SNR, we +have pminσ2 +S(AUS(θ)) ≥ pmin P +C′ > 2σ2 which ensures β > +pminσ2 +S(AUS(θ))/2 > 0. Notice that for ULA Mca + 1 = P +and from the fact that σ2 +S(AUS(θ)) ≥ +P +C′ , we can obtain the +following bound: +q = +C′ +S +√ +P +βσS(AUS(θ)) ≤ +2C′ +S +√ +P +pminσ3 +S(AUS(θ)) ≤ +C′′ +S +pminP +(36) +where C′′ +S = 2C′ +SC′1.5. Notice that: +CSβq = CSC′ +S +√Mca + 1 +σS(AUS(θ)) +≥ C1(S) +√ +P +√ +P += C1(S) +(37) +where the inequality follows from σS(AUS)≤∥AUS∥F / +√ +S = +√ +P. Using the fact that β ≤ pminσ2 +S(AUS(θ)), and the above +lower bound on σS(AUS(θ)), we obtain +q ≥ +C′ +S +√ +P +pminσ3 +S(AUS(θ)) ≥ +C′ +S +pminP . +(38) +Therefore, +qpminP +� +∆(Sula)/c2 +≥ +C′ +S +� +∆(Sula)/c2 +≥ +C′ +S +� +ln(P)/c2, where the last inequality follows from the +lower bound on ∆(Sula) in Lemma 3. Recall that c2 < 1 +2, and therefore for P ≥ 3, +√ +ln P/c2 > 1. This implies that +min(C1(S), C′ +S +� +ln(P)/c2) = C1(S). Combining this with +(37), we have ϵ ≤ C1(S) = min(C1(S), C′ +S +� +ln(P)/c2) ≤ +min(CSβq, qpminP√∆Sula/c2), which ensures that the as- +2The constant c2 = 3/16 +√ +2 is specified in the proof of Theorem 1 in +Appendix A. + +8 +sumption on ϵ in Corollary 1 holds. From Lemma 4, we have +∥Ry∥2 ≤ pmaxPS + σ2. Using this bound and (36), we get: +q1 +� +∆(Sula) ≤ +C′′ +S +pminP (PSpmax + σ2) +� +2 ln(P) += C′′ +S(S + +σ2 +pmaxP ) +�pmax +pmin +� � +2 ln(P) +≤ �C1(S, σ, pmax) +�pmax +pmin +� � +ln(8P/δ) +(39) +where �C1(S, σ, pmax) := (S + +σ2 +pmax ) +√ +2C′′ +S. The upper bound +follows from the observations that (S + +σ2 +pmaxP ) ≤ (S + +σ2 +pmax ) +for all P ≥ 1 and ln(P) ≤ ln(8P/δ) for any δ < 1. Notice +from (33), that Cula(S, σ, pmax) = c3/c2 �C2 +1(S, σ, pmax). From +(39), we have +c3 ln(8P +δ )q2 +1∆(Sula) +c2ϵ2 +≤ +c3 +c2ϵ2 �C2 +1(S, σ, pmax)(pmax +pmin +ln(8P/δ))2 += Cula(S, σ, pmax) +ϵ2 +�pmax +pmin +�2 +(ln(8P/δ))2 . +Therefore, (35) implies (27) and the proof is completed by +applying Corollary 1 since β > 0 and the conditions on ϵ and +L required for applying the corollary are satisfied. +Theorem 5. Let S = S(N1,N2) +nest +be a nested array with +N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋. Suppose the minimum +angular separation between the sources, and the SNR satisfy +the following conditions for some γ > 1: +∆min(θ) ≥ 5γ +P 2 , +pmin +σ2 +> 2C′ +n +P 2 , where C′ +n = 5γ/(γ − 1). +Under the assumptions [A1-A3], for any δ > 0 and 0 < +ϵ ≤ C2(S) := +� +1/5CSC′ +S, md(θ, �θ) ≤ ϵ is satisfied with +probability at least 1 − δ provided P ≥ 3 and +L ≥ Cnest(S, σ, pmax) +ϵ2 +�pmax +pmin +�2 +ln +�8P 2 +δ +� +. +(40) +Proof. From Proposition 2, if ∆min(θ) ≥ 5γ/P 2, we have +σ2 +S(AUS(θ)) ≥ P 2 +C′n . Following the same argument as Theo- +rem 4, this ensures that β > 0. Using the fact that Mca + 1 ≤ +P 2 (from (3)) and the lower bound on σ2 +S(AUS(θ)), we obtain +q ≤ +C′ +SP +βσS(AUS(θ)) ≤ +2C′ +SP +pminσ3 +S(AUS(θ)) ≤ +¯C′′ +S +pminP 2 +(41) +where +¯C′′ +S +:= +2C′ +SC′1.5 +n +. Notice that σS(AUS(θ)) +≤ +∥AUS∥F / +√ +S = √Mca + 1 ≤ P. Hence, similar to (37), +we can establish that CSβq +≥ +C2(S). Using the fact +P 2/5 ≤ Mca + 1 from (3), similar to (38) we obtain q ≥ +C′ +SP +√ +5pminσ3 +S(AUS(θ)) ≥ +C′ +S +√ +5pminP 2 . From Lemma 3, ∆(Snest) ≥ +P 2/16. It follows that qpminP +� +∆(Snest)/c2 ≥ +C′ +S +4c2 +√ +5. Since +4c2 < 1, it follows that min(C2(S), C′ +S/(4c2 +√ +5)) = C2(S) +and therefore ϵ ≤ C2(S) = min(C2(S), C′ +S/(4c2 +√ +5)) en- +sures that the assumption on ϵ in Corollary 1 holds. Using +∆(Snest) ≤ P 2 (from Lemma 3), Lemma 4, and (41), we get: +q1 +� +∆(Snest) ≤ �C1(S, σ, pmax)(pmax/pmin), +(42) +where �C1(S, σ, pmax)=(S + +σ2 +pmax ) ¯C′′ +S. By (42), we have +ln(8Mca +δ +)c3q2 +1∆(Snst) +c2ϵ2 +≤ +c3 +c2ϵ2 �C2 +1(S, σ, pmax) ln(8P 2/δ)(pmax +pmin +)2 += Cnest(S, σ, pmax) +ϵ2 +�pmax +pmin +�2 � +ln(8P 2/δ) +� +. +Therefore (40) implies (27) and the proof is again completed +by applying Corollary 1 since β > 0 and the conditions on ϵ +and L required for applying the corollary are satisfied. +Note that the range of values for ϵ where Theorem 4 and +5 are applicable differ slightly. However in the regime ϵ ≤ +min(C1(S), C2(S)) = C2(S) and P ≥ 3, we can fairly +compare the two array geometries. +Towards higher resolution with same snapshots: Theorem 4 +states that for a ULA, the matching distance error for Coarray +ESPRIT can be bounded by ϵ provided (i) the snapshots +scales only (poly)logarithmically in the dimension of the +coarray covariance matrix and (ii) the minimum separation +is ∆min ≥ γ/P. On the other hand, Theorem 5 guarantees +that for a nested array with P sensors, it is possible to bound +the matching distance error by the same ϵ with order wise the +same number of snapshots (L = Ω(ln(P 2)), but with a relaxed +separation condition that allows ∆min to be ∆min = Ω(1/P 2). +This validates the superior resolution properties of nested +arrays compared to ULA with the same budget of temporal +snapshots. This has been empirically observed in the literature, +but never theoretically established, until now. +Noise Resilience of Nested Arrays: If we consider the +separation regime ∆min = Ω(1/P) that is applicable for both +the ULA and nested array, Theorems 4 and 5 indicate that +the SNR (pmin/σ2) requirement for the nested array can be +P times smaller than that of the ULA, in order to achieve the +same DOA error bound with order-wise the same number of +snapshots (L = Ω(ln P)). This brings out another advantage of +nested arrays in terms of robustness against noise, especially +in the low-SNR regime [8]. +Effect of Dynamic Range: Our analysis also reveals the chal- +lenge posed by sources with higher dynamic range pmax/pmin +as also observed in [22]. Theorem 5 suggests that at the +same SNR (defined with respect to the weakest source pmin), +more snapshots maybe needed for resolving sources with +disproportionately varying powers (higher pmax compared to +the fixed pmin). As will be shown, the numerical results are +indeed consistent with the prediction made by our analysis. +B. The Myth of Large Snapshots: Correlation Error vs. Angle +Estimation Error +Since nested (and other) sparse arrays realize the virtual +difference coarray by correlation-processing, it is commonly +believed that one needs a large number (L = Ω(P 2)) of +temporal snapshots to estimate Θ(P 2) (cross) correlation +values between sensor pairs. This ‘myth’ of large snapshots +(that grows quadratically in the number of sensors P) is +partially true, if our goal is to estimate the coarray covariance +matrix Tca. If we only allow L to scale as L = Θ(log P) + +9 +(the so-called sample-starved regime), then one may indeed +incur large error in covariance estimation. However, Theorem +5 shows that the angle estimation error can be made arbitrarily +small (ϵ) with high probability (1 − δ) provided L scales +only as Ω( 1 +ϵ2 ln(8P 2/δ)), despite the possibility of the coarray +covariance error of a nested array increasing with P in this +snapshot-starved regime. This surprising phenomenon is due +to the fact that the potentially large covariance estimation error +(which can even grow with P in this regime) can actually be +mitigated/counterbalanced by the enhanced aperture/difference +set of the nested array that results in a large restricted smallest +singular value σS(AUS). As long as ∆min(θ) ≥ 5γ +P 2 , σ2 +S(AUS) +scales as cP 2 (for some constant c), and this helps us obtain +reliable angle estimation, although the covariance estimates +may be unreliable. +V. SIMULATIONS +We numerically investigate the useful SNR regime for coarray +processing (Section V-A), the impact of SNR and the number +of snapshots on DOA estimation error (V-B and V-C), the +relationship between DOA and covariance estimation error +(V-D), and the effect of the dynamic range of source powers +on resolving two closely spaced sources (V-E). +A. When is Coarray-Based DOA Estimation Beneficial? +We begin by examining under which circumstances coarray- +based algorithms offer an advantage over more conventional +DOA estimation methods. Specifically, in case of the ULA, +we could apply MUSIC or ESPRIT directly to the sample +covariance matrix �Ry in (7) instead of the averaged coarray +covariance matrix �Tca in (9). Fig. 1 shows the matching +distance error of coarray ESPRIT and direct ESPRIT, averaged +over 103 Monte Carlo trials, in case of the ULA, and, for +comparison, coarray ESPRIT in case of the nested array with +the same number of sensors (P = 20). We consider L = 100 +snapshots, and S = 4 equipower sources equally spaced by +∆ = 2/P. At medium to low SNR, the advantage of coarray- +based processing is apparent. At high SNR, the situation is +reversed, as the error of direct ESPRIT continues decreasing +as a function of SNR, whereas the error of coarray ESPRIT +saturates3. However, coarray-based processing—including re- +dundancy averaging (8)—can clearly offer significant benefits +in SNR or snapshot-limited conditions. As mostly such chal- +lenging scenarios are of interest in many applications, we focus +on coarray ESPRIT herein. +B. Improving Resolution by Increasing SNR or Snapshots +Next, we compare the probability of resolution as a function of +the minimum separation for the nested array and ULA with +the same number of sensors, P = 20. Coarray ESPRIT is +employed for both array geometries. We consider two sources +with equal power (p1 = p2) and (normalized) angles ω = +{0.1, 0.1 + ∆}. The sources are declared to be successfully +resolved when the estimated DOAs satisfy maxi |ˆωi − ωi| ≤ +∆/10. Fig. 2 shows the empirical probability of resolution +(averaged over 1000 Monte-Carlo trials) for varying separation +3This well-known and fundamental phenomenon is due to the finite- +snapshot error of the coarray covariance matrix, see [5]–[7]. +-20 +-10 +0 +10 +20 +30 +40 +50 +SNR (dB) +10-6 +10-4 +10-2 +100 +Average Matching Distance +ULA (coarray ESPRIT) +Nested (coarray ESPRIT) +ULA (direct ESPRIT) +Fig. 1: Comparison of ESPRIT applied to the sample covariance +matrix (7) (direct ESPRIT) and the estimated coarray covariance +matrix (9) (coarray ESPRIT). Coarray ESPRIT achieves lower angle +estimation error than direct ESPRIT at medium to low SNR. +10-3 +10-2 +10-1 +Angular Separation +0 +0.2 +0.4 +0.6 +0.8 +1 +Probability of resolution +# Sensors = 20, Snapshots = 55 += (1/P) += (1/P2) +ULA (SNR = 0 dB) +Nested (SNR = 0 dB) +ULA (SNR = -16 dB) +Nested (SNR = -16 dB) +10-3 +10-2 +10-1 +Angular Separation +0 +0.2 +0.4 +0.6 +0.8 +1 +Probability of resolution +# Sensors = 20, SNR = 0 dB += (1/P) += (1/P2) +ULA (L = 55) +Nested (L = 55) +ULA (L = 600) +Nested (L = 600) +Fig. 2: Probability of resolution vs. source separation for different +SNR levels (top) and number of snapshots (bottom). Increasing either +improves resolution for both arrays. +∆ and a fixed number of snapshots L = 55 and SNR = 0 and +−16 dB. We observe that both array geometries can operate +at a smaller separation at a higher SNR, i.e., smaller σ/pmin +ratio. Indeed, the transition from low to high probability of +resolution occur around ∆ ∝ 1/P for the ULA and ∆ ∝ 1/P 2 +for the nested array, as predicted by Theorems 4 and 5. It is +also possible to enhance resolution by increasing the number +of snapshots, as Fig. 2 demonstrates. Here, the SNR is fixed +at 0 dB and the number of snapshots is L = 55 and L = 600, +respectively. +C. Snapshot and SNR Trade-off +Section V-B showed that SNR and the number of temporal +snapshots can be exchanged for improved resolution. We now +study this trade-off in further detail. We consider S = 2 +equipowered sources located at ω = {0.1, 0.1 + ∆}, where +∆ ∈ {2/P, 2/P 2} and P = 20. Fig. 3 shows the separation- +relative matching distance error md(θ, �θ)/∆ (averaged over +103 Monte Carlo trials) as a function of both the number +of snapshots and SNR. Firstly, fewer snapshots are required +at higher SNR (and vice versa) to obtain the same recovery +error, both in case of the ULA (left column) and nested array +(right column). This supports Theorem 3, where the match- + +10 +ing distance depends on the number of snapshots and SNR +through (22) and (26), respectively. Secondly, the nested array +displays a more advantageous trade-off between snapshots and +SNR compared to the ULA for both source separation 2/P +(top row) and 2/P 2 (bottom row). The benefit is especially +apparent for ∆ = 2/P 2, where the nested array has a greatly +larger range of operating points where the relative matching +distance is low, as predicted by Theorem 5. Note that the gray +pixels correspond to a relative error of approximately 10% of +the separation, whereas white corresponds ≤ 1% error. +Fig. 3: Relative matching distance error md(θ, �θ)/∆ as a function of +snapshots and SNR. The nested array (right column) achieves lower +error than the ULA (left column) for both source separation ∆ = 2/P +(top row) and ∆ = 2/P 2 (bottom row). +D. DOA and Covariance Estimation Error +Next, we illustrate an intriguing benefit of coarray-based DOA +estimation in case of the nested array. We consider the average +DOA matching distance and average covariance estimation +error defined as ∥Tca − �Tca∥2 for a varying number of +sensors P and S = 4 equipower sources equally spaced by +∆ ∈ {1/P 1.5, 1/P 2}. The number of snapshots is L = 50 +and SNR = 0 dB. Fig. 4 shows that the nested array incurs a +larger covariance estimation error compared to the ULA with +the same number of sensors. However, despite obtaining a +worse estimate of the covariance matrix �Tca, the nested array +achieves superior DOA estimation performance when coarray +ESPRIT is applied to �Tca. In fact, when the separation is +∆ = 1/P 2, the average matching distance no longer decays +with P for the ULA, whereas it continues to do so for the +nested array. This is enabled by the larger coarray aperture +of the nested array, which offsets the effect of finite snapshot +covariance estimation error as discussed in Section IV-B. Note +that for a fixed number of snapshots and a growing number of +sensors P, the entries of the coarray covariance matrix Tca +become increasingly challenging to estimate, since the size of +Tca is proportional to the number of coarray elements Mca, +which is ∝ P for the ULA and ∝ P 2 for the nested array. +E. Effect of Dynamic Range of Source Powers +In the final experiment, we investigate the ability of coarray +ESPRIT to resolve two sources with unequal powers. We set +10 +15 +20 +25 +30 +35 +40 +Number of sensors (P) +10-4 +10-3 +10-2 +10-1 +Average Matching +Distance Error +# Snapshot = 50, SNR = 0 dB +Nested ( +=1/P2) +ULA ( +=1/P2) +Nested ( +=1/P1.5) +ULA ( +=1/P1.5) +10 +15 +20 +25 +30 +35 +40 +Number of sensors (P) +101 +102 +Average Covariance +Estimation Error +# Snapshot = 50, SNR = 0 dB +Fig. 4: Average matching distance (top) and covariance estimation +error (bottom) as a function of the number of sensors P. The +DOA estimation error of the nested array decays despite the larger +covariance estimation error compared to the ULA. +the dynamic range to pmax/pmin ∈ {1, 10} by fixing the power +of the weaker source to pmin = 0.2 and varying pmax. Fig. 5 +shows that the number of snapshots required to distinguish +two sources (separated by ∆ = 1/P) is significantly larger +when pmax/pmin = 10 compared to pmax/pmin = 1. This +is consistent with Theorems 4 and 5, which imply that the +sufficient number of snapshots for resolving two sources (with +high probability) grows with pmax if pmin and σ are held +fixed, irrespective of the array geometry. This brings out a +non-trivial dependence of the dynamic range pmax/pmin on +the sample complexity. Hence, distinguishing two sources with +greatly different powers is more challenging and requires more +snapshots than when the powers are equal. +101 +102 +103 +104 +Snapshots +0 +0.2 +0.4 +0.6 +0.8 +1 +Probability of resolution +ULA (pmax/pmin = 10) +Nested ULA (pmax/pmin = 10) +ULA (pmax/pmin = 1) +Nested ULA (pmax/pmin = 1) +Fig. 5: Effect of dynamic range of source powers on probability of +resolution. Coarray ESPRIT requires more snapshots to detect two +sources with larger dynamic range pmax/pmin. +VI. CONCLUSION +This paper investigated angle estimation error of coarray +ESPRIT. We considered both additive noise and finite-snapshot +covariance estimation error, which we probabilistically char- +acterized in the case of Toeplitz covariance matrices. Our +results show that if the number temporal snapshots scales +logarithmically with the number of sensors, coarray ESPRIT +achieves arbitrarily low estimation error with high probability. +This also shows that the DOA estimation error can be small +even though the covariance estimation error may be large. + +ULA,Separation=2/P +10 +100 +Matchingdistance/separation +5 +0 +-5 +10~1 +-15 +-20 +10~2 +101 +102 +103 +104 +SnapshotsNested,Separation=2/P +10 +100 +Matchingdistance/separation +5 +SNR (in dB) +0 +-5 +10~1 +-10 +-15 +-20 +10~2 +101 +102 +103 +104 +SnapshotsULA, Separation=2/p? +10 +100 +Matchingdistance/separation +5 +SNR (in dB) +0 +-5 +10~1 +-10 +-15 +-20 +10~2 +101 +102 +103 +104 +SnapshotsNested, Separation=2/p? +10 +100 +Matchingdistance/separation +5 +SNR (in dB) +0 +-5 +10~1 +-10 +-15 +-20 +102 +101 +102 +103 +104 +Snapshots11 +Finally, our theoretical and simulation results demonstrate that +sparse arrays can provide higher resolution and better noise +resilience compared to the ULA with the same number of +sensors and snapshots. +APPENDIX A +A. Intermediate Results +We will first state the complex extension of Hanson-Wright +inequality [33], which is obtained by applying [34, Theorem +1.1] with the strategy described on [34, Section 3.1, Page 9]. +Lemma 6. Let A ∈ Cn×n be a fixed Hermitian matrix. +Consider the random vector x = [x1, x2, · · · , xn]⊤ ∈ Cn with +independent real and imaginary components Re(xi), Im(xi) +satisfying E(Re(xi)) = E(Im(xi)) = 0, and ∥Re(xi)∥ψ2 ≤ K, +∥Im(xi)∥ψ2 ≤ K. Then for any ϵ > 0, we have +P(|xHAx − E(xHAx)|>ϵ) ≤ 2 exp +� +− c min( +ϵ2 +2K4∥A∥2 +F +, +ϵ +K2∥A∥2 +) +� +where c > 0 is a universal constant. +Proof. Let z = [Re(x)⊤, Im(x)⊤]⊤ ∈ R2n and define : ˜A = +�Re(A) +−Im(A) +Im(A) +Re(A) +� +. It is easy to see that for any Hermitian +A, we have the following equality xHAx = zT ˜Az. Further, +it can be verified that ∥˜A∥F = +√ +2∥A∥F and ∥˜A∥2 = ∥A∥2. +Now, we can apply [34, Theorem 1.1], to obtain the desired +probability bound. +Lemma 7. Let wi ∈ Cn, 1 ≤ i ≤ T be i.i.d com- +plex circularly symmetric Gaussian random variable with +distribution CN(0, Σ). Let A ∈ Cn×n be a fixed Her- +mitian matrix, then for any ϵ > 0 and universal constant +c, we have P(| 1 +L +�L +i=1 wH +i Awi − E[wH +i Awi]| ≥ ϵ) ≤ +2 exp +� +−cL min +� +ϵ2 +2K4∥Σ∥2 +2∥A∥2 +F , +ϵ +K2∥Σ∥2∥A∥2 +�� +. +Proof. Since wi is a complex circularly symmetric Gaussian +random variable distributed according to CN(0, Σ), we define +a new transformed variable ui = Σ−1/2wi where Σ1/2 is the +square root of the covariance matrix Σ. It can be verified that +ui ∼ CN(0, In), i.e., it is also a complex circularly symmetric +Gaussian +random +variable +with +independent +real +and +imaginary components. Define block-wise diagonal matrices +˜A += +diag(A, . . . , A), ˜Σ1/2 += +diag(Σ1/2, . . . , Σ1/2) +∈ +CnL×nL and ˜u = [uT +1 , . . . , uT +L]T +∈ CnL. Next, we can +observe that �L +i=1 wH +i Awi = �L +i=1 uH +i Σ1/2AΣ1/2ui = +˜uH ˜Σ1/2 ˜A˜Σ1/2˜u. +We +have +E(�L +i=1 wH +i Awi) += +LE +� +wH +i Aw +� +, +since +it +is +a +sum +of +L +i.i.d +random +variables. The desired probability can be re-written as: +P(| 1 +L +�L +i=1 Re(wH +i Awi) − E[Re(wH +i Awi)]| +≥ +ϵ) += +P(|Re(˜uH ˜Σ1/2 ˜A ˜Σ1/2˜u) − E[Re(˜uH ˜Σ1/2 ˜A ˜Σ1/2˜u)]| ≥ Lϵ). +Recall +that +Re(˜ui), Im(˜ui) +are +i.i.d +distributed +as +N(0, 1/2) and hence sub-Gaussian with K += +2/ +√ +3. +Note that due to the block-diagonal structure we have +∥ ˜Σ1/2 ˜A ˜Σ1/2∥2 +F = L∥Σ1/2AΣ1/2∥2 +F ≤ L∥A∥2 +F ∥Σ∥2 +2 +and +∥ ˜Σ1/2 ˜A ˜Σ1/2∥2 = ∥Σ1/2AΣ1/2∥2 ≤ ∥A∥2∥Σ∥2. The proof +is completed by applying Lemma 6 with ϵ = ϵL. +B. Proof of Theorem 1 +From Lemma 2, we have P(∥EL∥2 ≥ ϵ)≤P(sup |fe(θ)|≥ϵ). +In general, it is not straightforward to evaluate this supremum, +however, we exploit the following result from [28]that bounds +it by using the function value evaluated at a few grid points. +Lemma 8. +[28, Theorem 7.28, Chapter 10, Vol.2, Pg. 33] +Let f(θ) be a trigonometric polynomial of order N. Then, +supθ∈[−π,π] |f(θ)| ≤ 2 max1≤k≤4N |f(θk)|, +θk = k−2N +4N π. +From Proposition 1, we have fe(θ) = tr(EyΛ(θ)). However, +we want to relate it to the sample covariance matrix �Ry. In +order to do this, we show that tr(RavΛ(θ)) = tr( �RyΛ(θ)) +where recall from (7) that �Ry is the sample covariance matrix: +tr(RavΛ(θ)) = +P +� +m=1 +P +� +n=1 +[Rav]m,n[Λ(θ)]n,m += +Mca +� +s=−Mca +� +m,n: +dm−dn=s +ˆts +exp(−jsθ) +|Ωs| += +Mca +� +s=−Mca +ˆts|Ωs| exp(−jsθ) +|Ωs| += +(a) +Mca +� +s=−Mca +� +m,n: +dm−dn=s +[ �Ry]m,n[Λ(θ)]n,m = Tr( �RyΛ(θ)), +where (a) follows from the redundancy averaged estimator +where for all m, n such that dm − dn = s, we have |Ωs|ˆts = +� +dm−dn=s[ �Ry]m,n. Therefore, we have the following rela- +tion: fe(θ)=tr (EyΛ(θ))=tr ((Ry − Rav)Λ(θ)) = tr((Ry − +�Ry)Λ(θ)) = 1 +L +�L +t=1 +� +E[y(t)HΛ(θ)y(t)] − y(t)HΛ(θ)y(t) +� +. +Since the snapshots are i.i.d, we can define i.i.d ran- +dom variables {Zt(θ)}L +t=1 as Zt(θ) ≜ y(t)HΛ(θ)y(t) − +E(y(t)HΛ(θ)y(t)) with y(t) ∼ CN(0, Ry). Note that Λ(θ) +is Hermitian. Hence, we can apply Lemma 7 with Σ = Ry +and A = Λ(θ) to obtain ∀ϵ > 0, +P +� +1 +L | +L +� +t=1 +Zt(θ)| ≥ ϵ +� +≤ +(43) +2 exp +� +−cL min +� +ϵ2 +2K4∥Ry∥2 +2∥Λ(θ)∥2 +F +, +ϵ +K2∥Ry∥2∥Λ(θ)∥2 +�� +. +We want to obtain a universal upper bound that is similar +to (43) but not dependent on θ. Notice, ∥Λ(θ)∥2 +F = +1 +|Ω0| + +�Mca +s=1 +2 +|Ωs| ≤ 2∆(S). Similarly, we can also bound ∥Λ(θ)∥2 ≤ +∥Λ(θ)∥F ≤ +� +2∆(S). This gives us the following bound: +P +� +1 +L | +L +� +t=1 +Zt(θ)| ≥ ϵ +� +≤ +(44) +2 exp +� +−cL min +� +ϵ2 +4K4∥Ry∥2 +2∆(S) , +ϵ +K2∥Ry∥2 +� +2∆(S) +�� +. +Note fe is a trigonometric polynomial of order Mca. Now, +we will use Lemma 8 to bound the spectral function |fe(θ)|. +P(supθ∈[−π,π] |fe(θ)| ≥ ϵ) ≤ P(2 max1≤k≤4Mca |fe(θk)| ≥ +ϵ) +≤ +�4Mca +k=1 P +� +|fe(θk)| ≥ ϵ +2 +� +≤ +8Mca exp +� +− +c1L min +� +c2ϵ2 +∥Ry∥2 +2∆(S), +ϵ +∥Ry∥2√ +∆(S) +�� +, where c1 = c/(2 +√ +2K2) +(c was given in Lemma 7) and c2 += +1/(4 +√ +2K2) += +3/(16 +√ +2) < 1. The first inequality follows due to Lemma +8, the second inequality follows from union bound. The last +inequality is a consequence of the bound computed in (44). + +12 +APPENDIX B +A. Proof of Theorem 2 +The proof uses several results from [20]. However, unlike [20] +the underlying subspace of interest is the coarray subspace +and the perturbation is due to covariance estimation error and +noise. We provide key intermediate steps to make the results +self-contained. +Recall that columns of U and �U are orthonormal bases +for the subspaces R(U) and R( �U). Let the principal an- +gles between the subspaces R(U) and R( �U) be denoted as +Θ(R(U), R( �U)) := [ψ1, ψ2, · · · , ψS]T where 0 ≤ ψ1 ≤ +ψ2 ≤ · · · ≤ ψS ≤ π/2. Then from [35], we have cos(ψi) = +σi(UH �U) i = 1, 2, · · · , S. Recall from Lemma 1, the output +of ESPRIT is invariant to the choice of the basis. For ease +of analysis, we will choose a pair of basis for R(U) and +R( �U), which are also known as “canonical bases” [20]. Let +the SVD of the matrix UH �U be of the form UH �U := +LΣcRH, L, R ∈ CS×S, where Σc = diag(σc +1, σc +2, · · · , σc +S) +where σc +i = σi(UH �U) are arranged in descending order. The +canonical basis U(c) and �U(c) are given by: +U(c) := UL, +�U(c) := �UR +(45) +Using the canonical basis, we define the following matrices: +Ψ(c) :=U(c)† +0 +U(c) +1 , �Ψ(c) := �U(c)† +0 +�U(c) +1 . Since R(U)=R(U(c)) +and R( �U) = R( �U(c)), we have Θ(R(U(c)), R( �U(c))) = +Θ(R(U), R( �U)). Notice that the canonical basis has the +following property: cos(ψi) = σi(U(c)H �U(c)) = u(c)H +i +�u(c) +i . +We will use [20, Lemma 2] that relates the matching distance +error to the quantity ∥ �Ψ(c) − Ψ(c)∥2 and holds universally: +md(θ, ˆθ) ≤ π S3/2√Mca + 1 +σS(AUS(θ)) ∥ �Ψ(c) − Ψ(c)∥2. +(46) +B. Relating ∥ �Ψ(c) − Ψ(c)∥2 to ∥EL∥2 +Let B = A + N ∈ CM×N, where rank(A) ≥ L. Suppose ψL +is the largest principal angle between the subspace spanned +by L principal singular vectors (corresponding to L largest +singular values) of A and B, respectively. If σL+1(A) ≤ α +and σL(B) ≥ α + δ for some α ≥ 0 and δ > 0 then, Wedin’s +Theorem [36] states that: +sin(ψL) ≤ ∥N∥2/δ +(47) +Lemma +9. Suppose σS(Tca) +≥ +2∥EL∥2 +and β += +pminσ2 +S(AUS(θ)) − σ2 > 0. Then +sin(ψS) ≤ 2∥EL∥2/β +(48) +Proof. Recall that ˆTca = Tca − EL. To apply Wedin’s +theorem, we need to characterize quantities α and δ such +that: σS( ˆTca) ≥ δ + α, +σS+1(Tca) ≤ α. From (13), we +have σS+1(Tca) = σ2. We choose α = σ2. Using Weyl’s +inequality, σS(�Tca) ≥ σS(Tca) − ∥EL∥2 +(a) +≥ σS(Tca)/2 +(b) += +(σS(AUS(θ)PAUS(θ)H) + σ2)/2, where (a) follows from +the assumption 2∥EL∥2 ≤ σS(Tca) and (b) follows from +(13). Combining with the preceding inequality, we obtain +σS( ˆTca) − σ2 ≥ (σS(AUS(θ)PAUS(θ)H) − σ2)/2 ≥ β/2 > +0, where the last term is positive due to the given condition. +Then we can choose δ = σS(�Tca) − σ2 which satisfies +σS(�Tca) = α + δ with δ > 0. The proof is completed by +using (47). +Lemma 10. If pminσ2 +S(AUS(θ)) > σ2 and +∥EL∥2 ≤ σS(U(c) +0 )(σS(AUS(θ)PAUS(θ)H) − σ2) +4 +√ +2 +(49) +then ∥Ψ(c) − ˆΨ(c)∥2 ≤ +14 +√ +2∥EL∥2 +σ2 +S(U(c) +0 )(pminσ2 +S(AUS(θ))−σ2). +Proof. From the definition of U(c), �U(c) we have: +∥U(c) − �U(c)∥2 +2 = ∥(U(c) − �U(c))H(U(c) − �U(c))∥2 += 2(1 − cos(ψS)) ≤ 2(1 − cos2(ψS)) = 2 sin2(ψS). +(50) +By +the +assumption +of +this +lemma, +2∥EL∥2 +≤ +σS(U(c) +0 )(σS(AUS(θ)PAUS(θ)H) − σ2) and σS(U(c) +0 ) ≤ 1, +we have 2∥EL∥2 ≤ σS(Tca)σS(U(c) +0 ) ≤ σS(Tca). This +together with the assumption pminσ2 +S(AUS(θ)) > σ2 enables +us to apply Lemma 9. Combining (48) with (50) we obtain +the following bound: +∥ �U(c) − U(c)∥2 ≤ +2 +√ +2∥EL∥2 +σS(AUS(θ)PAUS(θ)H) − σ2 . +(51) +Notice that +∥ �Ψ +(c) − Ψ(c)∥2 = ∥( �U(c)† +0 +− U(c)† +0 +) �U(c) +1 ++ U +(c)† +0 +( �U(c) +1 +− U(c) +1 )∥2 +≤ ∥ �U(c)† +0 +− U(c)† +0 +∥2∥ �U(c) +1 ∥2 + ∥U(c)† +0 +∥2∥ �U(c) +1 +− U(c) +1 ∥2 +≤ ∥ �U(c)† +0 +− U(c)† +0 +∥2 + ∥U(c)† +0 +∥2∥ �U(c) − U(c)∥2 +(52) +where the last inequality follows from the fact that +�U(c) +1 , �U(c) +1 +− U(c) +1 +are submatrices of �U(c) and �U(c) − U(c), +respectively. Therefore, we have ∥ �U(c) +1 ∥2 ≤ ∥ �U(c)∥2 = 1, +and ∥ �U(c) +1 +− U(c) +1 ∥2 ≤ ∥ �U(c) − U(c)∥2. We use a result from +[37, Theorem 3.2] which states that a matrix F with rank S, +and its perturbed matrix �F = F + �E satisfy the following +inequality: ∥F† − �F†∥2 ≤ 3∥�E∥2/(σS(F)(σS(F) − ∥�E∥2)) +provided ∥�E∥2 < σS(F). From (51), and using the assumption +of the lemma we have: +∥ �U(c) +0 +− U(c) +0 ∥2 ≤ ∥ �U(c) − U(c)∥2 ≤ +2 +√ +2∥EL∥2 +σS(AUS(θ)PAUS(θ)H) − σ2 +≤ σS(U(c) +0 )/2. +(53) +We +can +use +the +aforementioned +result +by +substi- +tuting +F +with +U(c) +0 , +and +�F +with +�U(c) +0 : +∥( �U(c)† +0 +− +U(c)† +0 +)∥2 ≤ +3∥( � +U(c) +0 −U(c) +0 )∥2 +σS(U(c) +0 )(σS(U(c) +0 )−∥ � +U(c) +0 −U(c) +0 ∥2) ≤ 6∥ � +U(c) +0 −U(c) +0 ∥2 +σ2 +S(U(c) +0 ) +≤ +6∥ � +U(c)−U(c)∥2 +σ2 +S(U(c) +0 ) +, where the second inequality follows from (53). +Combining this with (52), we get the final bound: ∥Ψ(c) − +ˆΨ(c)∥2 ≤ +6∥( � +U(c)−U(c))∥2 +σ2 +S(U(c) +0 ) ++ +1 +σS(U(c) +0 )∥( �U(c) − U(c))∥2 ≤ +7∥( � +U(c)−U(c))∥2 +σ2 +S(U(c) +0 ) +≤ +14 +√ +2∥EL∥2 +σ2 +S(U(c) +0 )(σS(AUS(θ))PAUS(θ))H)−σ2) +≤ +14 +√ +2∥EL∥2/σ2 +S(βU(c) +0 ). +Next, we state the following Lemma from [20] that can be +used to obtain a lower bound on σ2 +S(U(c) +0 ). +Lemma 11 (Lemma 3, [20]). Let U(a) be any orthonormal +basis for R(AUS(θ)). Then the following holds: σ2 +S(U(a) +0 ) ≥ +max(1 − +S +σ2 +S(AUS(θ)), 4−S) + +13 +Proof of Theorem 2. Define +CS += +2−S +4 +√ +2 +and +C′ +S += +14π +√ +2S3/24S. Under Assumption A3, these quantities are +constants since S is held fixed. If β > 0 and the assump- +tion ∥EL∥2 ≤ CSβ ensures that condition (49) holds since +σS(U(c) +0 ) ≥ 2−S from Lemma 11. Now, we can apply +Lemma 10 to bound ∥Ψ(c) − ˆΨ(c)∥2. We plug this bound +on ∥Ψ(c) − ˆΨ(c)∥2 in (46): md(θ, �θ) ≤ 14 +√ +2π S3/2q∥EL∥2 +σ2 +S(U(c) +0 )C′ +S ≤ +q∥EL∥2, where the last inequality follows from the bound +σ2 +S(U(c) +0 ) ≥ 4−S in Lemma 11. +C. Proof of Theorem 3 +We will utilize Theorem 1 and Theorem 2 to prove Theorem 3. +One can see from Theorem 2 that under the assumptions β > 0 +and ∥EL∥2 ≤ CSβ we can bound md(θ, �θ) ≤ q∥EL∥2. For a +given ϵ > 0, two cases arise: +Case I (ϵ ≤ CSβq): In this case, min(CSβ, ϵ +q) = ϵ/q. There- +fore, ∥EL∥2 ≤ ϵ +q ⇒ ∥EL∥2 ≤ CSβ, and from Theorem 2 the +matching distance error is less than md(θ, �θ) ≤ ϵ. This means +P(md(θ, �θ) ≤ ϵ) ≥ P +� +∥EL∥2 ≤ ϵ +q +� +. From Theorem 1, we +can obtain the following tail bound: +P(∥EL∥2 ≤ ϵ +q ) ≥ +1 − 8Mca exp +� +−c1L min +� +c2ϵ2 +q2∥Ry∥2 +2∆(S) , +ϵ +q∥Ry∥2 +√ +∆(S) +�� +. (54) +Case II (ϵ > CSβq): For values of ϵ satisfying ϵ > Csβq, +we have min(CSβ, ϵ/q) = CSβ. Therefore, if ∥EL∥2 ≤ CSβ, +then from Theorem 2 we have md(θ, �θ) ≤ CSβq. We obtain +the following bound on the tail probability due to Theorem 1, +P(∥EL∥2 ≤ CSβ) ≥ +1 − 8Mca exp +� +−c1L min +� +c2C2 +Sβ2 +∥Ry∥2 +2∆(S) , +CSβ +∥Ry∥2 +√ +∆(S) +�� +. +(55) +If the number of snapshots L satisfy the following bound: +L ≥ c3 ln +� 8Mca +δ +� +max +� q2 +1∆(S) +c2ϵ2 , +q1√ +∆(S) +ϵ +, L2 +0 +c2 , L0 +� +, where q1 = +q∥Ry∥2, c3 = 1/c1 and L0 = +∥Ry∥2√ +∆(S) +CSβ +then combining +(54) and (55) we obtain the following bound P +� +md(θ, �θ) ≤ +min(ϵ, CSβq)) ≥ 1 − δ. +REFERENCES +[1] P. Pal and P. P. Vaidyanathan, “Nested arrays: A novel approach to array +processing with enhanced degrees of freedom,” IEEE Trans. on Signal +Processing, vol. 58, no. 8, pp. 4167–4181, 2010. +[2] P. P. Vaidyanathan and P. Pal, “Sparse sensing with co-prime samplers +and arrays,” IEEE Trans. on Signal Processing, vol. 59, no. 2, pp. 573– +586, Feb 2011. +[3] S. Haghighatshoar and G. Caire, “Low-complexity massive mimo sub- +space estimation and tracking from low-dimensional projections,” IEEE +Trans. on Signal Processing, vol. 66, no. 7, pp. 1832–1844, 2018. +[4] Y. I. Abramovich, D. A. Gray, A. Y. Gorokhov, and N. K. Spencer, +“Positive-definite Toeplitz completion in DOA estimation for nonuni- +form linear antenna arrays. I. fully augmentable arrays,” IEEE Trans. +on Signal Processing, vol. 46, no. 9, pp. 2458–2471, Sep 1998. +[5] M. Wang and A. Nehorai, “Coarrays, MUSIC, and the Cram´er–Rao +bound,” IEEE Trans. on Signal Processing, vol. 65, no. 4, pp. 933–946, +Feb. 2017. +[6] A. Koochakzadeh and P. Pal, “Cram´er-Rao bounds for underdetermined +source localization,” IEEE Signal Processing Letters, vol. 23, no. 7, pp. +919–923, July 2016. +[7] C.-L. Liu and P. P. Vaidyanathan, “Cram´er-Rao bounds for coprime +and other sparse arrays, which find more sources than sensors,” Digital +Signal Processing, vol. 61, pp. 43 – 61, Feb 2017. +[8] S. Shahsavari, J. Millhiser, and P. Pal, “Fundamental trade-offs in noisy +super-resolution with synthetic apertures,” in ICASSP 2021-2021 IEEE +International Conference on Acoustics, Speech and Signal Processing +(ICASSP). +IEEE, 2021, pp. 4620–4624. +[9] R. Schmidt, “Multiple emitter location and signal parameter estimation,” +IEEE trans. on antennas and prop., vol. 34, no. 3, pp. 276–280, 1986. +[10] R. Roy and T. Kailath, “Esprit-estimation of signal parameters via +rotational invariance techniques,” IEEE Trans. on acoustics, speech, and +signal processing, vol. 37, no. 7, pp. 984–995, 1989. +[11] Y. Hua and T. K. Sarkar, “Matrix pencil method for estimating pa- +rameters of exponentially damped/undamped sinusoids in noise,” IEEE +Trans. on Acoustics, Speech, and Signal Processing, vol. 38, no. 5, pp. +814–824, 1990. +[12] J. Steinwandt, F. Roemer, and M. Haardt, “Performance analysis of +esprit-type algorithms for co-array structures,” in 2017 IEEE 7th Interna- +tional Workshop on Computational Advances in Multi-Sensor Adaptive +Processing (CAMSAP). +IEEE, 2017, pp. 1–5. +[13] F. Li, H. Liu, and R. Vaccaro, “Performance analysis for doa estimation +algorithms: unification, simplification, and observations,” IEEE Trans. +on Aero. and Electronic Systems, vol. 29, no. 4, pp. 1170–1184, 1993. +[14] S. Sun and Y. D. Zhang, “4D automotive radar sensing for autonomous +vehicles: A sparsity-oriented approach,” IEEE Journal of Selected Topics +in Signal Processing, vol. 15, no. 4, pp. 879–891, 2021. +[15] Z. Tan, Y. C. Eldar, and A. Nehorai, “Direction of arrival estimation +using co-prime arrays: A super resolution viewpoint,” IEEE Trans. on +Signal Processing, vol. 62, no. 21, pp. 5565–5576, 2014. +[16] E. J. Cand`es and C. Fernandez-Granda, “Super-resolution from noisy +data,” Journal of Fourier Analysis and Applications, vol. 19, no. 6, pp. +1229–1254, 2013. +[17] ——, “Towards a mathematical theory of super-resolution,” Comm. on +pure and applied Mathematics, vol. 67, no. 6, pp. 906–956, 2014. +[18] H. Qiao and P. Pal, “Guaranteed localization of more sources than +sensors with finite snapshots in multiple measurement vector models +using difference co-arrays,” IEEE Trans. on Signal Processing, vol. 67, +no. 22, pp. 5715–5729, 2019. +[19] C. Zhou, Y. Gu, X. Fan, Z. Shi, G. Mao, and Y. D. Zhang, “Direction- +of-arrival estimation for coprime array via virtual array interpolation,” +IEEE Trans. on Signal Processing, vol. 66, no. 22, pp. 5956–5971, 2018. +[20] W. Li, W. Liao, and A. Fannjiang, “Super-resolution limit of the esprit +algorithm,” 2019. +[21] W. Li and W. Liao, “Stable super-resolution limit and smallest singular +value of restricted fourier matrices,” Applied and Computational Har- +monic Analysis, vol. 51, pp. 118–156, 2021. +[22] W. Liao and A. Fannjiang, “Music for single-snapshot spectral es- +timation: Stability and super-resolution,” Applied and Computational +Harmonic Analysis, vol. 40, no. 1, pp. 33–67, 2016. +[23] W. Liao, “Music for multidimensional spectral estimation: stability and +super-resolution,” IEEE trans. on signal processing, vol. 63, no. 23, pp. +6395–6406, 2015. +[24] A. Moitra, “Super-resolution, extremal functions and the condition +number of vandermonde matrices,” in Proceedings of the 47th annual +ACM symposium on Theory of computing. +ACM, 2015, pp. 821–830. +[25] W. Li, Z. Zhu, W. Gao, and W. Liao, “Stability and super-resolution of +music and esprit for multi-snapshot spectral estimation,” IEEE Trans. +on Signal Processing, vol. 70, pp. 4555–4570, 2022. +[26] C.-L. Liu and P. P. Vaidyanathan, “Remarks on the spatial smoothing +step in coarray music,” IEEE Signal Processing Letters, vol. 22, no. 9, +pp. 1438–1442, 2015. + +14 +[27] Y. C. Eldar, J. Li, C. Musco, and C. Musco, “Sample efficient toeplitz +covariance estimation,” in Proceedings of the 14th Annual ACM-SIAM +Symposium on Discrete Algorithms. +SIAM, 2020, pp. 378–397. +[28] A. Zygmund, Trigonometric series. +Cambridge university press, 2002. +[29] R. M. Gray et al., “Toeplitz and circulant matrices: A review,” Founda- +tions and Trends® in Communications and Information Theory, vol. 2, +no. 3, pp. 155–239, 2006. +[30] R. O’Donnell, “Lecture 1: Asymptotics,” A Theorist’s Toolkit, 2016. +[31] D. Batenkov, L. Demanet, G. Goldman, and Y. Yomdin, “Conditioning of +partial nonuniform fourier matrices with clustered nodes,” SIAM Journal +on Matrix Analysis and Applications, vol. 41, no. 1, pp. 199–220, 2020. +[32] C. Aubel and H. B¨olcskei, “Vandermonde matrices with nodes in the +unit disk and the large sieve,” Applied and Computational Harmonic +Analysis, vol. 47, no. 1, pp. 53–86, 2019. +[33] D. L. Hanson and F. T. Wright, “A bound on tail probabilities for +quadratic forms in independent random variables,” The Annals of Math- +ematical Statistics, vol. 42, no. 3, pp. 1079–1083, 1971. +[34] M. Rudelson, R. Vershynin et al., “Hanson-wright inequality and +sub-gaussian concentration,” Electronic Communications in Probability, +vol. 18, 2013. +[35] G. W. Stewart, “Matrix perturbation theory,” 1990. +[36] P.- ˚A. Wedin, “Perturbation bounds in connection with singular value +decomposition,” BIT Numerical Mathematics, vol. 12, no. 1, pp. 99– +111, 1972. +[37] P. C. Hansen, “The truncatedsvd as a method for regularization,” BIT +Numerical Mathematics, vol. 27, no. 4, pp. 534–553, 1987. + diff --git a/FtAzT4oBgHgl3EQfxP5f/content/tmp_files/load_file.txt b/FtAzT4oBgHgl3EQfxP5f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2647c3c83d8a6bf4b4940a761b9231382318fb82 --- /dev/null +++ b/FtAzT4oBgHgl3EQfxP5f/content/tmp_files/load_file.txt @@ -0,0 +1,917 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf,len=916 +page_content='1 Super-resolution with Sparse Arrays: A Non- Asymptotic Analysis of Spatio-temporal Trade-offs Pulak Sarangi, Mehmet Can H¨uc¨umeno˘glu, Robin Rajam¨aki, and Piya Pal Abstract—Sparse arrays have emerged as a popular alternative to the conventional uniform linear array (ULA) due to the enhanced degrees of freedom (DOF) and superior resolution offered by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In the passive setting, these advantages are realized by leveraging correlation between the received signals at different sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This has led to the belief that sparse arrays require a large number of temporal measurements to reliably estimate parameters of interest from these correlations, and therefore they may not be preferred in the sample-starved regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In this paper, we debunk this myth by performing a rigorous non- asymptotic analysis of the Coarray ESPRIT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This seemingly counter-intuitive result is a consequence of the scaling of the singular value of the coarray manifold, which compensates for the potentially large covariance estimation error in the limited snapshot regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Specifically, we show that for a nested array operating in the regime of fewer sources than sensors (S = O(1)), it is possible to bound the matching distance error between the estimated and true directions of arrival (DOAs) by an arbitrarily small quantity (ϵ) with high probability, provided (i) the number of temporal snapshots (L) scales only logarithmically with the number of sensors (P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' L = Ω(ln(P)/ϵ2), and (ii) a suitable separation condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our results also formally prove the well-known empirical resolution benefits of sparse arrays, by establishing that the minimum separation between sources can be Ω(1/P 2), as opposed to separation Ω(1/P) required by a ULA with the same number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In addition to the array geometry, our sample complexity expression reveals the dependence on other key model parameters such as Signal to Noise Ratio (SNR) and the dynamic range of the source powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This enables us to establish the superior noise-resilience of nested arrays both theoretically and empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1 Index Terms—Sparse Arrays, Nested Sampling, Super-resolution, Toeplitz Covariance Matrix, Non-Asymptotic Guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' INTRODUCTION The problem of source localization arises in different contexts ranging from target detection in sonar and radar, hybrid mmWave channel estimation, and DOA estimation in array signal processing [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Traditionally, these applications consider ULAs, which are known to resolve up to S = O(P) sources with P sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, deterministic sparse array geometries, such as nested and coprime arrays [1], [2], have recently gained significant attention primarily due to two attractive properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Firstly, sparse arrays are able to identify up to S = O(P 2) uncorrelated sources using only P sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Secondly, sparse arrays enjoy a performance gain showcased by lower Cram´er-Rao bound and higher angular resolution [4]– [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Both of these properties can be attributed to the enhanced spatial DOF enabled by the so-called difference coarray, which can be as large as Θ(P 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1This work was supported by Grants ONR N00014-19-1-2256, ONR N00014-19-1-2227, NSF 2124929, and NSF CAREER ECCS 1700506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The enhanced DOF of the coarray are realized by comput- ing temporal correlations between the spatial measurements and constructing an augmented covariance matrix called the “coarray covariance matrix”, whose size is determined by the size of the difference coarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Following the construction of the coarray covariance matrix, it is possible to fully harness the power of the difference coarray and identify the unknown source directions using classical subspace techniques, such as MUSIC, ESPRIT or the matrix pencil method [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Despite the success of coarray-based algorithms, a common belief is that they require a large number of temporal snapshots to fully utilize the number of DOFs provided by the coarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The root of this belief mainly lies in the inadequacy of existing performance analyses, which are primarily based on characterizing the asymptotic Mean Squared Error (MSE) of the Coarray MUSIC [5] and Coarray ESPRIT algorithms [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In particular, such asymptotic results primarily rely on the first-order perturbation analysis framework proposed in [13], which leaves two key questions unanswered regarding the performance of coarray algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Firstly, the perturbation framework fails to theoretically explain the improvement in resolution offered by sparse arrays over the ULA—a phe- nomenon that has been extensively observed in numerical ex- periments [5], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Secondly, the analysis does not adequately reveal the dependence of temporal snapshots on key model parameters such as the array geometry, number of sensors, SNR and dynamic range of the source powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The aforementioned shortcomings are partially addressed in [15], which adapts recent advances in the theory of super- resolution [16], [17] to the coarray setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The analysis, which is based on Total-Variational norm minimization, is indeed non-asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, it is possible to show that the snapshot requirement in this setting scales quadratically (rather than linearly) with the number of sensors P, which is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In a parallel line of work using a grid-based model, we recently showed that Ω(P 2) snapshots are sufficient for ensuring exact support recovery with high probability even for closely-spaced sources, where the smallest source separation scales as Ω(1/P 2) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Although the analysis is applicable for scenarios where S > P (more sources than sensors), the sample complexity Ω(P 2) is still conservative when S ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In [19], an atomic norm formulation is adopted to exploit the Toeplitz structure of the coarray covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The analysis provides a characterization of the covari- ance matrix estimation error, but not of the sample complexity required to achieve a desired DOA estimation error, which is often the main quantity of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Indeed, common folklore arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='01734v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='SP] 4 Jan 2023 2 suggests that the benefits of sparse arrays necessarily come at the cost of a large number of snapshots, since the coarray covariance matrix, which typically needs to be estimated, is of size Θ(P 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, one might be tempted to falsely conclude that sparse arrays are at a disadvantage compared to ULAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In this paper, our goal is to dispel this belief by providing new non-asymptotic results on the performance of Coarray ESPRIT with a focus on nested arrays in the regime S ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our analysis is motivated by contemporary applications such as autonomous sensing and mmWave channel estimation [3], [14], where identifying more sources than sensors may not be necessary, and the number snapshots may be restricted either due to coherent multipaths or a rapidly varying environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' While subspace-based algorithms have been around for several decades and actively used in practice, performance guarantees characterizing their precise resolution limit were obtained only recently [20]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This analysis has also been extended to multi-snapshot setting in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The key factor enabling these guarantees is the characterization of the smallest singular value of Vandermonde matrices [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, all the aforemen- tioned results are only applicable to the ULA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Furthermore, no statistical assumptions are made on the source signals, and hence, the coarray perspective is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The key difference between deterministic and random sources is that in the latter case, the perturbation to the subspace of interest is a con- sequence of both noise as well as finite-snapshot covariance estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, extending the analysis in [20], [25] to the stochastic case requires non-trivial modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Contributions: Our first main contribution is to probabilis- tically characterize the coarray covariance matrix estimation error due to finite snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our second main contribution is a non-asymptotic performance analysis for the Coarray ESPRIT algorithm in terms of the matching distance error metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Specifically, we characterize the number of temporal snapshots (sample complexity) required to bound the matching distance error by a specified parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' To the best of our knowledge, our sample complexity expression (in terms of snapshots) is the first to explicitly bring out the dependence on key model parameters such as the array geometry, SNR and dynamic range of the source powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Furthermore, we establish that it is possible to bound the matching distance error with an arbi- trarily small quantity for both the nested array and ULA, using the (order-wise) same number of snapshots L = Ω(ln P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, a nested array can achieve this in a much smaller separation regime ∆min = Ω(1/P 2) compared to the ULA, for which ∆min = Ω(1/P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our analysis dispels the widely-held belief that sparse arrays require significantly more snapshots compared to ULAs when the number of sources is less than the number of sensors, and at the same time establishes the superior resolution capabilities of nested arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In addition to advancing the theoretical understanding, this analysis could also serve as a guiding principle for practitioners to determine suitable operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notations: Symbol ⊙ represents the Khatri-Rao (columnwise Kronecker) product, whereas ∥·∥2 and ∥·∥F denote the spectral and Frobenius norm of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Moreover, σi(A) is the i-th largest singular value of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For a set real numbers {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , pK}, pmin and pmax denote the minimum and maximum numbers in the set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The symbol T := [0, 1) denotes the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For a sub-Gaussian random variable X, ∥X∥ψ2 denotes its sub-Gaussian norm defined as ∥X∥ψ2 := inf{t > 0 | E[exp X2/t2] ≤ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' BACKGROUND ON SPARSE ARRAYS Consider a sparse linear array (SLA) with P sensors located at {dpλ/2}P p=1, where λ is the wavelength of the incoming far-field narrow-band source signals and dp belongs to an integer set S (|S| = P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose S sources with distinct DOAs θ = {θ1, θ2, · · · , θS} impinge on the array where θi ∈ (−π/2, π/2] for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The signal received at the P sensors at time instance t is given by: y(t) = AS(θ)x(t) + n(t), t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (1) The matrix AS(θ) = [aS(θ1), aS(θ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , aS(θS)] ∈ CP ×S is the array manifold matrix where: aS(θi) = [ejπd1 sin(θi), ejπd2 sin(θi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' ejπdP sin(θi)]⊤, represents the steering vector corresponding to the direction θi, L denotes the total number of temporal snapshots, x(t) ∈ CS is the tth temporal snapshot of the source signal vector and n(t) ∈ CP is an additive noise term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We define the normalized spatial frequencies (which we refer to as normalized DOAs) as ωi = sin(θi)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Throughout this paper, we make the following statistical assumptions on the source signals and noise: [A1] Uncorrelated Gaussian Sources: The source signals x(t) are assumed to be uncorrelated white circularly sym- metric Gaussian CN(0, P) where P = diag(p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , pS) represents a diagonal covariance matrix of source powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [A2] Gaussian Noise: The noise n(t) follows a zero-mean circularly symmetric complex Gaussian distribution n(t) ∼ CN(0, σ2I), and is uncorrelated with x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Under assumptions [A1-A2], the measurements follow y(t) ∼ CN(0, Ry), where Ry is given by: Ry = AS(θ)PAH S (θ) + σ2IP ∈ CP ×P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (2) By vectorizing Ry, we obtain the “virtual measurements”: ry = (AS(θ)∗ ⊙ AS(θ))p + σ2i, where i = vec(IP ) and p = [p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , pS]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The matrix AS(θ)∗ ⊙ AS(θ) can be viewed as a “virtual array” with sensor locations given by the difference set of the SLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1 (Difference Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Given a SLA S = {d1, d2, · · · , dP }, its difference set DS is defined as: DS = {dm − dn|dm, dn ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The difference set DS of S is also called its virtual difference coarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let Mca > 0 be the largest integer such that the set US := {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , Mca} satisfies US ⊆ DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This set US denotes the largest contiguous non-negative segment of the difference set and is essentially a ULA with Mca + 1 sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' By harnessing the structure of US, sparse arrays enjoy enhanced degrees of freedom over the physical SLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' An array is called hole-free if its difference set is a ULA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', DS = {−Mca, · · · , Mca}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We now introduce the notation for a “generalized nested array”, which is a special hole-free array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 3 Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2 (Nested array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A generalized nested array S(N1,N2) with N1 ≥ N2 > 0, is defined as: S(N1,N2) = {n}N1 n=1 ∪ {m(N1 + 1)}N2 m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It can be shown that any nested array S(N1,N2) is hole-free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', US = {0, 1, · · · , Mca} with Mca = N2(N1 + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Furthermore, Sula = S(P −1,1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', choosing N1 = P − 1 and N2 = 1, yields a ULA with P sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For a given P, if N1 = ⌈ P 2 ⌉, N2 = ⌊ P 2 ⌋, then Mca + 1 = ⌊ P 2 ⌋(⌈ P 2 ⌉ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It can be verified that for P ≥ 3, we have: P 2/5 ≤ Mca + 1 ≤ P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (3) Therefore, Mca = Θ(P 2) is indeed achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Next, we introduce an important quantity that is essential for describing correlation-based processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='3 (Weight Function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider a hole-free array S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For every i ∈ DS, its weight function is defined as |Ωi|: Ωi = {(m, n)|dm − dn = i, 1 ≤ m, n ≤ P} where the set Ωi essentially captures all pairs (dm, dn) of sensor locations that generate the difference of i = dm − dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Due to symmetry, it can be verified that |Ωi| = |Ω−i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Next, we review the widely-used “redundancy averaging” technique used for correlation-domain processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Following [1], [5], [26], the virtual ULA measurements are given by: t = Favry, (4) where t = [t−Mca, · · · , t−1, t0, t1, · · · , tMca]⊤ and Fav is the redundancy averaging matrix given by: [Fav]i+Mca+1,m+P (n−1) = � 1 |Ωi| If dm − dn = i 0 Otherwise, (5) with −Mca ≤ i ≤ Mca and 1 ≤ m, n ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The element ti is obtained by averaging all entries [Ry]m,n whose indices (m, n) generate a difference of i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', dm − dn = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Define a Toeplitz operator TMca : C2Mca+1 → CMca+1×Mca+1 as: [TMca(z)]m,n = zMca+1+m−n, 1 ≤ m, n ≤ Mca + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If the vector z ∈ C2Mca+1 is conjugate symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', zMca+1+i = z∗ Mca+1−i, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , Mca, then TMca(z) is a Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using the virtual measurement t, an augmented vir- tual co-array covariance matrix Tca ∈ C(Mca+1)×(Mca+1) is constructed as follows: Tca := TMca(t) = AUS(θ)PAUS(θ)H + σ2IMca+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (6) Once this virtual coarray covariance matrix has been obtained, any subspace-based algorithm [9], [10] applied to Tca can exactly recover the source DOAs provided Mca ≥ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, this also reveals that by efficiently designing sparse arrays, we can resolve up to Θ(P 2) sources with only P sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In the next section, we describe how the correlation processing is modified in the finite snapshot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Finite-Snapshot Coarray Covariance Estimation Let �Ry be the sample covariance matrix given by: �Ry := 1 L L � t=1 y(t)y(t)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (7) With a finite L, all the operations on the true covariance matrix are replaced by operations on the sample covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' First, we apply the redundancy averaging on ˆry: ˆt := Favˆry, where ˆry := vec( �Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (8) Here ˆt = [ˆt−Mca, · · · , ˆt−1, ˆt0, ˆt1, · · · , ˆtMca]⊤ with ˆti = 1 |Ωi| � dm−dn=i[ �Ry]m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Next, the estimated coarray covari- ance matrix is obtained by constructing a Toeplitz Hermitian matrix from ˆt as follows: �Tca = TMca(ˆt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (9) For a hole-free sparse array S, from (4), the elements of the matrix Ry are given by: [Ry]m,n = tdm−dn 1 ≤ m, n ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (10) Similarly, using the estimated coarray covariance matrix �Tca, we define matrix Rav ∈ CP ×P as [Rav]m,n := �tdm−dn, 1 ≤ m, n ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (11) This essentially maps the entries ˆti into a P × P matrix with the assignments specified by the difference set of the array S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since the sample covariance matrix �Ry is imperfect, the estimate �Tca also incurs an error due to a finite number of snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We denote the covariance estimation error as: EL = Tca − �Tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (12) The error in estimating the coarray covariance matrix naturally causes errors in DOA estimation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since subspace based algorithms are typically applied to this estimated covariance matrix �Tca, it becomes crucial to probabilistically characterize the estimation error EL and how it affects the DOA estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This paper provides such a rigorous theoretical charac- terization of the DOA estimation error with limited snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Review of Existing Performance Analysis of Coarray-Based Angle Estimation The existing performance analyses for coarray-based algo- rithms are largely asymptotic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In particular, they rely on the first-order perturbation analysis framework proposed in [13], which has been used to obtain expressions for the mean square error (MSE) of coarray MUSIC [5], and coarray ESPRIT [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the eigen decomposition Tca = UΓsU + U⊥ΓnUH ⊥, where U ∈ CMca+1×S and U⊥ ∈ CMca+1×Mca+1−S denote the eigenvectors corresponding to the signal and noise subspaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The correspond- ing perturbed matrices are denoted as �Tca = Tca + ∆Tca, �U⊥ = U⊥ + ∆U⊥ and �Γn = Γn + ∆Γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The perturbed matrices satisfy: (Tca + ∆Tca)(U⊥ + ∆U⊥) = (U⊥ + ∆U⊥)(Γn+∆Γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The perturbation analysis in [5] hinges on (i) the perturbations being “small enough” and (ii) ignoring the higher order perturbation terms such as ∆Tca∆U⊥ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' One of the key drawbacks of this analysis is that a rigorous characterization of an upper bound on the “small enough perturbation” ∥∆Tca∥2 ≤ ϵ1 has not been provided explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Secondly, [5, Theorem 1] makes a critical assumption that “the signal subspace and the noise subspace are well-separated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4 This assumption leaves open the possibility of problematic (unidentifiable) source configurations, which have not been explicitly addressed in their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We address both of the aforementioned issues by adopting a non-asymptotic analysis framework that is free from any approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our analysis also explicitly characterizes source configurations that ensure separation between the so-called signal and noise subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In [18], the first rigorous non-asymptotic probabilistic guarantees were provided for support recovery using a grid-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Although their analysis is valid for S > P, the sample complexity L = Ω(P 2) is conservative when S < P as our analysis in Section IV will show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' PERFORMANCE ANALYSIS OF COARRAY ESPRIT WITH FINITE SNAPSHOTS The Coarray ESPRIT algorithm, an adaptation of ESPRIT in the coarray domain, was introduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It applies ESPRIT on the estimated coarray covariance matrix �Tca as opposed to covariance matrix �Ry of the physical measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For a self-contained exposition, we review the Coarray ESPRIT al- gorithm and point out certain invariance properties of Coarray ESPRIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We describe Coarray ESPRIT for the ideal coarray covariance matrix Tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The extension to the sample covariance estimate is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The Coarray ESPRIT Algorithm The coarray signal subspace is defined as the span of the steering vectors: Sca := R (AUS(θ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Matrix T0 := AUS(θ)PAUS(θ)H is positive semi-definite and permits the following eigendecompostion: T0 = BΓBH, where the di- agonal of Γ comprises of the eigenvalues ordered in non- increasing fashion and B is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We can partition B as B = [U, U⊥], where the columns of U ∈ C(Mca+1)×S denote the eigenvectors of T0 corresponding to its non-zero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Following this decomposition, we write Tca as: Tca = T0 + σ2IMca+1 = B(Γ + σ2IMca+1)BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (13) If Mca ≥ S, the Vandermonde structure of AUS(θ) allows us to argue that rank(AUS(θ)PAUS(θ)H) = S, and hence: Sca = R(AUS(θ)) = R(AUS(θ)PAUS(θ)H) = R(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (14) As a result of (14), ∃ an invertible Q ∈ CS×S such that U = AUS(θ)Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (15) Let U0 ∈ CMca×S and U1 ∈ CMca×S denote the submatrices corresponding to the first and last Mca rows of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Similarly, let V0, V1 ∈ CMca×S be the submatrices corresponding to the first and last Mca rows of AUS(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Due to the Vandermonde structure of AUS(θ), the following holds: V1 = V0D, where D = diag(ejπ sin(θ1), ejπ sin(θ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , ejπ sin(θS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' By (15), ma- trices U0 and U1 satisfy: U0 = V0Q, U1 = V0DQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (16) Now, consider the matrix Ψ = U† 0U1 ∈ CS×S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (17) Since U0 has full column rank (17) implies U† 0 = Q−1V† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Plugging this in (17) and combining with (16), we have: Ψ = Q−1DQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, the DOAs can be inferred from the eigenvalues of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since L is finite, we do not have access to Tca and Coarray ESPRIT is instead applied on its estimate �Tca defined in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If we can ensure that the error EL is small enough (which we will rigorously specify using Weyl’s inequality), �Tca will be at least rank-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let ˆU be the matrix of eigenvectors corresponding to the largest S eigenvalues of �Tca (which is well-defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We can consider �U as a basis of the perturbed coarray signal space ˆSca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From �U, we compute the matrices �U0, �U1, �Ψ following the same construction as U0, U1 and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let ˆλi = riej �φi be the polar representation of the eigenvalues of the matrix ˆΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The estimated normalized frequencies ˆΩ = {�ωi}S i=1 are then given by �ωi = �φi 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Basis Invariance Property of ESPRIT In the previous section, ESPRIT is performed using the basis given by the singular vectors �U (U) of �Tca (Tca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, the following Lemma shows that the output of ESPRIT is invariant to the choice of the basis for the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let �U ∈ C(Mca+1)×S be another basis for R( �U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then, the matrix �Ψ := �U† 0 �U1 is similar to the matrix �Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', �Ψ and �Ψ share the same eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since R( �U) = R( �U), there exists an invertible matrix W ∈ CS×S such that �U := �UW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Thus, the following holds: �U0 = �U0W, �U1 = �U1W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since W is an invertible matrix, �U† 0 = W−1 �U† 0 and �Ψ = �U† 0 �U1 = W−1 �U† 0 �U1W = W−1 �ΨW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Covariance Estimation Error In this section, we obtain tail bounds on ∥EL∥2 in terms of array parameters in a finite snapshot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Such a bound brings out the effect of the array geometry on the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our analysis leverages recent results derived in [27] which we specialize for complex Toeplitz Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Some of our intermediate steps depart from [27] by invoking a result on the bounding the supremum of a certain spectral function from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We first introduce the key quantities and intermediate results on bounding the spectral norm of a Toeplitz Hermitian matrix from [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let M ∈ CN×N be any Hermitian symmetric Toeplitz matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Such a matrix can be completely described by only its first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the “spectral function” associated with m = [m−(N−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , m−1, m0, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , mN−1]⊤ [29]: fm(θ) = �N−1 k=−(N−1) mk exp(−jkθ), where mk = Mk+1,1 and m−k = m∗ k as a result of the Hermitian Toeplitz structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Evidently, the spectral function is a trigonometric polynomial of order N − 1 [28] whose coefficients are determined by the vector m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This spectral function fm(θ) can be used to bound ∥M∥2 as indicated by the following lemma from [27]–[29]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let M∈CN×N be a Hermitian symmetric Toeplitz matrix and fm be the associated spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then, ∥M∥2 ≤ supθ∈[−π,π] |fm(θ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5 Lemma 2 indicates that the spectral norm of a Hermitian symmetric Toeplitz matrix can be bounded by the supre- mum of its associated spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note that the co- variance estimation error EL = Tca − �Tca is a Toeplitz Hermitian matrix, satisfying EL = T (e), where e = [e−Mca, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , e−1, e0, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , eMca]T is conjugate symmetric and ei = ti − ˆti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, to bound ∥EL∥2 using Lemma 2, we need to investigate the spectral function fe(θ): fe(θ) := Mca � k=−Mca ek exp(−jθk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (18) Towards this purpose, define Λ(θ), for 1 ≤ m, n ≤ P, [Λ(θ)]m,n = 1 |Ωdm−dn| exp(j(dm − dn)θ), (19) and Ey := Ry − Rav, where Ry and Rav are defined in (10) and (11), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The elements of Ey are given by: [Ey]m,n = tdm−dn − �tdm−dn = edm−dn, 1 ≤ m, n ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (20) Proposition 1 provides a compact representation of fe(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let fe(θ) be the spectral function defined in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then, the following equality holds: fe(θ) = tr (EyΛ(θ)) where Λ(θ) and Ey are defined in (19) and (20), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' tr (EyΛ(θ)) = P � m,n=1 [Ey]m,n[Λ(θ)]n,m = P � m,n=1 edm−dn e−j(dm−dn)θ |Ωdm−dn| = Mca � s=−Mca � m,n dm−dn=s es exp(−jsθ) |Ωs| = Mca � s=−Mca es exp(−jsθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We introduce a quantity referred to as “Redundancy coeffi- cient” that will play an important role in bounding ∥EL∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Definition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1 (Redundancy Coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Given a hole-free sparse array S, let Mca be the largest element in its differ- ence set DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The redundancy coefficient ∆(S) is defined as: ∆(S) := �Mca i=0 1 |Ωi|, where set Ωi is defined in Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The quantity ∆(S) is controlled by the redundancy pattern of the sparse array S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', the number of times an element repeats in the difference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We provide an illustrative example to show how the quantity ∆(S) grows as a function of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Given a generalized nested array S(N1,N2) nest with P := N1 + N2 ≥ 3 sensors, the following holds: ln(P) ≤ ∆(S(N1,N2) nest ) ≤ 2 ln(P), if N2 = 1, and P 2/16 ≤ ∆(S(N1,N2) nest ) ≤ P 2, if N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Case I (N2 = 1): The choice N2 = 1 corresponds to a ULA, with P = N1 + 1 sensors and |Ωi| = P − i, i ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, ∆(S(P −1,1) nest ) = �P −1 i=0 1 P −i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Such a harmonic sum can be bounded as ln(P) ≤ �P −1 i=0 1 P −i ≤ 1+ln(P) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For P ≥ 3, we get the desired bound since 1 + ln(P) ≤ 2 ln(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Case II (N2 = ⌊P/2⌋ ≥ 2): The differences between the elements of the outer and inner ULA which are of the form k = i(⌈P/2⌉ + 1) − j, 2 ≤ i ≤ ⌊P/2⌋ and 1 ≤ j ≤ ⌈P/2⌉, satisfy |Ωk| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, we have ∆(S(N1,N2) nest ) ≥ ⌈P/2⌉⌊P/2⌋/2 ≥ (P 2/8 − P/8) ≥ P 2/16, where the first inequality follows from ⌊P/2⌋ − 1 ≥ ⌊P/2⌋/2 and the last inequality uses P ≤ P 2/2 for P ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since S(N1,N2) nest is hole free, it implies 1/|Ωi| ≤ 1 for all 0 ≤ i ≤ Mca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, we can bound ∆(S(N1,N2) nest ) ≤ Mca + 1 ≤ P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' As the following Theorem will show, ∆(S) determines the sample complexity for controlling the covariance estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, with the same number of sensors, two differ- ent array geometries could require drastically different sample complexity for ensuring that the covariance estimation error is bounded by the same quantity with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the measurement model (1) obeying assumptions [A1-A2], where S is a hole-free sparse array with redundancy coefficient ∆(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let Tca ∈ CMca+1×Mca+1 be the coarray covariance matrix defined in (6) and �Tca be its estimate given by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For any ϵ ≥ 0, we have P � ∥Tca − �Tca∥2 ≥ ϵ � ≤ 8Mca exp � −c1L min � c2ϵ2 ∥Ry∥2 2∆(S) , ϵ ∥Ry∥2 � ∆(S) �� , where c1 and c2 are a positive universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The proof is in Appendix A-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Frequency/Angle Estimation Error of Coarray ESPRIT We next bound the DOA estimation error in terms of the covariance estimation error EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Finally, we will combine this bound with the probabilistic bounds on ∥EL∥2 in Theorem 1 to obtain the main sample complexity result (in Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We will use the matching distance metric, defined as follows [20]: md(θ, ˆθ) := min Π∈P max j min k∈Z |ˆωΠ(j) − ωj + k| (21) where ωi (ˆωi) are the normalized DOAs and P denotes the set of all possible permutations on {1, 2, · · · , S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For our analysis, we will use an additional assumption that will be invoked whenever suitable: [A3] The number of sources S = O(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', S is held constant and does not grow with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Eigen Gap condition: Define: β := pminσ2 S(AUS(θ)) − σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (22) Henceforth, we will refer the condition β > 0 as the “eigen gap condition” and it will play an important role in our analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall, from the definition of Tca = AUS(θ)PAUS(θ)H + σ2, β > 0 ensures that there is a margin between the smallest singular value of AUS(θ)PAUS(θ)H and the (S+1)th singular value of Tca (determined by the noise σ) as pminσ2 S(AUS(θ)) is a lower bound on σS(Tca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The following theorem relates the DOA estimation error in terms of matching distance to the covariance estimation error EL, provided the latter is upper bounded by a suitable quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let S be a hole-free sparse linear array with P sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let Tca ∈ CMca+1×Mca+1 be the coarray covariance 6 matrix defined in (6) and �Tca be its estimate given by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If assumption [A3] holds and the following conditions are satisfied: β > 0 and ∥EL∥2 ≤ CSβ (23) then the matching distance error of ESPRIT algorithm satisfies md(θ, ˆθ) ≤ q∥EL∥2 (24) where EL, β are defined in (12), (22), q = (C′ S √Mca + 1)/(βσS(AUS(θ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Quantities CS, C′ S are dependent only on S which is assumed to be O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' See Appendix B-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The following Lemma obtains both lower and upper bounds on the spectral norm ∥Ry∥2 that are valid regardless of the array geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the covariance matrix Ry given by (2), where S is any (sparse) array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Given a fixed S, signal powers p and noise power σ2, for all θ the following holds: pminP ≤ ∥Ry∥2 ≤ pmaxPS + σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For any S, we can bound the spectral norm ∥Ry∥2 as: ∥Ry∥2 = σ1(AS(θ)PAS(θ)H) + σ2 ≤ pmaxσ1(AS(θ))2 + σ2 ≤ pmaxPS + σ2 where the last inequality follows from the fact that σ1(AS(θ))2 ≤ ∥AS(θ)∥2 F = PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Similarly, we can lower bound the norm ∥Ry∥2 ≥ σ1(AS(θ)PAS(θ)H) ≥ pminσ2 1(AS(θ)) ≥ pmin∥AS(θ)∥2 F /S = pminP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Combining Theorem 1 and 2, we next present a sufficient condition on the number (L) of snapshots in terms of the model parameters (array geometry, SNR and source config- uration) that allows us to bound the matching distance error by a prescribed ϵ with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the measurement model (1), where S is a hole-free sparse array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose β > 0 and the statistical assumptions [A1-A3] hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then for any 0 < δ < 1 and ϵ > 0, the matching distance error satisfies md(θ, ˆθ) ≤ min(ϵ, CSβq) with probability at least 1 − δ, provided L≥c3 ln �8Mca δ � max � q2 1∆(S) c2ϵ2 ,q1 � ∆(S) ϵ ,L2 0 c2 ,L0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (26) Here q1 = q∥Ry∥2, L0 = ∥Ry∥2 � ∆(S)/(CSβ) and c2, c3 are universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' See Appendix B-C Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the measurement model (1), where S is a hole-free sparse array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose β > 0 and the statistical assumptions [A1-A3] hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then for any 0 < δ < 1 and 0 < ϵ ≤ q min(CSβ, pminP � ∆(S)/c2), the matching distance error satisfies md(θ, ˆθ) ≤ ϵ with probability at least 1 − δ provided L ≥ c3 ln (8Mca/δ) q2 1∆(S)/(c2ϵ2), (27) where q1, L0,c2, c3 are given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using the lower bound on ∥Ry∥2 from Lemma 4, we can see ϵ ≤ min(CSβq, q1 � ∆(S)/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since β ≥ ϵ/(CSq), this implies L0 ≤ q1 � ∆(S)/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This in- equality also implies L2 0/c2 ≤ q2 1∆(S)/(c2ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Us- ing ϵ ≤ q1 � ∆(S)/c2, we can conclude that L0 ≤ (q1 � ∆(S)/ϵ2)(q1 � ∆(S)/c2) = q2 1∆(S) c2ϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, (27) im- plies (26) since max( q2 1∆(S) c2ϵ2 , q1√ ∆(S) ϵ , L2 0 c2 , L0) = q2 1∆(S) c2ϵ2 , and the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Role of redundancy coefficient in determining Temporal Sample Complexity: Corollary 1 indicates that if the number of snapshots grows proportional to the redundancy coefficient ∆(S), then it is possible to bound the matching distance error by an arbitrarily small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall that ∆(S) is a function of the redundancy pattern of S and from Lemma 3 we have ∆(Sula) = Θ(ln(P)) and ∆(Snest) = Θ(P 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Based on this, at a cursory glance, one may be tempted to conclude from (27) that for the same number of sensors, the snapshot requirement for the nested array is significantly larger than for the ULA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This is also consistent with an existing misconception that co- array based processing requires a large number of snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, in reality the sample complexity is also controlled by the interaction of ∆(S) with other geometry dependent terms in (27) such as q1 = q∥Ry∥2, which in turn depend on both the physical array and coarray size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In the next section, we clarify this misconception regarding the seemingly higher snapshot requirement of nested arrays in the setting S = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Spatiotemporal trade-offs: The snapshot requirement in Corollary 1 is inversely proportional to β (since q ∝ 1 β ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If the array geometry and source configuration are kept fixed and we increase the SNR (either by increasing pmin or decreasing noise power σ), Corollary 1 suggests that it is possible to achieve the same probability of error with fewer snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our simulations also are consistent with this theoretical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This SNR and geometry dependent snapshot characterization is another novel contribution of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A CLOSER LOOK AT THE SEPARATION CONDITION FOR SUPER-RESOLUTION WITH SPARSE ARRAYS In order to understand the behavior of the smallest non- zero singular value σS(AUS(θ)), we consider the notion of minimum separation [20]: ∆min(θ) = min i,j∈Ω i̸=j min k∈Z ���ωi − ωj + k ��� (28) where ωi is the normalized spatial frequency corresponding to direction θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' By definition, for all θ we have 0 ≤ ∆min(θ) ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Instead of analyzing an arbitrary source configuration θ, one can obtain a more interpretable condition by representing 7 (23) as a function of the minimum separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The source configurations where ∆min(θ) is larger than some threshold inversely proportional to Mca +1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' ∆min(θ) > γ Mca+1, γ > 1) will be referred to as the “well-separated” regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We will inspect what this means for specific array geometries such as the ULA and nested array, and obtain tight bounds on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The “Well-Separated” Case In this section, we turn our attention to how the eigen gap condition can be utilized to obtain sufficient conditions on SNR for different array geometries in the “well-separated” regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let V ∈ CK×S be a Vandermonde matrix, with [V]m,n = zm−1 n where {zn}S n=1 are the so called “nodes” of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We begin by summarizing results from [21], [24], [31], [32] which characterize the minimum singular value of a Vandermonde matrix in the well-separated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The following Lemma follows from [32, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (32)] which is an intermediate result from [32, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let V(α) ∈ CK×S be a Vandermonde matrix with zn = ej2παn for 1 ≤ n ≤ S and S ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If αi ∈ [0, 1) are all distinct and satisfy: min i,j∈Ω i̸=j min k∈Z ���αi − αj + k ��� ≥ γ K (29) for some constant γ > 1, then the following holds: σS(V(α))2 ≥ K/C′, where C′ := γ/(γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (30) From Lemma 5, for S = Sula if the source configurations θ satisfies ∆min(θ) ≥ γ P for some γ > 1 and S ≤ P then we have the following lower bound: σS(AUS(θ))2 ≥ P/C′ (31) In the following Proposition, we apply Lemma 5 to character- ize lower bounds on σS(AUS) for the nested array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proposition 2 (Well-Separated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let S = S(N1,N2) nest be a nested array with N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋ with P ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose ∆min(θ) ≥ 5γ P 2 for some γ > 1 and S ≤ P 2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then, the following lower bound holds: σS(AUS(θ))2 ≥ P 2/C′ n, where C′ n = 5γ/(γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (32) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For the nested array with N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋, from (3) we have Mca + 1 ≥ P 2 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, ∆min(θ) ≥ 5γ P 2 implies ∆min(θ) ≥ γ Mca+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, the condition on ∆min(θ) in Lemma 5 holds and we have the desired lower bound: σS(AUS(θ))2 ≥ Mca+1 C′ ≥ ( γ−1 γ ) P 2 5 = P 2 C′n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proposition 2 shows that for a nested array, the sources are well-separated if ∆min(θ) ≥ 5γ/P 2 and in this case, σS(AUS(θ)) grows as Ω(P), owing to the the larger difference coarray of a nested array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In order to highlight the dependence of sample complexity only on key model parameters, we define quantities to combine parameters that are held fixed (such as S, pmin, pmax, σ): Cula(S, σ, pmax) := 8C ′2 S C ′3 c3 c2 (S + σ2 pmax )2 (33) Cnest(S, σ, pmax) := 4C ′2 S C ′3 n c3 c2 (S + σ2 pmax )2 (34) where C′, C′ n are universal constants and CS defined in Theorem 2 is dependent only on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using Proposition 2, we now specialize Corollary 1 for the ULA and nested array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let S = Sula be a ULA with P sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose the minimum angular separation between the sources, and the SNR satisfy the following conditions for some γ > 1: ∆min(θ) ≥ γ/P, pmin/σ2 > 2C′/P, where C′ = γ γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Under assumptions [A1-A3], for any 0 < δ < 1 and 0 < ϵ ≤ C1(S) := CSC′ S, md(θ, �θ) ≤ ϵ is satisfied with probability at least 1 − δ, provided P ≥ 3 and L ≥ Cula(S, σ, pmax) ϵ2 �pmax pmin �2 � ln �8P δ ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (35) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From Lemma 5, if ∆min(θ) ≥ γ/P, we have σ2 S(AUS(θ)) ≥ P C′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Under the assumption on the SNR, we have pminσ2 S(AUS(θ)) ≥ pmin P C′ > 2σ2 which ensures β > pminσ2 S(AUS(θ))/2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notice that for ULA Mca + 1 = P and from the fact that σ2 S(AUS(θ)) ≥ P C′ , we can obtain the following bound: q = C′ S √ P βσS(AUS(θ)) ≤ 2C′ S √ P pminσ3 S(AUS(θ)) ≤ C′′ S pminP (36) where C′′ S = 2C′ SC′1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notice that: CSβq = CSC′ S √Mca + 1 σS(AUS(θ)) ≥ C1(S) √ P √ P = C1(S) (37) where the inequality follows from σS(AUS)≤∥AUS∥F / √ S = √ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using the fact that β ≤ pminσ2 S(AUS(θ)), and the above lower bound on σS(AUS(θ)), we obtain q ≥ C′ S √ P pminσ3 S(AUS(θ)) ≥ C′ S pminP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (38) Therefore, qpminP � ∆(Sula)/c2 ≥ C′ S � ∆(Sula)/c2 ≥ C′ S � ln(P)/c2, where the last inequality follows from the lower bound on ∆(Sula) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall that c2 < 1 2, and therefore for P ≥ 3, √ ln P/c2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This implies that min(C1(S), C′ S � ln(P)/c2) = C1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Combining this with (37), we have ϵ ≤ C1(S) = min(C1(S), C′ S � ln(P)/c2) ≤ min(CSβq, qpminP√∆Sula/c2), which ensures that the as- 2The constant c2 = 3/16 √ 2 is specified in the proof of Theorem 1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 8 sumption on ϵ in Corollary 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From Lemma 4, we have ∥Ry∥2 ≤ pmaxPS + σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using this bound and (36), we get: q1 � ∆(Sula) ≤ C′′ S pminP (PSpmax + σ2) � 2 ln(P) = C′′ S(S + σ2 pmaxP ) �pmax pmin � � 2 ln(P) ≤ �C1(S, σ, pmax) �pmax pmin � � ln(8P/δ) (39) where �C1(S, σ, pmax) := (S + σ2 pmax ) √ 2C′′ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The upper bound follows from the observations that (S + σ2 pmaxP ) ≤ (S + σ2 pmax ) for all P ≥ 1 and ln(P) ≤ ln(8P/δ) for any δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notice from (33), that Cula(S, σ, pmax) = c3/c2 �C2 1(S, σ, pmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From (39), we have c3 ln(8P δ )q2 1∆(Sula) c2ϵ2 ≤ c3 c2ϵ2 �C2 1(S, σ, pmax)(pmax pmin ln(8P/δ))2 = Cula(S, σ, pmax) ϵ2 �pmax pmin �2 (ln(8P/δ))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, (35) implies (27) and the proof is completed by applying Corollary 1 since β > 0 and the conditions on ϵ and L required for applying the corollary are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let S = S(N1,N2) nest be a nested array with N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose the minimum angular separation between the sources, and the SNR satisfy the following conditions for some γ > 1: ∆min(θ) ≥ 5γ P 2 , pmin σ2 > 2C′ n P 2 , where C′ n = 5γ/(γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Under the assumptions [A1-A3], for any δ > 0 and 0 < ϵ ≤ C2(S) := � 1/5CSC′ S, md(θ, �θ) ≤ ϵ is satisfied with probability at least 1 − δ provided P ≥ 3 and L ≥ Cnest(S, σ, pmax) ϵ2 �pmax pmin �2 ln �8P 2 δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (40) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From Proposition 2, if ∆min(θ) ≥ 5γ/P 2, we have σ2 S(AUS(θ)) ≥ P 2 C′n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Following the same argument as Theo- rem 4, this ensures that β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using the fact that Mca + 1 ≤ P 2 (from (3)) and the lower bound on σ2 S(AUS(θ)), we obtain q ≤ C′ SP βσS(AUS(θ)) ≤ 2C′ SP pminσ3 S(AUS(θ)) ≤ ¯C′′ S pminP 2 (41) where ¯C′′ S := 2C′ SC′1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='5 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notice that σS(AUS(θ)) ≤ ∥AUS∥F / √ S = √Mca + 1 ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, similar to (37), we can establish that CSβq ≥ C2(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using the fact P 2/5 ≤ Mca + 1 from (3), similar to (38) we obtain q ≥ C′ SP √ 5pminσ3 S(AUS(θ)) ≥ C′ S √ 5pminP 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From Lemma 3, ∆(Snest) ≥ P 2/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It follows that qpminP � ∆(Snest)/c2 ≥ C′ S 4c2 √ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since 4c2 < 1, it follows that min(C2(S), C′ S/(4c2 √ 5)) = C2(S) and therefore ϵ ≤ C2(S) = min(C2(S), C′ S/(4c2 √ 5)) en- sures that the assumption on ϵ in Corollary 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using ∆(Snest) ≤ P 2 (from Lemma 3), Lemma 4, and (41), we get: q1 � ∆(Snest) ≤ �C1(S, σ, pmax)(pmax/pmin), (42) where �C1(S, σ, pmax)=(S + σ2 pmax ) ¯C′′ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' By (42), we have ln(8Mca δ )c3q2 1∆(Snst) c2ϵ2 ≤ c3 c2ϵ2 �C2 1(S, σ, pmax) ln(8P 2/δ)(pmax pmin )2 = Cnest(S, σ, pmax) ϵ2 �pmax pmin �2 � ln(8P 2/δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore (40) implies (27) and the proof is again completed by applying Corollary 1 since β > 0 and the conditions on ϵ and L required for applying the corollary are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note that the range of values for ϵ where Theorem 4 and 5 are applicable differ slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However in the regime ϵ ≤ min(C1(S), C2(S)) = C2(S) and P ≥ 3, we can fairly compare the two array geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Towards higher resolution with same snapshots: Theorem 4 states that for a ULA, the matching distance error for Coarray ESPRIT can be bounded by ϵ provided (i) the snapshots scales only (poly)logarithmically in the dimension of the coarray covariance matrix and (ii) the minimum separation is ∆min ≥ γ/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' On the other hand, Theorem 5 guarantees that for a nested array with P sensors, it is possible to bound the matching distance error by the same ϵ with order wise the same number of snapshots (L = Ω(ln(P 2)), but with a relaxed separation condition that allows ∆min to be ∆min = Ω(1/P 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This validates the superior resolution properties of nested arrays compared to ULA with the same budget of temporal snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This has been empirically observed in the literature, but never theoretically established, until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Noise Resilience of Nested Arrays: If we consider the separation regime ∆min = Ω(1/P) that is applicable for both the ULA and nested array, Theorems 4 and 5 indicate that the SNR (pmin/σ2) requirement for the nested array can be P times smaller than that of the ULA, in order to achieve the same DOA error bound with order-wise the same number of snapshots (L = Ω(ln P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This brings out another advantage of nested arrays in terms of robustness against noise, especially in the low-SNR regime [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Effect of Dynamic Range: Our analysis also reveals the chal- lenge posed by sources with higher dynamic range pmax/pmin as also observed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Theorem 5 suggests that at the same SNR (defined with respect to the weakest source pmin), more snapshots maybe needed for resolving sources with disproportionately varying powers (higher pmax compared to the fixed pmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' As will be shown, the numerical results are indeed consistent with the prediction made by our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The Myth of Large Snapshots: Correlation Error vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Angle Estimation Error Since nested (and other) sparse arrays realize the virtual difference coarray by correlation-processing, it is commonly believed that one needs a large number (L = Ω(P 2)) of temporal snapshots to estimate Θ(P 2) (cross) correlation values between sensor pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This ‘myth’ of large snapshots (that grows quadratically in the number of sensors P) is partially true, if our goal is to estimate the coarray covariance matrix Tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If we only allow L to scale as L = Θ(log P) 9 (the so-called sample-starved regime), then one may indeed incur large error in covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, Theorem 5 shows that the angle estimation error can be made arbitrarily small (ϵ) with high probability (1 − δ) provided L scales only as Ω( 1 ϵ2 ln(8P 2/δ)), despite the possibility of the coarray covariance error of a nested array increasing with P in this snapshot-starved regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This surprising phenomenon is due to the fact that the potentially large covariance estimation error (which can even grow with P in this regime) can actually be mitigated/counterbalanced by the enhanced aperture/difference set of the nested array that results in a large restricted smallest singular value σS(AUS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' As long as ∆min(θ) ≥ 5γ P 2 , σ2 S(AUS) scales as cP 2 (for some constant c), and this helps us obtain reliable angle estimation, although the covariance estimates may be unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' SIMULATIONS We numerically investigate the useful SNR regime for coarray processing (Section V-A), the impact of SNR and the number of snapshots on DOA estimation error (V-B and V-C), the relationship between DOA and covariance estimation error (V-D), and the effect of the dynamic range of source powers on resolving two closely spaced sources (V-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' When is Coarray-Based DOA Estimation Beneficial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We begin by examining under which circumstances coarray- based algorithms offer an advantage over more conventional DOA estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Specifically, in case of the ULA, we could apply MUSIC or ESPRIT directly to the sample covariance matrix �Ry in (7) instead of the averaged coarray covariance matrix �Tca in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1 shows the matching distance error of coarray ESPRIT and direct ESPRIT, averaged over 103 Monte Carlo trials, in case of the ULA, and, for comparison, coarray ESPRIT in case of the nested array with the same number of sensors (P = 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We consider L = 100 snapshots, and S = 4 equipower sources equally spaced by ∆ = 2/P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' At medium to low SNR, the advantage of coarray- based processing is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' At high SNR, the situation is reversed, as the error of direct ESPRIT continues decreasing as a function of SNR, whereas the error of coarray ESPRIT saturates3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, coarray-based processing—including re- dundancy averaging (8)—can clearly offer significant benefits in SNR or snapshot-limited conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' As mostly such chal- lenging scenarios are of interest in many applications, we focus on coarray ESPRIT herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Improving Resolution by Increasing SNR or Snapshots Next, we compare the probability of resolution as a function of the minimum separation for the nested array and ULA with the same number of sensors, P = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Coarray ESPRIT is employed for both array geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We consider two sources with equal power (p1 = p2) and (normalized) angles ω = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1 + ∆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The sources are declared to be successfully resolved when the estimated DOAs satisfy maxi |ˆωi − ωi| ≤ ∆/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2 shows the empirical probability of resolution (averaged over 1000 Monte-Carlo trials) for varying separation 3This well-known and fundamental phenomenon is due to the finite- snapshot error of the coarray covariance matrix, see [5]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 20 10 0 10 20 30 40 50 SNR (dB) 10-6 10-4 10-2 100 Average Matching Distance ULA (coarray ESPRIT) Nested (coarray ESPRIT) ULA (direct ESPRIT) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1: Comparison of ESPRIT applied to the sample covariance matrix (7) (direct ESPRIT) and the estimated coarray covariance matrix (9) (coarray ESPRIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Coarray ESPRIT achieves lower angle estimation error than direct ESPRIT at medium to low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 10-3 10-2 10-1 Angular Separation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='8 1 Probability of resolution # Sensors = 20, Snapshots = 55 = (1/P) = (1/P2) ULA (SNR = 0 dB) Nested (SNR = 0 dB) ULA (SNR = -16 dB) Nested (SNR = -16 dB) 10-3 10-2 10-1 Angular Separation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='8 1 Probability of resolution # Sensors = 20, SNR = 0 dB = (1/P) = (1/P2) ULA (L = 55) Nested (L = 55) ULA (L = 600) Nested (L = 600) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2: Probability of resolution vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' source separation for different SNR levels (top) and number of snapshots (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Increasing either improves resolution for both arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' ∆ and a fixed number of snapshots L = 55 and SNR = 0 and −16 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We observe that both array geometries can operate at a smaller separation at a higher SNR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', smaller σ/pmin ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Indeed, the transition from low to high probability of resolution occur around ∆ ∝ 1/P for the ULA and ∆ ∝ 1/P 2 for the nested array, as predicted by Theorems 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It is also possible to enhance resolution by increasing the number of snapshots, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2 demonstrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Here, the SNR is fixed at 0 dB and the number of snapshots is L = 55 and L = 600, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Snapshot and SNR Trade-off Section V-B showed that SNR and the number of temporal snapshots can be exchanged for improved resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We now study this trade-off in further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We consider S = 2 equipowered sources located at ω = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1 + ∆}, where ∆ ∈ {2/P, 2/P 2} and P = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 3 shows the separation- relative matching distance error md(θ, �θ)/∆ (averaged over 103 Monte Carlo trials) as a function of both the number of snapshots and SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Firstly, fewer snapshots are required at higher SNR (and vice versa) to obtain the same recovery error, both in case of the ULA (left column) and nested array (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This supports Theorem 3, where the match- 10 ing distance depends on the number of snapshots and SNR through (22) and (26), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Secondly, the nested array displays a more advantageous trade-off between snapshots and SNR compared to the ULA for both source separation 2/P (top row) and 2/P 2 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The benefit is especially apparent for ∆ = 2/P 2, where the nested array has a greatly larger range of operating points where the relative matching distance is low, as predicted by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note that the gray pixels correspond to a relative error of approximately 10% of the separation, whereas white corresponds ≤ 1% error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 3: Relative matching distance error md(θ, �θ)/∆ as a function of snapshots and SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The nested array (right column) achieves lower error than the ULA (left column) for both source separation ∆ = 2/P (top row) and ∆ = 2/P 2 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' DOA and Covariance Estimation Error Next, we illustrate an intriguing benefit of coarray-based DOA estimation in case of the nested array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We consider the average DOA matching distance and average covariance estimation error defined as ∥Tca − �Tca∥2 for a varying number of sensors P and S = 4 equipower sources equally spaced by ∆ ∈ {1/P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='5, 1/P 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The number of snapshots is L = 50 and SNR = 0 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4 shows that the nested array incurs a larger covariance estimation error compared to the ULA with the same number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, despite obtaining a worse estimate of the covariance matrix �Tca, the nested array achieves superior DOA estimation performance when coarray ESPRIT is applied to �Tca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In fact, when the separation is ∆ = 1/P 2, the average matching distance no longer decays with P for the ULA, whereas it continues to do so for the nested array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This is enabled by the larger coarray aperture of the nested array, which offsets the effect of finite snapshot covariance estimation error as discussed in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note that for a fixed number of snapshots and a growing number of sensors P, the entries of the coarray covariance matrix Tca become increasingly challenging to estimate, since the size of Tca is proportional to the number of coarray elements Mca, which is ∝ P for the ULA and ∝ P 2 for the nested array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Effect of Dynamic Range of Source Powers In the final experiment, we investigate the ability of coarray ESPRIT to resolve two sources with unequal powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We set 10 15 20 25 30 35 40 Number of sensors (P) 10-4 10-3 10-2 10-1 Average Matching Distance Error # Snapshot = 50, SNR = 0 dB Nested ( =1/P2) ULA ( =1/P2) Nested ( =1/P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='5) ULA ( =1/P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='5) 10 15 20 25 30 35 40 Number of sensors (P) 101 102 Average Covariance Estimation Error # Snapshot = 50, SNR = 0 dB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4: Average matching distance (top) and covariance estimation error (bottom) as a function of the number of sensors P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The DOA estimation error of the nested array decays despite the larger covariance estimation error compared to the ULA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' the dynamic range to pmax/pmin ∈ {1, 10} by fixing the power of the weaker source to pmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2 and varying pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5 shows that the number of snapshots required to distinguish two sources (separated by ∆ = 1/P) is significantly larger when pmax/pmin = 10 compared to pmax/pmin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This is consistent with Theorems 4 and 5, which imply that the sufficient number of snapshots for resolving two sources (with high probability) grows with pmax if pmin and σ are held fixed, irrespective of the array geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This brings out a non-trivial dependence of the dynamic range pmax/pmin on the sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, distinguishing two sources with greatly different powers is more challenging and requires more snapshots than when the powers are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 101 102 103 104 Snapshots 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='8 1 Probability of resolution ULA (pmax/pmin = 10) Nested ULA (pmax/pmin = 10) ULA (pmax/pmin = 1) Nested ULA (pmax/pmin = 1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5: Effect of dynamic range of source powers on probability of resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Coarray ESPRIT requires more snapshots to detect two sources with larger dynamic range pmax/pmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' CONCLUSION This paper investigated angle estimation error of coarray ESPRIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We considered both additive noise and finite-snapshot covariance estimation error, which we probabilistically char- acterized in the case of Toeplitz covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Our results show that if the number temporal snapshots scales logarithmically with the number of sensors, coarray ESPRIT achieves arbitrarily low estimation error with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This also shows that the DOA estimation error can be small even though the covariance estimation error may be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' ULA,Separation=2/P 10 100 Matchingdistance/separation 5 0 5 10~1 15 20 10~2 101 102 103 104 SnapshotsNested,Separation=2/P 10 100 Matchingdistance/separation 5 SNR (in dB) 0 5 10~1 10 15 20 10~2 101 102 103 104 SnapshotsULA, Separation=2/p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 10 100 Matchingdistance/separation 5 SNR (in dB) 0 5 10~1 10 15 20 10~2 101 102 103 104 SnapshotsNested, Separation=2/p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 10 100 Matchingdistance/separation 5 SNR (in dB) 0 5 10~1 10 15 20 102 101 102 103 104 Snapshots11 Finally, our theoretical and simulation results demonstrate that sparse arrays can provide higher resolution and better noise resilience compared to the ULA with the same number of sensors and snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' APPENDIX A A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Intermediate Results We will first state the complex extension of Hanson-Wright inequality [33], which is obtained by applying [34, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1] with the strategy described on [34, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1, Page 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let A ∈ Cn×n be a fixed Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Consider the random vector x = [x1, x2, · · · , xn]⊤ ∈ Cn with independent real and imaginary components Re(xi), Im(xi) satisfying E(Re(xi)) = E(Im(xi)) = 0, and ∥Re(xi)∥ψ2 ≤ K, ∥Im(xi)∥ψ2 ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then for any ϵ > 0, we have P(|xHAx − E(xHAx)|>ϵ) ≤ 2 exp � − c min( ϵ2 2K4∥A∥2 F , ϵ K2∥A∥2 ) � where c > 0 is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let z = [Re(x)⊤, Im(x)⊤]⊤ ∈ R2n and define : ˜A = �Re(A) −Im(A) Im(A) Re(A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It is easy to see that for any Hermitian A, we have the following equality xHAx = zT ˜Az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Further, it can be verified that ∥˜A∥F = √ 2∥A∥F and ∥˜A∥2 = ∥A∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Now, we can apply [34, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='1], to obtain the desired probability bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let wi ∈ Cn, 1 ≤ i ≤ T be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='d com- plex circularly symmetric Gaussian random variable with distribution CN(0, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let A ∈ Cn×n be a fixed Her- mitian matrix, then for any ϵ > 0 and universal constant c, we have P(| 1 L �L i=1 wH i Awi − E[wH i Awi]| ≥ ϵ) ≤ 2 exp � −cL min � ϵ2 2K4∥Σ∥2 2∥A∥2 F , ϵ K2∥Σ∥2∥A∥2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since wi is a complex circularly symmetric Gaussian random variable distributed according to CN(0, Σ), we define a new transformed variable ui = Σ−1/2wi where Σ1/2 is the square root of the covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' It can be verified that ui ∼ CN(0, In), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', it is also a complex circularly symmetric Gaussian random variable with independent real and imaginary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Define block-wise diagonal matrices ˜A = diag(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , A), ˜Σ1/2 = diag(Σ1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , Σ1/2) ∈ CnL×nL and ˜u = [uT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' , uT L]T ∈ CnL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Next, we can observe that �L i=1 wH i Awi = �L i=1 uH i Σ1/2AΣ1/2ui = ˜uH ˜Σ1/2 ˜A˜Σ1/2˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We have E(�L i=1 wH i Awi) = LE � wH i Aw � , since it is a sum of L i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='d random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The desired probability can be re-written as: P(| 1 L �L i=1 Re(wH i Awi) − E[Re(wH i Awi)]| ≥ ϵ) = P(|Re(˜uH ˜Σ1/2 ˜A ˜Σ1/2˜u) − E[Re(˜uH ˜Σ1/2 ˜A ˜Σ1/2˜u)]| ≥ Lϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall that Re(˜ui), Im(˜ui) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='d distributed as N(0, 1/2) and hence sub-Gaussian with K = 2/ √ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note that due to the block-diagonal structure we have ∥ ˜Σ1/2 ˜A ˜Σ1/2∥2 F = L∥Σ1/2AΣ1/2∥2 F ≤ L∥A∥2 F ∥Σ∥2 2 and ∥ ˜Σ1/2 ˜A ˜Σ1/2∥2 = ∥Σ1/2AΣ1/2∥2 ≤ ∥A∥2∥Σ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The proof is completed by applying Lemma 6 with ϵ = ϵL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof of Theorem 1 From Lemma 2, we have P(∥EL∥2 ≥ ϵ)≤P(sup |fe(θ)|≥ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In general, it is not straightforward to evaluate this supremum, however, we exploit the following result from [28]that bounds it by using the function value evaluated at a few grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [28, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='28, Chapter 10, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2, Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 33] Let f(θ) be a trigonometric polynomial of order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then, supθ∈[−π,π] |f(θ)| ≤ 2 max1≤k≤4N |f(θk)|, θk = k−2N 4N π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From Proposition 1, we have fe(θ) = tr(EyΛ(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, we want to relate it to the sample covariance matrix �Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' In order to do this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' we show that tr(RavΛ(θ)) = tr( �RyΛ(θ)) where recall from (7) that �Ry is the sample covariance matrix: tr(RavΛ(θ)) = P � m=1 P � n=1 [Rav]m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='n[Λ(θ)]n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='m = Mca � s=−Mca � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='n: dm−dn=s ˆts exp(−jsθ) |Ωs| = Mca � s=−Mca ˆts|Ωs| exp(−jsθ) |Ωs| = (a) Mca � s=−Mca � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='n: dm−dn=s [ �Ry]m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='n[Λ(θ)]n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='m = Tr( �RyΛ(θ)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' where (a) follows from the redundancy averaged estimator where for all m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' n such that dm − dn = s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' we have |Ωs|ˆts = � dm−dn=s[ �Ry]m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, we have the following rela- tion: fe(θ)=tr (EyΛ(θ))=tr ((Ry − Rav)Λ(θ)) = tr((Ry − �Ry)Λ(θ)) = 1 L �L t=1 � E[y(t)HΛ(θ)y(t)] − y(t)HΛ(θ)y(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since the snapshots are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='d, we can define i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='d ran- dom variables {Zt(θ)}L t=1 as Zt(θ) ≜ y(t)HΛ(θ)y(t) − E(y(t)HΛ(θ)y(t)) with y(t) ∼ CN(0, Ry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note that Λ(θ) is Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hence, we can apply Lemma 7 with Σ = Ry and A = Λ(θ) to obtain ∀ϵ > 0, P � 1 L | L � t=1 Zt(θ)| ≥ ϵ � ≤ (43) 2 exp � −cL min � ϵ2 2K4∥Ry∥2 2∥Λ(θ)∥2 F , ϵ K2∥Ry∥2∥Λ(θ)∥2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We want to obtain a universal upper bound that is similar to (43) but not dependent on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notice, ∥Λ(θ)∥2 F = 1 |Ω0| + �Mca s=1 2 |Ωs| ≤ 2∆(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Similarly, we can also bound ∥Λ(θ)∥2 ≤ ∥Λ(θ)∥F ≤ � 2∆(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This gives us the following bound: P � 1 L | L � t=1 Zt(θ)| ≥ ϵ � ≤ (44) 2 exp � −cL min � ϵ2 4K4∥Ry∥2 2∆(S) , ϵ K2∥Ry∥2 � 2∆(S) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Note fe is a trigonometric polynomial of order Mca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Now, we will use Lemma 8 to bound the spectral function |fe(θ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' P(supθ∈[−π,π] |fe(θ)| ≥ ϵ) ≤ P(2 max1≤k≤4Mca |fe(θk)| ≥ ϵ) ≤ �4Mca k=1 P � |fe(θk)| ≥ ϵ 2 � ≤ 8Mca exp � − c1L min � c2ϵ2 ∥Ry∥2 2∆(S), ϵ ∥Ry∥2√ ∆(S) �� , where c1 = c/(2 √ 2K2) (c was given in Lemma 7) and c2 = 1/(4 √ 2K2) = 3/(16 √ 2) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The first inequality follows due to Lemma 8, the second inequality follows from union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The last inequality is a consequence of the bound computed in (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 12 APPENDIX B A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof of Theorem 2 The proof uses several results from [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' However, unlike [20] the underlying subspace of interest is the coarray subspace and the perturbation is due to covariance estimation error and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We provide key intermediate steps to make the results self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall that columns of U and �U are orthonormal bases for the subspaces R(U) and R( �U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let the principal an- gles between the subspaces R(U) and R( �U) be denoted as Θ(R(U), R( �U)) := [ψ1, ψ2, · · · , ψS]T where 0 ≤ ψ1 ≤ ψ2 ≤ · · · ≤ ψS ≤ π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then from [35], we have cos(ψi) = σi(UH �U) i = 1, 2, · · · , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall from Lemma 1, the output of ESPRIT is invariant to the choice of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For ease of analysis, we will choose a pair of basis for R(U) and R( �U), which are also known as “canonical bases” [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let the SVD of the matrix UH �U be of the form UH �U := LΣcRH, L, R ∈ CS×S, where Σc = diag(σc 1, σc 2, · · · , σc S) where σc i = σi(UH �U) are arranged in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The canonical basis U(c) and �U(c) are given by: U(c) := UL, �U(c) := �UR (45) Using the canonical basis, we define the following matrices: Ψ(c) :=U(c)† 0 U(c) 1 , �Ψ(c) := �U(c)† 0 �U(c) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Since R(U)=R(U(c)) and R( �U) = R( �U(c)), we have Θ(R(U(c)), R( �U(c))) = Θ(R(U), R( �U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Notice that the canonical basis has the following property: cos(ψi) = σi(U(c)H �U(c)) = u(c)H i �u(c) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We will use [20, Lemma 2] that relates the matching distance error to the quantity ∥ �Ψ(c) − Ψ(c)∥2 and holds universally: md(θ, ˆθ) ≤ π S3/2√Mca + 1 σS(AUS(θ)) ∥ �Ψ(c) − Ψ(c)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (46) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Relating ∥ �Ψ(c) − Ψ(c)∥2 to ∥EL∥2 Let B = A + N ∈ CM×N, where rank(A) ≥ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose ψL is the largest principal angle between the subspace spanned by L principal singular vectors (corresponding to L largest singular values) of A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If σL+1(A) ≤ α and σL(B) ≥ α + δ for some α ≥ 0 and δ > 0 then, Wedin’s Theorem [36] states that: sin(ψL) ≤ ∥N∥2/δ (47) Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Suppose σS(Tca) ≥ 2∥EL∥2 and β = pminσ2 S(AUS(θ)) − σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then sin(ψS) ≤ 2∥EL∥2/β (48) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Recall that ˆTca = Tca − EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' To apply Wedin’s theorem, we need to characterize quantities α and δ such that: σS( ˆTca) ≥ δ + α, σS+1(Tca) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From (13), we have σS+1(Tca) = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We choose α = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Using Weyl’s inequality, σS(�Tca) ≥ σS(Tca) − ∥EL∥2 (a) ≥ σS(Tca)/2 (b) = (σS(AUS(θ)PAUS(θ)H) + σ2)/2, where (a) follows from the assumption 2∥EL∥2 ≤ σS(Tca) and (b) follows from (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Combining with the preceding inequality, we obtain σS( ˆTca) − σ2 ≥ (σS(AUS(θ)PAUS(θ)H) − σ2)/2 ≥ β/2 > 0, where the last term is positive due to the given condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then we can choose δ = σS(�Tca) − σ2 which satisfies σS(�Tca) = α + δ with δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' The proof is completed by using (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If pminσ2 S(AUS(θ)) > σ2 and ∥EL∥2 ≤ σS(U(c) 0 )(σS(AUS(θ)PAUS(θ)H) − σ2) 4 √ 2 (49) then ∥Ψ(c) − ˆΨ(c)∥2 ≤ 14 √ 2∥EL∥2 σ2 S(U(c) 0 )(pminσ2 S(AUS(θ))−σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From the definition of U(c), �U(c) we have: ∥U(c) − �U(c)∥2 2 = ∥(U(c) − �U(c))H(U(c) − �U(c))∥2 = 2(1 − cos(ψS)) ≤ 2(1 − cos2(ψS)) = 2 sin2(ψS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (50) By the assumption of this lemma, 2∥EL∥2 ≤ σS(U(c) 0 )(σS(AUS(θ)PAUS(θ)H) − σ2) and σS(U(c) 0 ) ≤ 1, we have 2∥EL∥2 ≤ σS(Tca)σS(U(c) 0 ) ≤ σS(Tca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This together with the assumption pminσ2 S(AUS(θ)) > σ2 enables us to apply Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Combining (48) with (50) we obtain the following bound: ∥ �U(c) − U(c)∥2 ≤ 2 √ 2∥EL∥2 σS(AUS(θ)PAUS(θ)H) − σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (51) Notice that ∥ �Ψ (c) − Ψ(c)∥2 = ∥( �U(c)† 0 − U(c)† 0 ) �U(c) 1 + U (c)† 0 ( �U(c) 1 − U(c) 1 )∥2 ≤ ∥ �U(c)† 0 − U(c)† 0 ∥2∥ �U(c) 1 ∥2 + ∥U(c)† 0 ∥2∥ �U(c) 1 − U(c) 1 ∥2 ≤ ∥ �U(c)† 0 − U(c)† 0 ∥2 + ∥U(c)† 0 ∥2∥ �U(c) − U(c)∥2 (52) where the last inequality follows from the fact that �U(c) 1 , �U(c) 1 − U(c) 1 are submatrices of �U(c) and �U(c) − U(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, we have ∥ �U(c) 1 ∥2 ≤ ∥ �U(c)∥2 = 1, and ∥ �U(c) 1 − U(c) 1 ∥2 ≤ ∥ �U(c) − U(c)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We use a result from [37, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='2] which states that a matrix F with rank S, and its perturbed matrix �F = F + �E satisfy the following inequality: ∥F† − �F†∥2 ≤ 3∥�E∥2/(σS(F)(σS(F) − ∥�E∥2)) provided ∥�E∥2 < σS(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From (51), and using the assumption of the lemma we have: ∥ �U(c) 0 − U(c) 0 ∥2 ≤ ∥ �U(c) − U(c)∥2 ≤ 2 √ 2∥EL∥2 σS(AUS(θ)PAUS(θ)H) − σ2 ≤ σS(U(c) 0 )/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (53) We can use the aforementioned result by substi- tuting F with U(c) 0 , and �F with �U(c) 0 : ∥( �U(c)† 0 − U(c)† 0 )∥2 ≤ 3∥( � U(c) 0 −U(c) 0 )∥2 σS(U(c) 0 )(σS(U(c) 0 )−∥ � U(c) 0 −U(c) 0 ∥2) ≤ 6∥ � U(c) 0 −U(c) 0 ∥2 σ2 S(U(c) 0 ) ≤ 6∥ � U(c)−U(c)∥2 σ2 S(U(c) 0 ) , where the second inequality follows from (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Combining this with (52), we get the final bound: ∥Ψ(c) − ˆΨ(c)∥2 ≤ 6∥( � U(c)−U(c))∥2 σ2 S(U(c) 0 ) + 1 σS(U(c) 0 )∥( �U(c) − U(c))∥2 ≤ 7∥( � U(c)−U(c))∥2 σ2 S(U(c) 0 ) ≤ 14 √ 2∥EL∥2 σ2 S(U(c) 0 )(σS(AUS(θ))PAUS(θ))H)−σ2) ≤ 14 √ 2∥EL∥2/σ2 S(βU(c) 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Next, we state the following Lemma from [20] that can be used to obtain a lower bound on σ2 S(U(c) 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Lemma 11 (Lemma 3, [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Let U(a) be any orthonormal basis for R(AUS(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Then the following holds: σ2 S(U(a) 0 ) ≥ max(1 − S σ2 S(AUS(θ)), 4−S) 13 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Define CS = 2−S 4 √ 2 and C′ S = 14π √ 2S3/24S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Under Assumption A3, these quantities are constants since S is held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' If β > 0 and the assump- tion ∥EL∥2 ≤ CSβ ensures that condition (49) holds since σS(U(c) 0 ) ≥ 2−S from Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Now, we can apply Lemma 10 to bound ∥Ψ(c) − ˆΨ(c)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We plug this bound on ∥Ψ(c) − ˆΨ(c)∥2 in (46): md(θ, �θ) ≤ 14 √ 2π S3/2q∥EL∥2 σ2 S(U(c) 0 )C′ S ≤ q∥EL∥2, where the last inequality follows from the bound σ2 S(U(c) 0 ) ≥ 4−S in Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Proof of Theorem 3 We will utilize Theorem 1 and Theorem 2 to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' One can see from Theorem 2 that under the assumptions β > 0 and ∥EL∥2 ≤ CSβ we can bound md(θ, �θ) ≤ q∥EL∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' For a given ϵ > 0, two cases arise: Case I (ϵ ≤ CSβq): In this case, min(CSβ, ϵ q) = ϵ/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' There- fore, ∥EL∥2 ≤ ϵ q ⇒ ∥EL∥2 ≤ CSβ, and from Theorem 2 the matching distance error is less than md(θ, �θ) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' This means P(md(θ, �θ) ≤ ϵ) ≥ P � ∥EL∥2 ≤ ϵ q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' From Theorem 1, we can obtain the following tail bound: P(∥EL∥2 ≤ ϵ q ) ≥ 1 − 8Mca exp � −c1L min � c2ϵ2 q2∥Ry∥2 2∆(S) , ϵ q∥Ry∥2 √ ∆(S) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (54) Case II (ϵ > CSβq): For values of ϵ satisfying ϵ > Csβq, we have min(CSβ, ϵ/q) = CSβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Therefore, if ∥EL∥2 ≤ CSβ, then from Theorem 2 we have md(θ, �θ) ≤ CSβq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' We obtain the following bound on the tail probability due to Theorem 1, P(∥EL∥2 ≤ CSβ) ≥ 1 − 8Mca exp � −c1L min � c2C2 Sβ2 ∥Ry∥2 2∆(S) , CSβ ∥Ry∥2 √ ∆(S) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' (55) If the number of snapshots L satisfy the following bound: L ≥ c3 ln � 8Mca δ � max � q2 1∆(S) c2ϵ2 , q1√ ∆(S) ϵ , L2 0 c2 , L0 � , where q1 = q∥Ry∥2, c3 = 1/c1 and L0 = ∥Ry∥2√ ∆(S) CSβ then combining (54) and (55) we obtain the following bound P � md(θ, �θ) ≤ min(ϵ, CSβq)) ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Pal and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Vaidyanathan, “Nested arrays: A novel approach to array processing with enhanced degrees of freedom,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4167–4181, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Vaidyanathan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Pal, “Sparse sensing with co-prime samplers and arrays,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 573– 586, Feb 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Haghighatshoar and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Caire, “Low-complexity massive mimo sub- space estimation and tracking from low-dimensional projections,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1832–1844, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Abramovich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Gorokhov, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Spencer, “Positive-definite Toeplitz completion in DOA estimation for nonuni- form linear antenna arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' fully augmentable arrays,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2458–2471, Sep 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Wang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Nehorai, “Coarrays, MUSIC, and the Cram´er–Rao bound,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 933–946, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Koochakzadeh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Pal, “Cram´er-Rao bounds for underdetermined source localization,” IEEE Signal Processing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 919–923, July 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liu and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Vaidyanathan, “Cram´er-Rao bounds for coprime and other sparse arrays, which find more sources than sensors,” Digital Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 61, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 43 – 61, Feb 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Shahsavari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Millhiser, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Pal, “Fundamental trade-offs in noisy super-resolution with synthetic apertures,” in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4620–4624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on antennas and prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 276–280, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Roy and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Kailath, “Esprit-estimation of signal parameters via rotational invariance techniques,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on acoustics, speech, and signal processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 984–995, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hua and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Sarkar, “Matrix pencil method for estimating pa- rameters of exponentially damped/undamped sinusoids in noise,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Acoustics, Speech, and Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 814–824, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Steinwandt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Roemer, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Haardt, “Performance analysis of esprit-type algorithms for co-array structures,” in 2017 IEEE 7th Interna- tional Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Vaccaro, “Performance analysis for doa estimation algorithms: unification, simplification, and observations,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Aero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' and Electronic Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1170–1184, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Sun and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Zhang, “4D automotive radar sensing for autonomous vehicles: A sparsity-oriented approach,” IEEE Journal of Selected Topics in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 879–891, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [15] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Tan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Eldar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Nehorai, “Direction of arrival estimation using co-prime arrays: A super resolution viewpoint,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 21, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5565–5576, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Cand`es and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fernandez-Granda, “Super-resolution from noisy data,” Journal of Fourier Analysis and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1229–1254, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [17] ——, “Towards a mathematical theory of super-resolution,” Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on pure and applied Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 906–956, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Qiao and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Pal, “Guaranteed localization of more sources than sensors with finite snapshots in multiple measurement vector models using difference co-arrays,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5715–5729, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Gu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Shi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Mao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Zhang, “Direction- of-arrival estimation for coprime array via virtual array interpolation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 22, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 5956–5971, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liao, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fannjiang, “Super-resolution limit of the esprit algorithm,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Li and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liao, “Stable super-resolution limit and smallest singular value of restricted fourier matrices,” Applied and Computational Har- monic Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 51, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 118–156, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Fannjiang, “Music for single-snapshot spectral es- timation: Stability and super-resolution,” Applied and Computational Harmonic Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 33–67, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [23] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liao, “Music for multidimensional spectral estimation: stability and super-resolution,” IEEE trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on signal processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 23, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 6395–6406, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Moitra, “Super-resolution, extremal functions and the condition number of vandermonde matrices,” in Proceedings of the 47th annual ACM symposium on Theory of computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' ACM, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 821–830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [25] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Gao, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liao, “Stability and super-resolution of music and esprit for multi-snapshot spectral estimation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' on Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4555–4570, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Liu and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Vaidyanathan, “Remarks on the spatial smoothing step in coarray music,” IEEE Signal Processing Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1438–1442, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 14 [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Eldar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Musco, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Musco, “Sample efficient toeplitz covariance estimation,” in Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' SIAM, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 378–397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Zygmund, Trigonometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Cambridge university press, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [29] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', “Toeplitz and circulant matrices: A review,” Founda- tions and Trends® in Communications and Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 155–239, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' O’Donnell, “Lecture 1: Asymptotics,” A Theorist’s Toolkit, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Batenkov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Demanet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Goldman, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Yomdin, “Conditioning of partial nonuniform fourier matrices with clustered nodes,” SIAM Journal on Matrix Analysis and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 199–220, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Aubel and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' B¨olcskei, “Vandermonde matrices with nodes in the unit disk and the large sieve,” Applied and Computational Harmonic Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 53–86, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [33] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hanson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Wright, “A bound on tail probabilities for quadratic forms in independent random variables,” The Annals of Math- ematical Statistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1079–1083, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Rudelson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Vershynin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=', “Hanson-wright inequality and sub-gaussian concentration,” Electronic Communications in Probability, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 18, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [35] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Stewart, “Matrix perturbation theory,” 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [36] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content='- ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Wedin, “Perturbation bounds in connection with singular value decomposition,” BIT Numerical Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 99– 111, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' Hansen, “The truncatedsvd as a method for regularization,” BIT Numerical Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} +page_content=' 534–553, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfxP5f/content/2301.01734v1.pdf'} diff --git a/JtAzT4oBgHgl3EQfVPz7/content/tmp_files/2301.01283v1.pdf.txt b/JtAzT4oBgHgl3EQfVPz7/content/tmp_files/2301.01283v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..17354c1629571f7cf24b21655a6ca0f45d5a3d01 --- /dev/null +++ b/JtAzT4oBgHgl3EQfVPz7/content/tmp_files/2301.01283v1.pdf.txt @@ -0,0 +1,1182 @@ +Cross Modal Transformer via Coordinates Encoding for 3D Object Dectection +Junjie Yan +Yingfei Liu +Jianjian Sun +Fan Jia +Shuailin Li +Tiancai Wang +Xiangyu Zhang +MEGVII Technology +Abstract +In this paper, we propose a robust 3D detector, named +Cross Modal Transformer (CMT), for end-to-end 3D multi- +modal detection. +Without explicit view transformation, +CMT takes the image and point clouds tokens as inputs +and directly outputs accurate 3D bounding boxes. +The +spatial alignment of multi-modal tokens is performed im- +plicitly, by encoding the 3D points into multi-modal fea- +tures. The core design of CMT is quite simple while its +performance is impressive. CMT obtains 73.0% NDS on +nuScenes benchmark. Moreover, CMT has a strong robust- +ness even if the LiDAR is missing. Code will be released at +https://github.com/junjie18/CMT. +1. Introduction +Multi-sensor fusion has shown its great superiority in au- +tonomous driving system [1,8,20,24,28]. Different sensors +usually provide the complementary information for each +other. For instance, the camera captures information in a +perspective view and the image contains rich semantic fea- +tures while point clouds provide much more localization +and geometry information. Taking full advantage of differ- +ent sensors helps reduce the uncertainty and makes accurate +and robust prediction. +Sensor data of different modalities usually has large +discrepancy in distribution, making it hard to merge the +multi-modalities. State-of-the-art (SoTA) methods tend to +fuse the multi-modality by constructing unified bird’s-eye- +view (BEV) representation [20,24,28] or querying from to- +kens [1,8]. For example, BEVFusion [28] explores a unified +representation by BEV transformation for BEV feature fu- +sion (see Fig. 1(a)). TransFusion [1] follows a two-stage +pipeline and the camera images in second stage provide +supplementary information for prediction refinement (see +Fig. 1(b)). However, exploring a truly end-to-end pipeline +for multi-sensor fusion remains to be a question. +Recently, the effectiveness of end-to-end object detec- +tion with transformer (DETR) [3, 56] has been proved in +many perception tasks, such as instance segmentation [10, +(a) BEVFusion +quries +PC +Transformer +image +quries +PC +Transformer +step1 +image +Transformer +topk +step2 +PC +image +BEV +Encoder +VT +(b) TransFusion +(c) CMT (Ours) +PE +Figure 1. +Comparison between BEVFusion, TransFusion, and +our proposed CMT. (a) In BEVFusion, the camera features are +transformed into BEV space by view transform. Two modality +features are concatenated in BEV space and the BEV encoder is +adopted for fusion. +(b) TransFusion first generates the queries +from the high response regions of LiDAR features. After that, ob- +ject queries interact with point cloud features and image features +separately. (c) In CMT, the object queries directly interact with +multi modality features simultaneously. Position encoding (PE) is +added to the multi-modal features for alignment. ”VT” is the view +transformation from image to 3D space. +12], multi-object tracking [30, 51] and visual 3D detec- +tion [26, 27, 45]. The DETR architecture is simple yet ef- +fective thanks to the object queries for representing different +instances and bipartite matching for one-to-one assignment. +Inspired by DETR, we aim to build an elegant end-to- +end pipeline for multi-modal fusion in 3D object detection. +In DETR, object queries directly interact with the image to- +kens through cross-attention in transformer decoder. For 3D +arXiv:2301.01283v1 [cs.CV] 3 Jan 2023 + +0 +10 +20 +30 +40 +50 +60 +70 +80 +NDS[%] +CMT +w/o Cams +w/o LiDAR +PETR +CMT-L +71.9 +67.7 +43.4 +45.5 +68.6 +drop +result +Figure 2. CMT has a strong robustness under sensor missing con- +dition. During inference, CMT without LiDAR achieves similar +detection performance compared to the SoTA camera-only detec- +tor PETR [26]. CMT without camera input only introduce a slight +drop, compared to our LiDAR-only baseline CMT-L. (Note: we +evaluate without any finetune process) +object detection, one intuitive way is to concatenate the im- +age and point cloud tokens together for further interaction +with object queries. However, the concatenated tokens are +disordered and unaware of their corresponding locations in +3D space. Therefore, it is necessary to provide the location +prior for multi-modal tokens and object queries. +In this paper, we propose Cross-Modal Transformer +(CMT), a simple yet effective end-to-end pipeline for high- +performance 3D object detection (see Fig. 1(c)). First, we +propose the Coordinates Encoding Module (CEM), which +produces position-aware features, by encoding 3D points +set implicitly into multi-modal tokens. +Specifically, for +camera images, 3D points sampled from frustum space are +used to indicate the probability of 3D positions for each +pixel. While for LiDAR, the BEV coordinates are simply +encoded into the point cloud tokens. Next, we introduce +the position-guided queries. Each query is initialized as a +3D reference point following PETR [26]. We transform the +3D coordinates of reference points to both image and Li- +DAR spaces, to perform the relative coordinates encoding +in each space. Moreover, for faster convergence, we in- +troduce the inductive bias of locality, by extending Query +Denoising [19] to a point-based formulation. +The proposed CMT framework brings many advantages +compared to existing methods. Firstly, our method is a sim- +ple and end-to-end pipeline and can be easily extended. The +3D positions are encoded into the multi-modal features im- +plicitly, which avoids introducing the bias caused by ex- +plicit cross-view feature alignment. Secondly, our method +only contains basic operations, without the feature sam- +pling or complex 2D-to-3D view transformation on multi- +modal features. Thirdly, the robustness of our CMT is much +stronger than other existing approaches. Extremely, under +the condition of LiDAR miss, our CMT with only image +tokens can achieve similar performance compared to those +visual 3D object detectors [23,26] (see Fig. 2). +To summarize, our contributions are: +• we propose a robust 3D detector, which is a truly end- +to-end framework without any post-process. It over- +comes the sensor missing problem. +• The 3D positions are implicitly encoded into the multi- +modal tokens, without any complex operations, like +grid sampling and voxel-pooling. +• CMT achieves state-of-the-art 3D detection perfor- +mance on nuScenes dataset. It provides a simple base- +line for future research. +2. Related Work +2.1. Camera Based 3D Object Detection +Camera-based 3D object detection is one of the basic +tasks in computer vision. Early works [41, 42] mainly fol- +low the dense prediction pipeline. They first localize the +objects on image plane and then predict their relevant 3D at- +tributes, such as depth, size and orientation. However, with +the surrounding cameras, the perspective-view based design +requires elaborate post-processes to eliminate the redundant +predictions of the overlapping regions. Recently, 3D ob- +ject detection under the BEV has attracted increasing atten- +tion. The BEV representation provides a unified coordinate +to fuse information from multiple camera views. LSS [32], +BEVDet [15] and BEVDepth [21] predict the depth distri- +bution to lift the image features to 3D frustum meshgrid. +Besides, inspired by DETR [4], DETR3D [45] and BEV- +Former [23] project the predefined BEV queries onto im- +ages and then employ the transformer attention to model +the relation of multi-view features. The above methods ex- +plicitly project the local image feature from 2D perspective +view to BEV. Different from them, PETR [26,27] and Spa- +tialDETR [9] adopt the positional embedding that depends +on the camera poses, allowing the transformer to implicitly +learn the projection from image views to 3D space. +2.2. LiDAR Based 3D Object Detection +LiDAR-based 3D object detection aims to predict 3D ob- +ject bounding boxes using the point clouds captured from +LiDAR. Existing methods process the point cloud into dif- +ferent representations. Point-based methods [22,33–36,49] +directly extract features from raw point clouds and pre- +dict 3D bounding boxes. PointNet [34] is the first archi- +tecture to process the point cloud in an end-to-end man- +ner, which preserves the spatial characteristics of the point +cloud. Other methods project the unordered, irregular Li- +DAR point clouds onto a regular feature space such as +3D voxels [54], feature pillars [17, 44, 50] and range im- +ages [11, 38]. Then the features are extracted in the BEV + +Figure 3. The architecture of Cross-Modal Transformer (CMT) paradigm. The multi-view images and point clouds are input to two back- +bone networks to extract feature tokens. In coordinates encoding module, coordinates of camera rays and BEV positions are transformed +into the image position encoding (Im PE) and point cloud position encoding (PC PE), respectively. The queries are generated by the +position-guided query generator. In query generator, 3D anchor points are projected to different modalities and the relative coordinates are +encoded (see the right part). Multi-modal tokens further interact with queries in the transformer decoder. The updated queries are further +used to predict the 3D bounding boxes. To accelerate the model convergence, the point-based query denoising is introduced. +plane using a standard 2D backbone. VoxelNet [54] first di- +vides the raw point clouds into regular voxel grids, and then +uses PointNet network to extract features from the points in +each voxel grid. +2.3. Multi-modal 3D Object Detection +Multi-sensor fusion in 3D detection has gained great at- +tention in recent years. +State-of-the-art (SoTA) methods +tend to find a unified representation for both modalities, or +define object queries to fuse the features for further predic- +tion. For example, BEVFusion [24, 28] applies a lift-splat- +shoot (LSS) operation to project image feature onto BEV +space and concatenates it with LiDAR feature. UVTR [20] +generates a unified representation in the 3D voxel space by +deformable attention [56]. While for query-based methods, +FUTR3D [8] defines the 3D reference points as queries and +directly samples the features from the coordinates of pro- +jected planes. TransFusion [1] follows a two-stage pipeline. +The proposals are generated by LiDAR features and further +refined by querying the image features. +2.4. Transformer-based Object Detection +The pioneering work DETR [3] proposes a transformer- +based detector paradigm without any hand-craft compo- +nents, and has achieved state-of-the-arts in both 2D and +3D detection [6, 23, 27, 53]. However, DETR-like meth- +ods usually suffer from the slow convergence. To this end, +many works [5, 16, 19, 25, 52, 53, 56] are proposed to im- +prove the training efficiency from various aspects. Other +improvements in 2D detection mainly focus on modifying +the transformer layers [52,56], designing informative object +queries [19,25,53], or exploring the label assignment mech- +anism [5, 16]. +Deformable DETR [56] proposes the de- +formable attention, which only attends to sampling points of +local regions. SAM-DETR [52] presents a semantic aligner +between object queries and encoded features to accelerate +the matching process. To alleviate the instability of bipartite +matching, DAB-DETR [25] formulates the object queries +as dynamic anchor boxes, while DN-DETR [19] auxillarily +reconstructs the ground-truths from the noisy ones. Based +on them, DINO [53] further improves the denoising anchor +boxes via a contrastive way. +3. Method +The overall architecture of the proposed CMT is illus- +trated in Fig. 3. Multi-view images and LiDAR points are +fed into two individual backbones to extract multi-modal +tokens. +The 3D coordinates are encoded into the multi- +modal tokens by the coordinates encoding. +The queries +from the position-guided query generator are used to in- +teract with the multi-modal tokens in transformer decoder +and then predict the object class as well as the 3D bounding +boxes. Point-based query denoising is further introduced +to accelerate the training convergence by introducing local +prior. The whole framework is learned in a fully end-to- +end manner and LiDAR backbone is trained from scratch +without pretraining. +3.1. Coordinates Encoding Module +The coordinates encoding module (CEM) is used to en- +code the 3D position information into multi-modal tokens. +It generates both the camera and BEV position encodings +(PEs), which are added to image tokens and point cloud +tokens respectively. With the help of CEM, multi-modal +tokens can be implicitly aligned in 3D space. +Let P(u, v) be the 3D points set corresponding to the + +feature map F(u, v) of different modalities. Here (u, v) in- +dicates the coordinate in the feature map. Specifically, F is +the image feature for camera while BEV feature for LiDAR. +Suppose the output position embedding of CEM is Γ(u, v), +its calculation can be formulated as: +Γ(u, v) = ψ(P(u, v)) +(1) +where ψ is a multi-layer perception (MLP) layer. +CE for Images. Since the image is captured from a per- +spective view, each pixel can be seen as an epipolar line +in 3D space. Inspired by PETR [26], for each image, we +encode a set of points in camera frustum space to per- +form the coordinates encoding. Given the image feature +Fim, each pixel can be formulated as a series of points +{pk(u, v) = (u ∗ dk, v ∗ dk, dk, 1)T , k = 1, 2, ..., d} in the +camera frustum coordinates. Here, d is the number of points +sampled along the depth axis. The corresponding 3D points +can be calculated by: +pim +k (u, v) = T l +ciK−1 +i +pk(u, v) +(2) +where T l +ci ∈ R4×4 is the transformation matrix from the i- +th camera coordinate to the LiDAR coordinate. Ki ∈ 4 × 4 +is the intrinsic matrix of i-th camera. The position encoding +of pixel (u, v) for image is formulated as: +Γim(u, v) = ψim({pim +k (u, v), +k = 1, 2, ..., d}) +(3) +CE for Point Clouds. +We choose VoxelNet [48, 54] or +PointPillar [17] as backbone to encode the point cloud to- +kens Fpc. +Intuitively, the point set P in Eq. (1) can be +sampled along the Z-axis. Suppose (u, v) is the coordi- +nates in BEV feature map, the sampled point set is then +pk(u, v) = (u, v, hk, 1)T , where hk indicates the height of +k-th points and h0 = 0 as default. The corresponding 3D +points of BEV feature map can be calculated by: +ppc +k (u, v) =(u ∗ ud, v ∗ vd, hk, 1) +(4) +where (ud, vd) is the size of each BEV feature grid. To +simplify, we only sample one point along the height axis. It +is equivalent to the 2D coordinate encoding in BEV space. +The position embedding of point cloud can be obtained by: +Γpc(u, v) = ψpc({ppc +k (u, v), +k = 1, 2, ..., h}) +(5) +3.2. Position-guided Query Generator +Following Anchor-DETR [46] and PETR [26], we firstly +initialize the queries with n anchor points A = {ai = +(ax,i, ay,i, az,i), i = 1, 2, ..., n} sampled from uniform dis- +tribution between [0, 1]. Then these anchor points are trans- +formed into 3D world space by linear transformation: +� +� +� +� +� +ax,i = +ax,i∗(xmax − xmin) + xmin +ay,i = +ay,i∗(ymax − ymin) + ymin +az,i = +az,i∗(zmax − zmin) + zmin +(6) +box center +anchor point +add +noise +noisy query +decoder +box center +anchor point +add +noise +noisy query +decoder +None +positive +negative +Figure 4. Illustration of the proposed point-based query denoising. +The noise queries are generated from the box center of ground- +truths. The positive and negative queries are split by the noise +scale. Positive queries are used to reconstruct the ground-truths +boxes, while negative queries predict the “no object”. +where [xmin, ymin, zmin, xmax, ymax, zmax] is the region +of interest (RoI) of 3D world space. After that, we project +the 3D anchor points A to different modalities and encode +the corresponding point sets by CEM. Then the positional +embedding Γq of object queries can be generated by: +Γq = ψpc(Apc) + ψim(Aim) +(7) +where Apc and Aim are the point set projected on BEV +plane and image plane, respectively. The positional embed- +ding Γq are further added with the query content embedding +to generate the initial position-guided queries Q0. +3.3. Point-based Query Denoising (PQD) +For fast convergence, we extend the query denoising +strategy in DN-DETR [19] to 3D object detection as shown +in Fig. 4. Different from DN-DETR [19], we generate the +noisy anchor points by center shifting since the box scale +is not that important in 3D object detection. +For each +3D ground-truths box (x, y, z, w, l, h, θ), we first sample +the random ratio λ1, λ2, λ3 within (−λ, λ), where λ is the +hyper-parameter to control the noise scale. Since 3D space +is sparse and unobstructed, we adopt a larger tolerance (e.g. +λ = 1.0). Then the center noise (∆x, ∆y, ∆z) can be cal- +culated as: +∆x = λ1w +2 , ∆y = λ2l +2 , ∆z = λ3h +2 . +(8) +The center noise is added to the center of ground-truths +to obtain the noise anchor points. +Then each noise an- +chor point can be converted into noise query, as described +in the last section. Inspired by DINO [53], we also intro- +duce the negative noisy queries to predict the “no object”. +To simplify the pipeline, the positive and negative queries +are simply split by the random ratios λ1, λ2, λ3 and a given +threshold ξ. For each noise query, it is a positive query if +� +λ2 +1 + λ2 +2 + λ2 +3 < ξ, otherwise a negative query. + +Table 1. Performance comparison on the nuScenes test set. “L” is LiDAR and “C” is camera. +Methods +Modality NDS↑ mAP↑ mATE↓ mASE↓ mAOE↓ mAVE↓ mAAE↓ +BEVDet [15] +C +0.488 0.424 +0.524 +0.242 +0.373 +0.950 +0.148 +DETR3D [45] +C +0.479 0.412 +0.641 +0.255 +0.394 +0.845 +0.133 +PETR [26] +C +0.504 0.441 +0.593 +0.249 +0.383 +0.808 +0.132 +CenterPoint [50] +L +0.673 0.603 +0.262 +0.239 +0.361 +0.288 +0.136 +UVTR [20] +L +0.697 0.639 +0.302 +0.246 +0.350 +0.207 +0.123 +TransFusion [1] +L +0.702 0.655 +0.256 +0.240 +0.351 +0.278 +0.129 +PointPainting [39] +LC +0.610 0.541 +0.380 +0.260 +0.541 +0.293 +0.131 +PointAugmenting [40] +LC +0.711 0.668 +0.253 +0.235 +0.354 +0.266 +0.123 +MVP [7] +LC +0.705 0.664 +0.263 +0.238 +0.321 +0.313 +0.134 +FusionPainting [47] +LC +0.716 0.681 +0.256 +0.236 +0.346 +0.274 +0.132 +UVTR [20] +LC +0.711 0.671 +0.306 +0.245 +0.351 +0.225 +0.124 +TransFusion [1] +LC +0.717 0.689 +0.259 +0.243 +0.359 +0.288 +0.127 +BEVFusion [28] +LC +0.729 0.702 +0.261 +0.239 +0.329 +0.260 +0.134 +CMT-L +L +0.701 0.646 +0.298 +0.242 +0.330 +0.222 +0.124 +CMT +LC +0.730 0.704 +0.299 +0.241 +0.323 +0.240 +0.112 +Table 2. Performance comparison on the nuScenes val set. “L” is +LiDAR and “C” is camera. +Methods +modality +NDS↑ +mAP↑ +FUTR3D [8] +L +0.655 +0.593 +UVTR [20] +L +0.676 +0.608 +TransFusion [1] +L +0.702 +0.655 +FUTR3D [8] +LC +0.683 +0.645 +UVTR [20] +LC +0.702 +0.654 +TransFusion [1] +LC +0.713 +0.675 +BEVFusion [28] +LC +0.714 +0.685 +CMT-L +L +0.686 +0.624 +CMT +LC +0.719 +0.694 +3.4. Decoder and Loss +As for the decoder, we follow the original transformer +decoder in DETR [46] and use L decoder layers. For each +decoder layer, the position-guided queries interact with the +multi-modal tokens and update their representations. Two +feed-forward networks (FFNs) are used to predict the 3D +bounding boxes and the classes using updated queries. We +formulate the prediction process of each decoder layer as +follows: +ˆbi = Ψreg(Qi), ˆci = Ψcls(Qi), +(9) +where Ψreg and Ψcls respectively represent the FFN for +regression and classification. Qi is the the updated object +queries of the i-th decoder layer. +For set prediction, the bipartite matching is applied for +one-to-one assignment between predictions and ground- +Figure 5. We analyze the system robustness of CMT at test period +under three simulated sensor errors: (a) single camera miss, (b) all +camera miss and (c) LiDAR miss. +truths. We adopt the focal loss for classification and L1 +loss for 3D bounding box regression: +L(y, ˆy) = ω1Lcls(c, ˆc) + ω2Lreg(b,ˆb) +(10) +where ω1 and ω2 are the hyper-parameter to balance the two +loss terms. Note that for positive and negative queries in +query denoising, the loss is calculated in the same way. +3.5. Masked-Modal Training for Robustness +Security is the most important concern for autonomous +driving systems. +An ideal system requires solid perfor- +mance even if part of them fails, as well as not relying on +any input of a specific modality. Recently, BEVFusion [24] +has explored the robustness of LiDAR sensor failure. How- +ever, the exploration is limited to restricted scan range and +model need be retrained. In this paper, we try more extreme +failures, including single camera miss, camera miss and Li- +DAR miss, as shown in Fig. 5. It is consistent with the +actual scene and ensures the safety of autonomous driving. + +Table 3. Quantitative results on the nuScenes val with LiDAR or camera miss. With the masked-modal training, the efficacy and robustness +of our CMT is significantly improved, especially when the LiDAR camera is missed. +Metric +Vanilla training +Masked-modal training +CMT +only LiDAR +only Cams +CMT +only LiDAR +only Cams +NDS ↑ +0.716 +0.594 +0.067 +0.719 (↑0.3%) +0.677 (↑8.3%) +0.434 (↑36.7%) +mAP ↑ +0.685 +0.472 +0.000 +0.694 (↑0.9%) +0.613 (↑14.1%) +0.386 (↑38.6%) +To improve the robustness of the model, we propose a +training strategy, called masked-modal training. In training +process, we randomly use only a single modality for train- +ing, such as camera or LiDAR, with the ratio of η1 and η2. +This strategy ensures that the model are fully trained with +both single modal and multi-modal. Then the model can be +tested with single modal or multi-modal, without modify- +ing the model weight. The experimental results show that +masked-modal training will not affect the performance of +our fusion model. Even if LiDAR is damaged, it can still +achieve similar performance compared to SoTA range-view +3D detectors [15,26]. +4. Experiments +4.1. Datasets and Metrics +We evaluate our method on nuScenes [2]. nuScenes is a +large-scale multi-modal dataset, which is composed of data +from 6 cameras, 1 LiDAR and 5 radars. The dataset has +1000 scenes totally and is divided into 700/150/150 scenes +as train/validation/test sets, respectively. +Cameras. Each scene has 20s video frames with 12 +FPS. 3D bounding boxes are annotated every 0.5s. We only +use these key frames. In each frame, nuScenes provides +images from six cameras. +LiDAR. NuScenes provides a 32-beam LiDAR with 20 +FPS. The key frames are also annotated every 0.5s, the same +as cameras. We follow the common practice to transform +the points from the past 9 frames to the current frame for +training and evaluation. +Metrics. We follow the nuScenes official metrics. We +report the nuScenes Detection Score (NDS), mean Average +Precision (mAP), mean Average Translation Error (mATE), +mean Average Scale Error (mASE), mean Average Orien- +tation Error(mAOE), mean Average Velocity Error (mAVE) +and mean Average Attribute Error (mAAE). +4.2. Implementation Details +We use ResNet [13] or VoVNet [18] as image backbone +to extract the 2D image features. The C5 feature is up- +sampled and fused with C4 feature to produce P4 feature. +We use VoxelNet [54] or PointPillars [17] as the backbone +to extract the point-cloud features. We set the region-of- +interest (RoI) to [−54.0m, 54.0m] for X and Y axis, and +[−5.0m, 3.0m] for Z axis. The 3D coordinates in the world +space are normalized to [0, 1]. All the feature dimension +is set to 256, including the LiDAR feature, image feature +and query embedding. Six decoder layers are adopted in +transformer decoder. Voxel size of 0.075 and image size of +1600 × 640 are adopted as default in our experiments. +Our model is trained with the batch size of 16 on 8 A100 +GPUs. It is trained for total 20 epochs with CBGS [55]. We +adopt the AdamW [29] optimizer for optimization. The ini- +tial learning rate is 1.0×10−4 and we follow the cycle learn- +ing rate policy [37]. The mask ratios η1 and η2 are both set +to 0.25 for masked-modal training. The threshold ξ is set to +0.75 to divide the noise queries into positives and negatives +for training. The tolerance λ that controls the noise scale is +set to 1. The GT sample augmentation is employed for the +first 15 epochs and closed for the rest epochs. As for the +loss weights, we follow the default setting in DETR3D [45] +and set the ω1 and ω2 to 2.0 and 0.25, respectively. +4.3. State-of-the-Art Comparison +As shown in Tab. 1, CMT achieves comparable results +compared to several state-of-the-art methods on nuScenes +test set. Our LiDAR-only baseline, named CMT-L, achieves +the 70.1% NDS, which is a nearly SoTA performance +among all existing LiDAR-only methods. Our multi-modal +method CMT achieves 73.0% NDS and 70.4% mAP and +outperforms existing SoTA BEVFusion. Benefits from the +large receptive field, CMT gains better results on some met- +rics like mAVE. We also compare the performance with +other SoTA methods on nuScenes val set (see Tab. 2). +It shows that our proposed CMT with multi-modal fu- +sion outperforms the BEVFusion by 0.5% NDS and 0.9% +mAP. CMT introduces large performance improvements +compared to our LiDAR-only CMT-L by 2.9%/5.8% and +3.3%/7.0% NDS/mAP on test and validation set, showing +that camera images bring complementary information. +4.4. Strong Robustness +We evaluate the robustness of our framework under var- +ious harsh environments, including LiDAR miss and cam- +era miss. Tab. 3 shows the results when the sensor miss +occurs, by simulating the scenarios of any modality totally +broken. The performance is compared between the vanilla +training and masked-modal training. It validates the effect + +Table 4. The ablation studies of different components in the proposed CMT. +Im +PC NDS mAP mATE mASE mAOE +✓ +0.595 0.554 0.515 0.258 +0.429 +✓ 0.665 0.626 0.372 0.255 +0.347 +✓ +✓ 0.669 0.641 0.377 0.254 +0.375 +(a) Position encoding for query. +PQD +NDS mAP mATE mASE mAOE +0.626 0.584 0.429 0.259 +0.420 +✓ +0.669 0.641 0.377 0.254 +0.375 +(b) Point-based Query Denoise (PQD) +NDS mAP mATE mASE mAOE +0.075 +0.669 0.641 0.377 0.254 +0.375 +0.1 +0.671 0.638 0.378 0.252 +0.334 +0.125 +0.655 0.624 0.396 0.255 +0.397 +(c) Voxel size of LiDAR backbone. +NDS mAP mATE mASE mAOE +ResNet-50 +0.658 0.623 0.376 0.253 +0.399 +ResNet-101 0.664 0.629 0.383 0.254 +0.363 +VoV-99 +0.669 0.641 0.377 0.254 +0.375 +(d) Image backbone. +NDS mAP mATE mASE mAOE +800 × 320 0.654 0.609 0.374 0.256 +0.389 +1600 × 640 0.669 0.641 0.377 0.254 +0.375 +(e) Input size of image backbone. +NDS mAP mATE mASE mAOE +PointPillars 0.628 0.598 0.430 0.252 +0.455 +VoxelNet +0.669 0.641 0.377 0.254 +0.375 +(f) Lidar backbone +of masked-modal training. Note that the model are only +trained with multi-modality and evaluated without any fine- +tune process. With vanilla training, the model fails to pre- +dict anything meaningful (only Cams with mAP=0) when +LiDAR is missing. With masked-modal training, the ab- +sence of LiDAR or camera modalities lead to 4.2% and +28.5% NDS drop compared to CMT, respectively. +It is +observed that losing one modality still remains similar re- +sults compared to single-modal training settings. It over- +comes the drawback that multi-modal method usually rely +on one major modality and performance would degrade sig- +nificantly if losing the major modality. Especially, for the +case of LiDAR missing, the performance is still compara- +ble to the SoTA camera-only method PETR [26], validating +the strong robustness of our method. +Moreover, we also investigate the case when any one +of cameras fails. Experimental result shows slight perfor- +mance drop, indicating the tolerable to single camera miss +of our method. Six sensors brings an average decrease of +0.7% NDS, no more than 1% performance of the oracle ver- +sion. The front and back sensor relatively play the important +role among camera sensors, with 1.1% and 0.8% decrease +respectively, due to their distant or large field of view. Com- +pared to the camera-only setting, our multi-modal frame- +work facilitate the compensation between LiDAR and im- +age domains, thus presenting a robust performance. +4.5. Ablations +We present the ablation studies in Tab. 4. All the experi- +ments are conducted for 20 epochs without CBGS [31]. +We first ablate the effect of Im PE and PC PE on the gen- +eration of position-guided queries (see Tab. 4 (a)). It shows +that removing PC PE introduces a 7.4%/8.70% NDS/mAP +performance drop, which is much larger than the drop of re- +moving Im PE 0.4%/1.5%. Next, we explore the effective- +ness of our proposed PQD, as shown in Tab. 4(b). We can +easily find that PQD can greatly improve the overall perfor- +mance by 4.3%/5.7% NDS/mAP. With PQD, the training +convergence can be boosted, which is similar to the practice +in DN-DETR [19]. Further, Tab. 4 (c-f) illustrates the effect +of scaling up the CMT model as well as the input size. Over- +all, CMT can benefit from the scaling model size. Interest- +ingly, we find increasing the voxel number (smaller voxel +size) and image size achieves similar improvements ≈ 1.5% +in NDS. While scaling the image size increases more mAP +than the voxel number(+3.2% vs. +1.7%). When increasing +the image size from 800 × 320 to 1600 × 640, we find the +performance improvements are mainly from these small ob- +jects, such as pedestrian and motorcycle. We also conduct +experiments on replacing image and LiDAR backbones, +we use VoV-99 [18] and ResNet [13] as our image back- +bones. Experiments show that our proposed CMT can ben- +efit from larger backbones. For image, VoV-99 backbone +achieves the best result and outperforms the ResNet-50 by +1.1%/1.8% in NDS/mAP. While for LiDAR, VoxelNet out- +performs the PointPillar by 4.1%/4.3% in NDS/mAP. +4.6. Analysis +CMT is a direct and easy pipeline for multi-modal fu- +sion and can be easily extended. +Moreover, benefiting +from DETR [3] framework and our training schedule, CMT +shows strong robustness under sensor miss conditions. We +present some attempted experiments in this section. +Data extension. Multi-frame is now a common setting in + +Figure 6. Visualization of attention maps on multi-view images and BEV point clouds. The blue points (•) are the initialized anchor points +while red points (•) are the corresponding centers of box predictions. It can be easily found that the high response regions of attention +maps mainly focus on the foreground objects, closest to the anchor points. +camera-based 3D object detection [14,23,27]. Using multi +frames often outperforms the single frame by a clear margin +and can solve some typical occlusion problem. We follow +the multi-frame alignment in PETRv2 [27]. Considering +the high memory cost of multi-frames, we conduct our ex- +periment with a 800 × 320 image resolution. As shown in +Tab. 5, adding image frame only improves the NDS/mAP +by 0.2%/0.7%. More image frames rarely improve the per- +formance with multi-modal fusion. +Radar has advantages in long range detection and the ro- +bustness of extreme weather. Following FUTR3D [8], we +stack points from 5 radars together to generate the point +cloud. We use several MLP layers to perform coordinates +encoding on Radar features, the same as LiDAR. Tab. 5 +shows that adding Radar data to our pipeline degrades the +performance by 0.9% NDS and 0.6% mAP. +Visualization. For better understanding on querying from +multi-modal tokens, we visualize the attention map of +cross-attention on different modalities (see Fig. 6). We can +clearly find that the attention maps have higher response on +images and point clouds. It shows that our method can im- +plicitly achieve the cross-modal interaction. We visualize +the initial anchor points and the center points of predictions. +Most anchor points focus on the closest foreground objects. +After the interaction with queries in decoder, anchor points +gradually access the accurate center points. +Table 5. Results with more image frames or with Radar points. ++Frame ++ Radar +NDS +mAP +0.698 +0.662 +✓ +0.700 +0.669 +✓ +0.689 +0.656 +5. Conclusions +In this paper, we propose a fully end-to-end framework +for multi-modal 3D object detection. It implicitly encodes +the 3D coordinates into the tokens of images and point +clouds. With the coordinates encoding, the simple yet effec- +tive DETR pipeline can be adopted for multi-modal fusion +and end-to-end learning. With masked-modal training, our +multi-modal detector can be learned with strong robustness, +even if one of multi-modalities are missed. We hope such +a simple pipeline design could provide more insights on the +end-to-end 3D object detection. +Limitation: Though our CMT brings some advantages +compared to those existing approaches, it also reveals some +limitations. The computation cost is relatively larger due +to the large number of multi-modal tokens and the global +attention employed in transformer decoder. To solve this +problem, some efforts in two directions maybe taken. The +first one is to reduce the redundancy of multi-modal tokens. +The foreground tokens can be roughly selected with another +individual network [43]. The foreground tokens are then in- +put to our network for high-speed inference. Another pos- +sible solution is to replace the global attention with other +efficient attentions, like deformable attention [56]. One can +also employ a small set of object queries since most queries +correspond to the empty objects. +References +[1] Xuyang Bai, Zeyu Hu, Xinge Zhu, Qingqiu Huang, Yilun +Chen, Hongbo Fu, and Chiew-Lan Tai. Transfusion: Robust +lidar-camera fusion for 3d object detection with transform- +ers. In Proceedings of the IEEE/CVF Conference on Com- +puter Vision and Pattern Recognition, pages 1090–1099, +2022. 1, 3, 5 + +[2] Holger Caesar, Varun Bankiti, Alex H Lang, Sourabh Vora, +Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Gi- +ancarlo Baldan, and Oscar Beijbom. +nuscenes: A multi- +modal dataset for autonomous driving. In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 11621–11631, 2020. 6 +[3] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas +Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to- +end object detection with transformers. In European confer- +ence on computer vision, pages 213–229. Springer, 2020. 1, +3, 7 +[4] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas +Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to- +end object detection with transformers. In European confer- +ence on computer vision, pages 213–229. Springer, 2020. 2 +[5] Qiang Chen, Xiaokang Chen, Gang Zeng, and Jingdong +Wang. +Group detr: Fast training convergence with de- +coupled one-to-many label assignment. +arXiv preprint +arXiv:2207.13085, 2022. 3 +[6] Qiang Chen, Jian Wang, Chuchu Han, Shangang Zhang, +Zexian Li, Xiaokang Chen, Jiahui Chen, Xiaodi Wang, +Shumin Han, Gang Zhang, Haocheng Feng, Kun Yao, Junyu +Han, Errui Ding, and Jingdong Wang. Group detr v2: Strong +object detector with encoder-decoder pretraining. 2022. 3 +[7] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. +Multi-view 3d object detection network for autonomous +driving. In Proceedings of the IEEE conference on Computer +Vision and Pattern Recognition, pages 1907–1915, 2017. 5 +[8] Xuanyao Chen, Tianyuan Zhang, Yue Wang, Yilun Wang, +and Hang Zhao. Futr3d: A unified sensor fusion framework +for 3d detection. arXiv preprint arXiv:2203.10642, 2022. 1, +3, 5, 8 +[9] Simon Doll, Richard Schulz, Lukas Schneider, Viviane Ben- +zin, Markus Enzweiler, and Hendrik Lensch. +Spatialdetr: +Robust scalable transformer-based 3d object detection from +multi-view camera images with global cross-sensor atten- +tion. In European Conference on Computer Vision, pages +230–245. Springer, 2022. 2 +[10] Bin Dong, Fangao Zeng, Tiancai Wang, Xiangyu Zhang, +and Yichen Wei. +Solq: Segmenting objects by learning +queries. +Advances in Neural Information Processing Sys- +tems, 34, 2021. 1 +[11] Lue Fan, Xuan Xiong, Feng Wang, Naiyan Wang, and +Zhaoxiang Zhang. +Rangedet: In defense of range view +for lidar-based 3d object detection. In Proceedings of the +IEEE/CVF International Conference on Computer Vision, +pages 2918–2927, 2021. 2 +[12] Yuxin Fang, Shusheng Yang, Xinggang Wang, Yu Li, Chen +Fang, Ying Shan, Bin Feng, and Wenyu Liu. Instances as +queries. Proc. IEEE Conf. Comp. Vis. Patt. Recogn., 2021. 1 +[13] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. +Deep residual learning for image recognition. In Proceed- +ings of the IEEE conference on computer vision and pattern +recognition, pages 770–778, 2016. 6, 7 +[14] Junjie Huang and Guan Huang. Bevdet4d: Exploit tempo- +ral cues in multi-camera 3d object detection. arXiv preprint +arXiv:/2203.17054, 2021. 8 +[15] Junjie Huang, Guan Huang, Zheng Zhu, and Dalong Du. +Bevdet: High-performance multi-camera 3d object detection +in bird-eye-view. arXiv preprint arXiv:2112.11790, 2021. 2, +5, 6 +[16] Ding Jia, Yuhui Yuan, Haodi He, Xiaopei Wu, Haojun Yu, +Weihong Lin, Lei Sun, Chao Zhang, and Han Hu. Detrs with +hybrid matching. arXiv preprint arXiv:2207.13080, 2022. 3 +[17] Alex H Lang, Sourabh Vora, Holger Caesar, Lubing Zhou, +Jiong Yang, and Oscar Beijbom. Pointpillars: Fast encoders +for object detection from point clouds. In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 12697–12705, 2019. 2, 4, 6 +[18] Youngwan Lee and Jongyoul Park. +Centermask: +Real- +time anchor-free instance segmentation. In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 13906–13915, 2020. 6, 7 +[19] Feng Li, Hao Zhang, Shilong Liu, Jian Guo, Lionel M Ni, +and Lei Zhang. Dn-detr: Accelerate detr training by intro- +ducing query denoising. In Proceedings of the IEEE/CVF +Conference on Computer Vision and Pattern Recognition, +pages 13619–13627, 2022. 2, 3, 4, 7 +[20] Yanwei Li, Yilun Chen, Xiaojuan Qi, Zeming Li, Jian +Sun, and Jiaya Jia. +Unifying voxel-based representation +with transformer for 3d object detection. +arXiv preprint +arXiv:2206.00630, 2022. 1, 3, 5 +[21] Yinhao Li, Zheng Ge, Guanyi Yu, Jinrong Yang, Zengran +Wang, Yukang Shi, Jianjian Sun, and Zeming Li. Bevdepth: +Acquisition of reliable depth for multi-view 3d object detec- +tion. arXiv preprint arXiv:2206.10092, 2022. 2 +[22] Zhichao Li, Feng Wang, and Naiyan Wang. Lidar r-cnn: An +efficient and universal 3d object detector. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pages 7546–7555, 2021. 2 +[23] Zhiqi Li, Wenhai Wang, Hongyang Li, Enze Xie, Chong- +hao Sima, Tong Lu, Qiao Yu, and Jifeng Dai. Bevformer: +Learning bird’s-eye-view representation from multi-camera +images via spatiotemporal transformers. +arXiv preprint +arXiv:2203.17270, 2022. 2, 3, 8 +[24] Tingting Liang, Hongwei Xie, Kaicheng Yu, Zhongyu Xia, +Zhiwei Lin, Yongtao Wang, Tao Tang, Bing Wang, and Zhi +Tang. Bevfusion: A simple and robust lidar-camera fusion +framework. arXiv preprint arXiv:2205.13790, 2022. 1, 3, 5 +[25] Shilong Liu, Feng Li, Hao Zhang, Xiao Yang, Xianbiao Qi, +Hang Su, Jun Zhu, and Lei Zhang. +Dab-detr: Dynamic +anchor boxes are better queries for detr. +arXiv preprint +arXiv:2201.12329, 2022. 3 +[26] Yingfei Liu, Tiancai Wang, Xiangyu Zhang, and Jian Sun. +Petr: Position embedding transformation for multi-view 3d +object detection. arXiv preprint arXiv:2203.05625, 2022. 1, +2, 4, 5, 6, 7 +[27] Yingfei Liu, Junjie Yan, Fan Jia, Shuailin Li, Qi Gao, Tian- +cai Wang, Xiangyu Zhang, and Jian Sun. Petrv2: A uni- +fied framework for 3d perception from multi-camera images. +arXiv preprint arXiv:2206.01256, 2022. 1, 2, 3, 8 +[28] Zhijian Liu, Haotian Tang, Alexander Amini, Xinyu Yang, +Huizi Mao, Daniela Rus, and Song Han. Bevfusion: Multi- +task multi-sensor fusion with unified bird’s-eye view repre- +sentation. arXiv preprint arXiv2205.13542, 2022. 1, 3, 5 + +[29] Ilya Loshchilov and Frank Hutter. Decoupled weight decay +regularization. arXiv preprint arXiv:1711.05101, 2017. 6 +[30] Tim Meinhardt, Alexander Kirillov, Laura Leal-Taixe, and +Christoph Feichtenhofer. Trackformer: Multi-object track- +ing with transformers. +arXiv preprint arXiv:2101.02702, +2021. 1 +[31] Chao Peng, Tete Xiao, Zeming Li, Yuning Jiang, Xiangyu +Zhang, Kai Jia, Gang Yu, and Jian Sun. Megdet: A large +mini-batch object detector. In Proceedings of the IEEE con- +ference on Computer Vision and Pattern Recognition, pages +6181–6189, 2018. 7 +[32] Jonah Philion and Sanja Fidler. Lift, splat, shoot: Encoding +images from arbitrary camera rigs by implicitly unprojecting +to 3d. In European Conference on Computer Vision, pages +194–210. Springer, 2020. 2 +[33] Charles R Qi, Wei Liu, Chenxia Wu, Hao Su, and Leonidas J +Guibas. Frustum pointnets for 3d object detection from rgb- +d data. In Proceedings of the IEEE conference on computer +vision and pattern recognition, pages 918–927, 2018. 2 +[34] Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. +Pointnet: Deep learning on point sets for 3d classification +and segmentation. In Proceedings of the IEEE conference +on computer vision and pattern recognition, pages 652–660, +2017. 2 +[35] Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J +Guibas. Pointnet++: Deep hierarchical feature learning on +point sets in a metric space. Advances in neural information +processing systems, 30, 2017. 2 +[36] Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. Pointr- +cnn: 3d object proposal generation and detection from point +cloud. In Proceedings of the IEEE/CVF conference on com- +puter vision and pattern recognition, pages 770–779, 2019. +2 +[37] Leslie N Smith. Cyclical learning rates for training neural +networks. In 2017 IEEE winter conference on applications +of computer vision (WACV), pages 464–472. IEEE, 2017. 6 +[38] Pei Sun, Weiyue Wang, Yuning Chai, Gamaleldin El- +sayed, Alex Bewley, Xiao Zhang, Cristian Sminchisescu, +and Dragomir Anguelov. +Rsn: Range sparse net for effi- +cient, accurate lidar 3d object detection. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern +Recognition, pages 5725–5734, 2021. 2 +[39] Sourabh Vora, Alex H Lang, Bassam Helou, and Oscar Bei- +jbom. +Pointpainting: Sequential fusion for 3d object de- +tection. +In Proceedings of the IEEE/CVF conference on +computer vision and pattern recognition, pages 4604–4612, +2020. 5 +[40] Chunwei Wang, Chao Ma, Ming Zhu, and Xiaokang Yang. +Pointaugmenting: Cross-modal augmentation for 3d object +detection. In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pages 11794– +11803, 2021. 5 +[41] Tai Wang, ZHU Xinge, Jiangmiao Pang, and Dahua Lin. +Probabilistic and geometric depth: Detecting objects in per- +spective. In Conference on Robot Learning, pages 1475– +1485. PMLR, 2022. 2 +[42] Tai Wang, Xinge Zhu, Jiangmiao Pang, and Dahua Lin. +Fcos3d: Fully convolutional one-stage monocular 3d object +detection. +In Proceedings of the IEEE/CVF International +Conference on Computer Vision, pages 913–922, 2021. 2 +[43] Yulin Wang, Zhaoxi Chen, Haojun Jiang, Shiji Song, Yizeng +Han, and Gao Huang. +Adaptive focus for efficient video +recognition. In Proceedings of the IEEE/CVF International +Conference on Computer Vision, pages 16249–16258, 2021. +8 +[44] Yue Wang, Alireza Fathi, Abhijit Kundu, David A Ross, +Caroline Pantofaru, Tom Funkhouser, and Justin Solomon. +Pillar-based object detection for autonomous driving. +In +European Conference on Computer Vision, pages 18–34. +Springer, 2020. 2 +[45] Yue Wang, Guizilini Vitor Campagnolo, Tianyuan Zhang, +Hang Zhao, and Justin Solomon. Detr3d: 3d object detection +from multi-view images via 3d-to-2d queries. In In Confer- +ence on Robot Learning, pages 180–191, 2022. 1, 2, 5, 6 +[46] Yingming Wang, Xiangyu Zhang, Tong Yang, and Jian Sun. +Anchor detr: Query design for transformer-based detector. +arXiv preprint arXiv:2109.07107, 2021. 4, 5 +[47] Shaoqing Xu, Dingfu Zhou, Jin Fang, Junbo Yin, Zhou Bin, +and Liangjun Zhang. +Fusionpainting: Multimodal fusion +with adaptive attention for 3d object detection. In 2021 IEEE +International Intelligent Transportation Systems Conference +(ITSC), pages 3047–3054. IEEE, 2021. 5 +[48] Yan Yan, Yuxing Mao, and Bo Li. Second: Sparsely embed- +ded convolutional detection. Sensors, 18(10):3337, 2018. 4 +[49] Zetong Yang, Yanan Sun, Shu Liu, and Jiaya Jia. +3dssd: +Point-based 3d single stage object detector. In Proceedings +of the IEEE/CVF conference on computer vision and pattern +recognition, pages 11040–11048, 2020. 2 +[50] Tianwei Yin, Xingyi Zhou, and Philipp Krahenbuhl. Center- +based 3d object detection and tracking. In Proceedings of +the IEEE/CVF conference on computer vision and pattern +recognition, pages 11784–11793, 2021. 2, 5 +[51] Fangao Zeng, Bin Dong, Yuang Zhang, Tiancai Wang, Xi- +angyu Zhang, and Yichen Wei. Motr: End-to-end multiple- +object tracking with transformer. In European Conference +on Computer Vision, pages 659–675. Springer, 2022. 1 +[52] Gongjie Zhang, Zhipeng Luo, Yingchen Yu, Kaiwen Cui, +and Shijian Lu. Accelerating detr convergence via semantic- +aligned matching. In Proceedings of the IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition (CVPR), +pages 949–958, June 2022. 3 +[53] Hao Zhang, Feng Li, Shilong Liu, Lei Zhang, Hang Su, Jun +Zhu, Lionel M Ni, and Heung-Yeung Shum. +Dino: Detr +with improved denoising anchor boxes for end-to-end object +detection. arXiv preprint arXiv:2203.03605, 2022. 3, 4 +[54] Yin Zhou and Oncel Tuzel. Voxelnet: End-to-end learning +for point cloud based 3d object detection. In Proceedings of +the IEEE conference on computer vision and pattern recog- +nition, pages 4490–4499, 2018. 2, 3, 4, 6 +[55] Benjin Zhu, Zhengkai Jiang, Xiangxin Zhou, Zeming Li, and +Gang Yu. Class-balanced grouping and sampling for point +cloud 3d object detection. arXiv preprint arXiv:1908.09492, +2019. 6 +[56] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang +Wang, and Jifeng Dai. Deformable detr: Deformable trans- + +formers for end-to-end object detection. +arXiv preprint +arXiv:2010.04159, 2020. 1, 3, 8 + diff --git a/KNE4T4oBgHgl3EQfiA0C/content/tmp_files/2301.05129v1.pdf.txt b/KNE4T4oBgHgl3EQfiA0C/content/tmp_files/2301.05129v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..64ff6d696e9ae9e1b7d6426c1d59169a8eb571e4 --- /dev/null +++ b/KNE4T4oBgHgl3EQfiA0C/content/tmp_files/2301.05129v1.pdf.txt @@ -0,0 +1,4446 @@ +arXiv:2301.05129v1 [math.RT] 12 Jan 2023 +Holomorphic Induction Beyond the Norm-Continuous Setting, +With Applications to Positive Energy Representations +Milan Niestijl +January 13, 2023 +Abstract +We extend the theory of holomorphic induction of unitary representations of a possibly infinite-dimensional +Lie group G beyond the setting where the to-be-induced representation is required to be norm-continuous. +We allow the group G to be a connected regular BCH(Baker-Campbell-Hausdorff) Fr´echet-Lie group. +Given a smooth R-action α on G, we proceed to show that the corresponding class of so-called positive +energy representations is intimately related with holomorphic induction. In particular, we show that if +ρ is a unitary ground-state representation of G ⋊α R for which the energy-zero subspace Hρ(0) admits +a dense set of G-analytic vectors, then ρ|G is holomorphically induced from the representation of the +connected subgroup H := (Gα)0 of α-fixed points on Hρ(0). As a consequence, we obtain an isomor- +phism B(Hρ)G ∼= B(Hρ(0))H between the corresponding commutants. We also find that any two such +ground-state representations are necessarily unitary equivalent if their energy-zero subspaces are unitarily +equivalent as H-representations. These results were previously only available under the assumption of +norm-continuity of the H-representation on Hρ(0). +Contents +1 +Introduction +1 +2 +Preliminaries +3 +2.1 +Analytic functions on locally convex vector spaces +. . . . . . . . . . . . . . . . . . . . . . . . +3 +2.1.1 +Homogeneous polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.1.2 +Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Smooth, analytic and strongly-entire representations . . . . . . . . . . . . . . . . . . . . . . . +6 +2.3 +Positive energy and ground-state representations. . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3 +The space HO +ρ of strongly-entire vectors +9 +3.1 +Necessary conditions for the existence of strongly-entire representations +. . . . . . . . . . . . +10 +3.2 +Properties of HO +ρ and holomorphic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +4 +A general approach to holomorphic induction +16 +4.1 +A substitute for holomorphic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +4.2 +Holomorphically induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +4.3 +Uniqueness +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +4.4 +Commutants +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +4.5 +Holomorphic induction in stages +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +5 +A geometric approach to holomorphic induction +24 +5.2 +Complex structures on Eσ = G ×H V O +σ +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +5.3 +Geometric holomorphic induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +6 +Arveson spectral theory +29 +6.1 +Certain classes of R-representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +6.2 +Arveson spectral subspaces +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +i + +7 +Positive energy representations and holomorphic induction +37 +7.1.1 +Notation and preliminary observations . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +7.1.2 +The spectral gap condition +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +7.1.3 +Ground-state representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +7.1.4 +Strongly-entire ground-state representations for T-actions . . . . . . . . . . . . . . . . +42 +8 +Examples +43 +A Representations on reproducing kernel Hilbert spaces +46 +1 +Introduction +This paper is concerned with unitary representations a possibly infinite-dimensional connected Lie group G +that is modeled on a locally convex vector space (cf. [Mil84, Nee06]). Let α : R → Aut(G) be a smooth +action of R on G. We consider those G-representations that extend to a unitary representation ρ of G ⋊α R +which is smooth, in the sense that it admits a dense set of smooth vectors, and which is of positive energy, +meaning that the self-adjoint generator −i d +dt +�� +t=0 ρ(1G, t) of the unitary 1-parameter group t �→ ρ(1G, t) has +non-negative spectrum. +For infinite-dimensional Lie groups, a full classification of all irreducible representations is typically not +tractable, and even less so for factor representations. The positive energy condition serves to isolate a class +of representations that are more susceptible to systematic study. It is also quite natural from a physical per- +spective, because the Hamiltonian in quantum physics is nearly always required to be a positive self-adjoint +operator. It is then no surprise that positive energy representations of Lie groups are abundant in physics +literature [SW64, Bor87, Bor66, Haa92, LM75, Ol’81, PS86, Seg81]. +Holomorphic induction has proven to be a particularly effective tool in the study of positive energy represen- +tations. Let us first describe the main idea of holomorphic induction in the case where G is finite-dimensional. +Let H := (Gα)0 be the connected subgroup of α-fixed points in G, with Lie algebra h = Lie(H). A unitary +G-representation ρ is typically called holomorphically induced from the unitary H-representation σ on Vσ +if the homogeneous Hermitian vector bundle V := G ×H Vσ over G/H can be equipped with a G-invariant +complex-analytic bundle structure, with respect to which the Hilbert space Hρ can be G-equivariantly em- +bedded into the space of holomorphic sections O(G/H; V) of V, in such a way that the corresponding point +evaluations Ex : Hρ → Vx are continuous and satisfy ExE∗ +x = idVx for every x ∈ G/H. In particular, these +conditions imply that Hρ is unitarily equivalent to the G-representation on a reproducing kernel Hilbert +space, and that Hρ contains Vσ as H-subrepresentation. +An important special case is obtained when Vσ is one-dimensional. If ρ is holomorphically induced from σ, +we may identify Vσ with a cyclic ray [v0] in Hρ, whose G-orbit in the projective space P(Hρ) is a complex +submanifold. This means that ρ is a so-called coherent state representation[Nee00, Def. XV.2.1]. In this +case, the G-homogeneous line bundle V is the pull-back of the tautological line bundle over P(Hρ) along +the map G/H → P(Hρ), gH �→ [ρ(g)v0], and elements in the image of the corresponding map V → Hρ are +usually called coherent states. This is also the setting of the well-known Borel-Weil Theorem [DK00, Thm. +4.12.5]. Such representations have been studied extensively [Per86, Nee00, Lis95], and are known to be tightly +related to highest-weight representations [Nee00, Def. X.2.9, Ch. XV]. In particular, every unitary highest +weight representation of G is a coherent state representation[Nee00, Prop. XV.2.6]. The converse is not true. +The Schr¨odinger representation of the Heisenberg group Heis(R2, ω) provides a counterexample [Nee00, Ex. +XV.3.5]. +Holomorphic induction, defined as above, was studied in [Nee13] in the context where G is a Banach-Lie group +and where σ is bounded, meaning that it is continuous with respect to the norm-topology on B(Vσ). Writing +g for the Lie algebra of G and gC for its complexification, invariant complex structures on G/H correspond to +closed Lie subalgebras b ⊆ gC satisfying b+b = gC, b∩b = hC and Adh(b) ⊆ b for all h ∈ H [Bel05, Thm. 15] +(cf. [Kir76, p. 203] for the case where G is finite dimensional). The corresponding G-invariant holomorphic +bundle structures on V then turn out to be parametrized by extensions of dσ : h → B(Vσ) to a Lie algebra +homomorphism χ : b → B(Vσ) satisfying χ(Adh(ξ)) = σ(h)χ(ξ)σ(h)−1 for all ξ ∈ b and h ∈ H [Nee13, Thm. +1.6], as is to be expected from the finite-dimensional setting [TW71, Thm. 3.6]. The holomorphic structure is +used to relate various important properties of the G-representation ρ with those of σ. For example, [Nee13, +1 + +Thm. 2.12] entails that the commutants B(Hρ)G ∼= B(Vσ)H,χ are isomorphic as von Neumann algebras, which +implies in particular that ρ is irreducible, multiplicity-free or of type I, II or III if and only if this is true for +σ [Nee13, Cor. 2.14]. Moreover, [Nee13, Cor. 2.16] states that there is up to unitary equivalence at most one +unitary G-representation ρ that is holomorphically induced from a given pair (σ, χ). The relation between +holomorphically induced representations and the positive energy condition is then explained by [Nee13, Thm. +3.12, 3.14], which essentially state that in the above context, and under suitable assumptions, holomorphi- +cally induced representations correspond to so-called semibounded ones, the semiboundedness condition being +a ‘stable’ and stronger version of the positive energy condition (cf. [Nee10b]). These observations suggest +that the class of holomorphically induced representations may well admit a fruitful classification theory of +its factor representations. This line of reasoning was pursued in [Nee14, Thm. 5.4, 5.10] and [Nee12, Thm. +6.1, 7.3, 8.1], resulting in a classification of the irreducible semibounded unitary representations of certain +double extensions of Hilbert Loop groups and of hermitian Lie groups corresponding to infinite-dimensional +irreducible symmetric spaces. +In [Nee14, Appendix C], the theory of holomorphic induction was further developed, allowing G to be a +connected regular BCH Fr´echet-Lie group, under certain additional assumptions. Still, σ was required to +be norm-continuous. Let us mention that a particular and well-known special case of such a situation had +already appeared in the study of smooth positive energy representations of loop groups. In fact, these had +been completely classified using holomorphic induction [PS86] (cf. [Nee01]). +Still, the assumption of norm-continuity of σ is too restrictive in numerous examples, some of which we +encounter in Section 8 below. It is typically only suitable for describing the class of semibounded unitary +representations of G. In order to obtain a theory that can be used to describe the possibly larger class of all +positive energy representations, one necessarily needs to go beyond the norm-continuity of σ. +The purpose of the present paper is to remove this assumption of norm-continuity of the representation σ +that is induced from, whilst still allowing G to be a connected regular BCH Fr´echet-Lie group. A main +difficulty in this direction is that of equipping the homogeneous vector bundle G ×H Vσ with a G-invariant +complex-analytic bundle structure. The proof of [Nee13, Thm. 1.6] breaks down beyond the norm-continuous +setting, so a new approach is required. +We provide two possible solutions to this problem. As in [Nee14, Appendix C], we assume that gC admits a +triangular decomposition of the form gC = n− ⊕ hC ⊕ n+, where n± and hC are closed Lie subalgebras of gC +satisfying n± ⊆ n∓, and where b = hC ⊕ n−. In the first, which we call the general approach, we avoid speci- +fying a complex-analytic vector bundle altogether. Instead we replace the space of holomorphic sections by a +suitable subspace Cω(G; Vσ)H,χ of the space of real-analytic H-equivariant maps Cω(G; Vσ)H, defined directly +in terms of an extension χ : b → L(D) of dσ to b with some domain D ⊆ V ω +σ consisting of analytic vectors. +This also avoids the need for a G-invariant complex structure on the homogeneous space G/H. In the second, +which we call the geometric approach, we define a stronger notion of holomorphic induction. In this case, +H∞ +ρ +actually embeds into a space of holomorphic mappings on a homogeneous vector bundle. It therefore +requires complex geometry. A significant drawback of this approach is that it requires a dense set of so-called +strongly-entire vectors, whose availability is usually not known, unless G happens to be finite-dimensional, +in which case it is completely understood by the results of [Goo69] and [Pen74], see also Theorem 3.1.6 below. +Let us also mention that this paper does not complete the story of holomorphic induction. The developed +theory still excludes regular Fr´echet-Lie groups that are not BCH, such as the Virasoro group. Yet, it is +known that holomorphic induction can be used to obtain a complete classification of the positive energy +representations of the Virasoro group [NS15]. Nevertheless, the present paper makes substantial progress +towards a more complete understanding of holomorphic induction in the infinite-dimensional context. In +relation to positive energy representations, progress was made in a different direction in [NR22], where the +class of ground-state representations is studied in the setting of topological groups. +Structure of the paper +— In Section 2, we first recall some preliminaries regarding analytic functions on locally convex spaces. We +proceed to define smooth, analytic and strongly-entire representations, which are increasingly regular. +We also recall some important results related to positive energy and ground-state representations. +— We proceed in Section 3 to define and study the space HO +ρ of so-called strongly-entire vectors. We +equip this space with a locally convex topology, and extend the results of [Goo69] from the setting +2 + +of finite-dimensional Lie groups to the present one, where G is allowed to be infinite-dimensional. In +particular, if GC is a complex 1-connected regular BCH Fr´echet-Lie group with Lie algebra gC, we +obtain that HO +ρ carries a representation of GC that has a holomorphic action GC × HO +ρ → HO +ρ . The +space HO +ρ plays an important role in the geometric approach to holomorphic induction. +— In Section 4.2 we present the general approach towards holomorphic induction. +After determining +a useful equivalent formulation, we characterize the inducibility of pairs (σ, χ) in terms of positive +definite functions on G, which leads to the uniqueness of the holomorphically induced representation +up to unitary equivalence. We then proceed to show that there is an isomorphism of von Neumann +algebras B(hρ)G ∼= B(Vσ)H,χ between the commutants, provided that Vσ ⊆ Hρ is invariant under +B(Hρ)G, in complete analogy with the previously described norm-continuous setting. We also briefly +discuss holomorphic induction in stages. +— After equipping the G-homogeneous vector bundle Vσ := G ×H V O +σ +with a complex-analytic bundle +structure, using an suitable extension χ of dσ with domain V O +σ , we define in Section 5.3 the geometric +notion of holomorphically induced representations, and compare it to the one presented in Section 4.2. +— In relating holomorphic induction with the positive energy condition, we shall have need for a suitably +general notion of Arveson spectral subspaces. We therefore generalize in Section 6 the results of [NSZ15, +Sec. A.3] and [Nee13, Sec. A.2] to the level of generality needed in the next section. +— In Section 7, we study the relation between holomorphic induction and the positive energy condition. In +particular, we show that if ρ is a unitary ground-state representation of G⋊αR for which the energy-zero +subspace Hρ(0) admits a dense set of G-analytic vectors, then ρ|G is holomorphically induced from the +H-representation on Hρ(0). As a consequence, we obtain an isomorphism B(Hρ)G ∼= B(Hρ(0))H of von +Neumann algebras between the corresponding commutants. We also find that any two such ground-state +representations are necessarily unitary equivalent if their energy-zero subspaces are unitarily equivalent +as H-representations. +— In Section 8, we consider numerous interesting examples of unitary representations that are holomor- +phically induced from representations that are not norm-continuous. +Acknowledgments: +This research is supported by the NWO grant 639.032.734 “Cohomology and representation theory of infinite- +dimensional Lie groups”. I would like to thank my PhD supervisor Bas Janssens for his guidance. I am more- +over grateful to Karl-Hermann Neeb, who has carefully read an earlier version of this manuscript and given +various suggestions for improvement. The conversations with Karl-Hermann Neeb were also enlightening. +2 +Preliminaries +2.1 +Analytic functions on locally convex vector spaces +Let us recall some definitions and properties of analytic functions between locally convex vector spaces. +The main references are [BS71b], [BS71a] and [Gl¨o02b]. Throughout the following, fix locally convex vector +spaces E and F over the field K that both are complete and Hausdorff, where K is either R or C. Let +∆k : E → Ek, ∆k(h) = (h, · · · , h) be the diagonal. +2.1.1 +Homogeneous polynomials +Definition 2.1.1. Suppose U ⊆ E is open and f ∈ C∞(U, F). For any x ∈ U, define δ0 +x(f) : E → F and +δk +x(f) : E → F by δ0 +x(f)(v) := f(x) and δk +x(f)(v) := dkf(x; v, · · · , v), where k ∈ N. +Definition 2.1.2. Let k ∈ N. A map f : E → F is called a homogeneous polynomial of degree k if there +exists a k-linear symmetric map �f : Ek → F such that f = �f ◦ ∆k. Let P k(E; F) denote the space of +continuous homogeneous polynomials E → F of degree k. For k = 0, we set P 0(E; F) := F. +Set E0 := K. For k ∈ N≥0, we write Mult(Ek; F) for the space of continuous k-linear maps Ek → F, +equipped with the topology of uniform convergence on products of compact sets in E. For the case k = 1, we +also write B(E; F) := Mult(E; F). Let Symk(E; F) ⊆ Mult(Ek; F) denote the closed subspace of continuous +symmetric k-linear maps Ek → F. Let E �⊗F denote the completed projective tensor product of E and +F [Tre67, Def. 43.2, 43.5]. Define E �⊗k := E �⊗ · · · �⊗E (k times). The topology on E �⊗k is defined by the +3 + +seminorms q1 ⊗ · · · ⊗ qk, where each qi is a continuous seminorms on E, see also [Tre67, Def. 43.3]. On +algebraic tensors t ∈ E⊗k, this seminorm is given by +(q1 ⊗ · · · ⊗ qk)(t) := inf + + + +� +j +k +� +i=1 +qi(ξ(j) +i +) : t = +� +j +ξ(j) +1 +⊗ · · · ⊗ ξ(j) +k , with ξ(j) +i +∈ E + + + . +(2.1) +On simple tensors we have (q1 ⊗ · · · ⊗ qk)(ξ1 ⊗ · · · ⊗ ξk) = �k +i=1 qi(ξi), where ξi ∈ E [Tre67, Prop. 43.1]. +Proposition 2.1.3 ([Tre67, Prop. 43.4, Cor. 3 on p. 465]). +There is a canonical linear isomorphism Mult(Ek; F) ∼= B(E �⊗k; F). It is a homeomorphism if E is Fr´echet. +Equip P k(E; F) with the topology of uniform convergence on compact sets. If p is a continuous seminorm +on F, B ⊆ E is a subset and f : E → F is a function, we write pB(f) := supx∈B p(f(x)). +Proposition 2.1.4. Let k ∈ N≥0. Then P k(E; F) ∼= Symk(E; F) as locally convex vector spaces. +Proof. If �f : Ek → F is a symmetric k-linear map and f = �f ◦ ∆k is the corresponding homogeneous +polynomial, then �f can be recovered from f using the formula [BS71b, Thm. A]: +�f(x1, · · · xk) = 1 +k! +1 +� +ǫ1,··· ,ǫk=0 +(−1)k−(ǫ1+···+ǫk)f(ǫ1x1 + · · · + ǫkxk). +(2.2) +This formula moreover shows that �f is continuous if and only if f is so, and there is a linear isomorphism +Symk(E; F) → P k(E; F) given by �f �→ �f ◦ ∆k =: f. It remains to show that this map is also a homeo- +morphism. Suppose that f = �f ◦ ∆k for some �f ∈ Symk(E; F). If B ⊆ E is a compact subset and p is a +continuous seminorm on F, then pB(f) ≤ pBk( �f). Hence Symk(E; F) → P k(E; F), �f �→ f is continuous. For +the continuity of the inverse, we use (2.2), from which it follows that if Bi ⊆ E are compact subsets for i ∈ N +and p is a continuous seminorm on F, then +sup +xi∈Bi +p( �f(x1, · · · , xk)) ≤ 2k +k! pB(f), +(2.3) +where +B = { ǫ1x1 + · · · + ǫkxk : ǫi ∈ {0, 1}, xi ∈ Bi +for i ∈ {1, · · · , k} } , +which is a compact subset of E. Consequently the map f �→ �f is continuous P k(E; F) → Symk(E; F). +Define the locally convex space P(E; F) := �∞ +k=0 P k(E; F), equipped with the product topology. If F = K, +we simply write P n(E) := P n(E; K). +2.1.2 +Analytic functions +Let U ⊆ E be open and let f : U → F be a function. +Definition 2.1.5. +— Suppose K = C. The function f : U → F is called complex-analytic or holomorphic if it is continuous, +and for every x ∈ U there exists a 0 neighborhood V in E with x+V ⊆ U and functions fk ∈ P k(E; F) +for k ∈ N≥0 such that: +f(x + h) = +∞ +� +k=0 +fk(h), +∀h ∈ V. +— Suppose K = R. The function f : U → F is called real-analytic if it extends to some complex-analytic +map fC : UC → FC for some open neighborhood UC of U in EC. +— Suppose K = C. The function f : U → F is called entire if it is continuous and there exist functions +fk ∈ P k(E; F) for k ∈ N≥0 such that f(x) = �∞ +k=0 fk(x) for all x ∈ E. +Remark 2.1.6. The above definition of a real-analytic map differs from the one used in [BS71a], where a +function f : U → F is called real-analytic if it is continuous and for every x ∈ U there exists a 0-neighborhood +V in U with x + V ⊆ U and homogeneous polynomials fk : E → F such that f(x + h) = �∞ +k=0 fk(h) holds +for all h ∈ V . The two notions are equivalent if E and F are Fr´echet spaces [Gl¨o02b, Rem. 2.9], [BS71a, +Thm. 7.1]. +4 + +Proposition 2.1.7 ([BS71a, Prop. 5.1]). +Suppose K = C. Let fk ∈ P k(E; F) for every k ∈ N≥0. Let U ⊆ E be a 0-neighborhood s.t. f(h) := � +k fk(h) +is convergent for every h ∈ U. Assume that f : U → F is continuous at 0 ∈ U. Then, for every continuous +seminorm p on F, there exists a 0-neighborhood V ⊆ U such that �∞ +k=0 pV (fk) < ∞. +Lemma 2.1.8. Suppose K = C. Let fn ∈ P n(E; F) for every n ∈ N≥0. Consider the following assertions: +1. f := �∞ +n=0 fn defines an entire function E → F. +2. �∞ +n=0 pB(fn) < ∞ for any compact subset B ⊆ E and continuous seminorm p on F. +We have that (1) =⇒ (2). If E is a Fr´echet space, then also (2) =⇒ (1) holds true. +Proof. Assume that f = �∞ +n=0 fn defines an entire function E → F. Let B ⊆ E be a compact subset and +let p be a continuous seminorm on F. We may assume that B is balanced. As f is continuous, f(2B) ⊆ F is +compact and hence bounded. So Mp := p2B(f) < ∞. As f is entire, we have f(zx) = �∞ +n=0 fn(x)zn for any +x ∈ E and z ∈ C. Let x ∈ 2B. Then also zx ∈ 2B for any z ∈ C with |z| ≤ 1, as B is balanced. Applying +[BS71a, Cor. 3.2] to the holomorphic map g : C → F, g(z) := f(zx), we find that fn(x) = +1 +2πi +� +|z|=1 +g(z) +zn+1 dz +and moreover that +p(fn(x)) ≤ sup +|z|=1 +p(g(z)) ≤ p2B(f) = Mp, +∀n ∈ N≥0. +Hence p2B(fn) ≤ Mp, so that pB(fn) ≤ Mp2−n for all n ∈ N≥0. Thus �∞ +n=0 pB(fn) ≤ Mp +�∞ +n=0 2−n < ∞. +Suppose that E is a Fr´echet space. Assume that (2) holds true. Then in particular the series �∞ +n=0 fn(x) is +convergent for any x ∈ E. So f := �∞ +n=0 fn defines a function E → F. To show f is entire, it remains only +to show that it is continuous. The condition (2) implies that sN → f uniformly on compact subsets, where +sN := �N +n=0 fn for any N ∈ N. As sN is continuous for every N ∈ N and E is Fr´echet by assumption, this +implies that f is continuous (by a standard 3ǫ argument). +Proposition 2.1.9 ([Gl¨o02b, Prop. 2.4]). +Every real- or complex-analytic map is smooth. +Proposition 2.1.10 ([BS71a, Prop. 5.5]). +Suppose K = C. If f : U → F is complex-analytic, then f(x + h) = �∞ +k=0 +1 +k!δk +x(f)(h) for all h ∈ V , where V +is the maximal balanced 0-neighborhood of E such that x + V ⊆ U. +Proposition 2.1.11 ([Gl¨o02b, Lem. 2.5]). +Suppose K = C. Then f is complex-analytic if and only if f is smooth and δ1 +x := df(x; −) : E → F is +complex-linear for every x ∈ U. +Proposition 2.1.12 ([Gl¨o02b, Lem. 2.6]). +Suppose K = C. If f : U → F is complex-analytic, then so is df : U × E → F. +With these definitions, the chain rule holds for both real- and complex-analytic mappings. One proceeds to +define real- and complex- analytic manifolds and Lie groups, see e.g. [Mil84] and [Nee06] for more details. +Definition 2.1.13. If M is a real-analytic manifold and V is a locally convex vector space, we write Cω(M; V ) +for the set of analytic functions M → V . If M is a complex-analytic manifold and V is complex, we write +O(M; V ) for the space of complex-analytic mappings M → V . +Proposition 2.1.14 (Identity Theorems [BS71a, Prop. 6.6]). +1. Suppose that E and F are complex. Let f : U → F be complex-analytic and assume that U is connected. +If f(x) = 0 for all x ∈ V for some open and non-empty V ⊆ U, then f = 0. +2. Suppose that E is real and F is complex. Let f : UC → F be complex-analytic, where UC ⊆ EC is open +and connected. If UC contains a non-empty subset V ⊆ E that is open in E and f(x) = 0 holds for +every x ∈ V , then f = 0. +Proposition 2.1.15. Let x ∈ U. The following linear map is continuous: +j∞ +x : C∞(U, F) → P(E; F), +f �→ +∞ +� +k=0 +1 +k!δk +x(f) +If U is connected, then its restriction to Cω(U; F) is injective. +5 + +Proof. The map j∞ +x is linear, as each δk +x : C∞(U, F) → P k(E; F) is so. As P(E; F) = �∞ +n=0 P n(E; F) carries +the product topology, to see j∞ +x is continuous it suffices to show that δk +x is continuous for every k ∈ N≥0. This +is immediate from the definition of the compact-open C∞-topology on C∞(U, F) [Nee06, Def. I.5.1(d)], and +the topology of uniform convergence on compact subsets carried by P k(E; F). Assume that U is connected. +Let f ∈ Cω(U; F) and suppose that j∞ +x (f) = 0. Using Proposition 2.1.10 it follows that f(x + h) = 0 for all +h in some 0-neighborhood of E. By Proposition 2.1.14 this implies that f = 0. +2.2 +Smooth, analytic and strongly-entire representations +Let G be a BCH(Baker-Campbell-Hausdorff) Fr´echet-Lie group with Lie algebra g. We write gC for the +complexification of g. Assume that G is regular in the sense of [Nee06, Def. II.5.2]. We refer to [Nee06] and +[Mil84] for an overview on locally convex Lie theory. +Let us first clarify some notation. If D is a pre-Hilbert space, we write L(D) for the set of linear operators +on D. We further define +L†(D) := { T ∈ L(D) : D ⊆ dom(T ∗) and T ∗D ⊆ D } . +Set T † := T ∗|D for T ∈ L†(D). Then (−)† is an involution on L†(D) (cf. [Sch90, Ch. 2]). We will also have +need for various involutions on U(gC). Let θ : gC → gC be defined by θ(ξ + iη) := ξ − iη for ξ, η ∈ g. +Definition 2.2.1. Extend the conjugation θ on gC to a complex conjugate-linear automorphism of U(gC). +Let τ denote the involutive anti-automorphism of U(gC) extending ξ �→ −ξ on gC. Define x∗ := τ(θ(x)) for +x ∈ U(gC). Explicitly, θ, τ and (−)∗ satisfy the following relations, where ξj ∈ gC for j ∈ N: +θ(ξ1 · · · ξn) = θ(ξ1) · · · θ(ξn), +τ(ξ1 · · · ξn) = (−1)nξn · · · ξ1 +and +(ξ1 · · · ξn)∗ = (−1)nθ(ξn) · · · θ(ξ1). +If (ρ, Hρ) is a unitary G-representation, we say that it is continuous if it is so with respect to the strong +operator topology on U(Hρ). +Definition 2.2.2. Let (ρ, Hρ) be a continuous unitary representation of G. A vector ψ ∈ Hρ is called smooth, +resp. analytic, if the orbit map G → Hρ, g �→ ρ(g)v is smooth, resp. analytic. We write H∞ +ρ and Hω +ρ for the +linear subspaces of smooth and analytic vectors, respectively. We say that the representation ρ is smooth if +H∞ +ρ is dense in Hρ and analytic if Hω +ρ is dense in Hρ. +Remark 2.2.3. If ρ is a smooth unitary representation of G, then the derived representation dρ of gC on H∞ +ρ +extends to an algebra homomorphism dρ : U(gC) → L†(H∞ +ρ ) satisfying dρ(x)† = dρ(x∗) for any x ∈ U(gC). +Definition 2.2.4. Let (ρ, Hρ) be a smooth unitary representation of G. +— Following [JN19, Def. 3.9], we define two locally convex topologies on the space H∞ +ρ : +– The weak topology on H∞ +ρ is defined by the seminorms pξ(ψ) := ∥dρ(ξ1 · · · ξn)ψ∥, where n ∈ N≥0 +and ξ = (ξ1, · · · , ξn) ∈ gn. +– The strong topology is defined by the seminorms pB(ψ) := supξ∈B ∥dρ(ξ1 · · · ξn)ψ∥, where B ⊆ gn +is bounded and n ∈ N≥0. +The space H∞ +ρ +is complete w.r.t. to either of these topologies [JN19, Prop. 3.19], where we used that +G is a regular Fr´echet-Lie group. +— A vector ψ ∈ H∞ +ρ is called entire if �∞ +n=0 +1 +n! supξ∈B ∥dρ(ξn)ψ∥ < ∞ for every compact B ⊆ gC. +— If ψ ∈ H∞ +ρ and B ⊆ gC, we define pn +B(ψ) := supξ1,··· ,ξn∈B ∥dρ(ξ1 · · · ξn)ψ∥ and qB(ψ) := �∞ +n=0 +1 +n!pn +B(ψ). +— A vector ψ ∈ H∞ +ρ is called strongly-entire if qB(ψ) < ∞ for every compact subset B ⊆ gC. +— We write HO +ρ ⊆ H∞ +ρ +for the linear subspace of strongly-entire vectors. Equip HO +ρ with the topology +defined by the seminorms qB for compact subsets B ⊆ gC. +— We say that the representation ρ strongly-entire if HO +ρ is dense in Hρ. +If ψ ∈ H∞ +ρ , we write f ψ : G → Hρ for the orbit map f ψ(g) = ρ(g)ψ. As f ψ is smooth, the homogeneous +polynomial f ψ +n (ξ) := +1 +n!dρ(ξn)ψ is continuous as a map gC → Hρ, so f ψ +n ∈ P n(gC; Hρ). Notice further that +j∞ +0 (f ψ) = �∞ +n=0 f ψ +n ∈ P(gC; Hρ). Let βψ +n be the unique element of Symn(gC; Hρ) satisfying f ψ +n = βψ +n ◦ ∆n. +Explicitly, βψ +n (ξ1, · · · , ξn) = +1 +(n!)2 +� +σ∈Sn dρ(ξσ1 · · · ξσn)v. +6 + +Lemma 2.2.5. Let ψ ∈ H∞ +ρ . Assume that qB(ψ) < ∞ for every compact subset B ⊆ g. +Then qB(ψ) < ∞ for every compact subset B ⊆ gC. +Proof. Let BC ⊆ gC be compact. Replacing BC by its balanced hull, we may assume that BC is balanced. Let +B := +� +ξ + ξ : ξ ∈ BC +� +⊆ g, which is compact in g. Then BC ⊆ B + iB and so qBC(ψ) ≤ q2B(ψ) < ∞. +Proposition 2.2.6. Let (ρ, Hρ) be a smooth unitary representation of G. +Let ψ ∈ H∞ +ρ . The following +assertions are equivalent: +1. ψ ∈ Hω +ρ . +2. There exists a 0-neighborhood V ⊆ g such that �∞ +n=0 +1 +n!dρ(ξn)ψ converges for every ξ ∈ V and the map +V → Hρ, ξ �→ �∞ +n=0 +1 +n!dρ(ξn)ψ is continuous. +3. �∞ +n=0 +1 +n!dρ(ξn)ψ converges for every ξ in a 0-neighborhood g. +4. There is a 0-neighborhood V ⊆ g such that � +n=0 +1 +n!pn +V (ψ) < ∞. +5. There is a 0-neighborhood V ⊆ g such that �∞ +n=0 +1 +n!⟨ψ, dρ(ξn)ψ⟩ converges for all ξ ∈ V . +6. The map G → C, g �→ ⟨ψ, ρ(g)ψ⟩ is analytic at 1 ∈ G. +Proof. Assume that ψ ∈ Hω +ρ . Then the orbit map f ψ : G → Hρ is real-analytic, and hence so is f ψ ◦ exp : +g → Hρ. Notice that f ψ(eξ) = ρ(eξ)ψ, so that δn +0 (f ψ ◦ exp) = dρ(ξn)ψ. Using Proposition 2.1.10, it follows +that f ψ(eξ) = �∞ +n=0 +1 +n!dρ(ξn)ψ on some balanced 0-neighborhood V ⊆ g. So (1) =⇒ (2). +We show that (2) =⇒ (1). Let V ⊆ g be a 0-neighborhood such that �∞ +n=0 +1 +n!dρ(ξn)ψ converges for every +ξ ∈ V and s.t. the map ξ �→ �∞ +n=0 +1 +n!dρ(ξn)ψ is continuous on V . Replacing V by some smaller balanced +open set, we may assume that V is balanced. Define hψ(ξ) := �∞ +n=0 +1 +n!dρ(ξn)ψ. In view of Remark 2.1.6, the +assumptions imply that hψ is real-analytic on V , where it was used that g is Fr´echet and Hρ is a Hilbert space. +Then hψ is smooth by Proposition 2.1.9. Let ξ ∈ V . We show that hψ(ξ) = ρ(eξ)ψ. Let s ∈ I := [−1, 1]. +Then sξ ∈ V , because V is balanced. Notice that +d +dt +���� +t=s +hψ(tξ)ψ = dρ(ξ)hψ(sξ), +and +d +dt +���� +t=s +ρ(etξ)ψ = dρ(ξ)ρ(esξ). +Let η ∈ H∞ +ρ . +Using dρ(ξ)∗η = −dρ(ξ)η it follows that +d +dt +�� +t=s ⟨ρ(etξ)η, hψ(tξ)⟩ = 0. +As a consequence, +⟨η, ρ(e−tξ)hψ(tξ)⟩ = ⟨η, ψ⟩ for all t ∈ I. +As this is valid for any η in the dense set H∞ +ρ +it follows that +ρ(e−tξ)hψ(tξ)ψ = ψ or equivalently that hψ(tξ)ψ = ρ(etξ)ψ for all t ∈ I. In particular, taking t = 1 we +conclude that hψ(ξ) = ρ(eξ)ψ for all ξ ∈ V . As hψ is real-analytic on V , so is ξ �→ ρ(eξ)ψ. Since G is BCH, +this implies that g �→ ρ(g)ψ is analytic at 1 ∈ G. In turn, this implies that it is analytic everywhere, where +we have used that G is a real-analytic Lie group and that the composition of real-analytic maps is again +real-analytic [Gl¨o02b, Proposition 2.8]. Thus ψ ∈ Hω +ρ . +The implication (2) +=⇒ +(3) is trivial whereas (3) +=⇒ +(4) follows from [BS71a, Prop. 5.2] because +V is absorbing and g is a Baire space, as it is Fr´echet. To see that (4) +=⇒ +(2), assume that V ⊆ g +is a 0-neighborhood such that �∞ +n=0 +1 +n!pn +V (ψ) < ∞. For ξ ∈ V , we write sN(ξ) := �N +n=0 +1 +n!dρ(ξn)ψ and +s(ξ) := �∞ +n=0 +1 +n!dρ(ξn)ψ. It remains only to prove that s is continuous on V . Let ξ ∈ V . Suppose that (ξk) +is a sequence in V with ξk → ξ. Let ǫ > 0. Let N ∈ N be such that �∞ +n=N+1 +1 +n!pn +V (ψ) < ǫ. Then for any +η ∈ V we have ∥s(η) − sN(η)∥ ≤ �∞ +n=N+1 +1 +n!pn +V (ψ) < ǫ. Using that sN is continuous, let N ′ ∈ N be s.t. +∥sN(ξ) − sN(ξk)∥ < ǫ and ξk ∈ V for all k ≥ N ′. Then +∥s(ξ) − s(ξk)∥ ≤ ∥s(ξ) − sN(ξ)∥ + ∥sN(ξ) − sN(ξk)∥ + ∥sN(ξk) − s(ξk)∥ < 3ǫ, +∀k ≥ N ′. +Thus s(ξk) → s(ξ). Hence s is sequentially continuous at 0. As g is Fr´echet, this implies that s is continuous +at ξ. Thus (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4). It is trivial that (3) =⇒ (5) whereas (5) =⇒ (3) follows +immediately from [Nee11, Prop. 3.4, 6.3] (by considering D := H∞ +ρ +and v := ψ). Finally, (6) ⇐⇒ (1) is +precisely [Nee11, Thm. 5.2]. This completes the proof. +Let us consider an analogous statements for entire vectors: +7 + +Proposition 2.2.7. Let ψ ∈ H∞ +ρ . The following assertions are equivalent: +1. The series �∞ +n=0 f ψ +n (ξ) = �∞ +n=0 +1 +n!dρ(ξn)ψ defines an entire function gC → Hρ, ξ �→ �∞ +n=0 f ψ +n (ξ). +2. ψ is an entire vector for ρ, i.e., �∞ +n=0 +1 +n! supξ∈B ∥dρ(ξn)ψ∥ < ∞ for every compact B ⊆ gC. +3. The map g → Hρ, ξ �→ ρ(eξ)ψ extends to an entire function gC → Hρ. +4. �∞ +n=0 supξi∈B ∥βψ +n (ξ1, · · · , ξn)∥ < ∞ for every compact B ⊆ g. +Proof. As gC is Fr´echet by assumption, we know using Lemma 2.1.8 that the series �∞ +n=0 f ψ +n (ξ) = �∞ +n=0 +1 +n!dρ(ξn)ψ +defines an entire function on gC if and only if +∞ +� +n=0 +1 +n! sup +ξ∈B +∥dρ(ξn)ψ∥ < ∞, +∀B ⊆ gC compact. +That is, if and only if (2) holds true. Thus (1) +⇐⇒ +(2). Assume next that (2) is valid. As singletons +are compact, it follows in particular that � +n=0 f ψ +n (ξ) converges for every ξ ∈ gC. By Proposition 2.2.6, this +implies that ψ ∈ Hω +ρ . Hence the orbit map f ψ : G → Hρ is real-analytic. As G is BCH, the exponential +map exp : g → G is real-analytic and hence ξ �→ f ψ(eξ) = ρ(eξ)ψ is a real-analytic map g → Hρ. Since +δn +0 (f ψ ◦ exp; ξ) = dρ(ξn)ψ for every n ∈ N, Proposition 2.1.10 implies that f ψ(eξ) = �∞ +n=0 +1 +n!dρ(ξn)ψ on +some 0-neighborhood in V . As (2) and hence (1) hold by assumption, it follows that �∞ +n=0 f ψ +n is an entire +function extending ξ �→ ρ(eξ)ψ. Thus (3) holds true. Suppose conversely that (3) is valid, so that f ψ ◦ exp +extends to an entire function F : gC → Hρ. By Proposition 2.1.10 and using that δn +0 (f ψ ◦ exp; ξ) = dρ(ξn)ψ +for n ∈ N, we find that F(ξ) = �∞ +n=0 +1 +n!dρ(ξn)ψ for every ξ ∈ gC. Thus (1) holds true. We have shown +(1) +⇐⇒ +(2) +⇐⇒ +(3). Next we show (2) +=⇒ +(4). Let B ⊆ gC be compact. As gC is complete, the +closed convex hull of B is again compact [Tre67, p. 67]. Thus we may assume that B is convex. Replacing +B further by its balanced hull, we may assume that B is balanced. Then B + · · · + B (n times) ⊆ nB. From +equation (2.3) it follows that +sup +ξi∈B +∥βψ +n (ξ1, · · · , ξn)∥ ≤ 2n +n! sup +ξ∈nB +∥f ψ +n (ξ)∥ = (2n)n +n! +sup +ξ∈B +∥f ψ +n (ξ)∥. +Choose some t > 2e. Since �∞ +n=0 supξ∈B ∥f ψ +n (ξ)∥ < ∞ for every compact B, it follows (by considering tB) +that there exists some C > 0 s.t. supξ∈B ∥f ψ +n (ξ)∥ ≤ Ct−n for every n ∈ N≥0. Then +∞ +� +n=0 +sup +ξi∈B +∥βψ +n (ξ1, · · · , ξn)∥ ≤ C +∞ +� +n=0 +1 +n! +�2n +t +�n +< ∞, +The implication (4) =⇒ (2) is trivial. +Remark 2.2.8. The characterization (4) of entire vectors in Proposition 2.2.7 makes the difference between +entire and strongly-entire vectors clear, namely whether one considers the symmetric n-linear maps βψ +n or their +non-symmetric analogues (ξ1, · · · , ξn) �→ 1 +n!dρ(ξ1 · · · ξn)ψ. Analogous to [Nee11, Rem. 3.7], it is in general not +known whether or not any entire vector is in fact strongly-entire. In the case where g is finite-dimensional, +this follows immediately from [Pen74, Thm. I.3, Rem. I.7]. +Corollary 2.2.9. HO +ρ ⊆ Hω +ρ ⊆ H∞ +ρ . +Proof. Any strongly-entire vector is entire. Consequently, the first inclusion follows by combining Proposi- +tion 2.2.7 and Proposition 2.2.6. The second one follows from the fact that if the orbit map f ψ : G → Hρ is +real-analytic, then it is smooth by Proposition 2.1.9. +The space HO +ρ of strongly-entire vectors will be considered in more detail in Section 3 below. +2.3 +Positive energy and ground-state representations. +Let G be a regular locally convex Lie group with Lie algebra g. If H is a Hilbert space and S ⊆ H is a subset, +we write �S� ⊆ H for the closed linear span of S. +Theorem 2.3.1 (Borchers-Arveson [BR87, Thm. 3.2.46], [BGN20, Lem. 4.17]). +Let M ⊆ B(H) be a von Neumann algebra on the Hilbert space H. Let (Ut)t∈R be a strongly continuous +unitary one-parameter group satisfying UtMU −1 +t +⊆ M for all t ∈ R. Assume that Ut = eitH with H ≥ 0. +Define α : R → Aut(M) by αt(x) := AdUt(x) := UtxU −1 +t +for t ∈ R and x ∈ M. Denote by Mα(S) ⊆ M the +Arveson spectral subspace for S ⊆ R. Then +8 + +1. There exists a strongly continuous unitary one-parameter group Vt = eitH0 in M with H0 ≥ 0 and +AdVt = αt for every t ∈ R. +2. � +t>0�Mα[t, ∞)H� = {0}. +3. Vt is uniquely determined by the additional requirement that for any other such V ′ +t = eitH′ +0, we have +H′ +0 ≥ H0. In this case, the spectral projection P corresponding to Vt is determined uniquely by +P[t, ∞)H = +� +s 0. Then �ρ0(T ) = idHρ and ρ is ground-state w.r.t α. +4. Let P denote the spectral measure associated to t �→ �ρ0(t). Let ǫ > 0. Then the projection P[0, ǫ) has +central support 1M = idHρ ∈ Z(M). In particular P[0, ǫ)Hρ is cyclic for M. +Proof. The first three assertions follow by [JN21, Cor. 3.9] and the last by [BGN20, Lem. 4.17]. +3 +The space HO +ρ of strongly-entire vectors +Let G be a regular BCH Fr´echet-Lie group with Lie algebra g. Let (ρ, Hρ) be a smooth unitary representation +of G. In this section, we extend some results of [Goo69] concerning the space of strongly-entire vectors HO +ρ +from the case where G is finite-dimensional to the present setting. +9 + +3.1 +Necessary conditions for the existence of strongly-entire representations +We first show that when dim(g) < ∞, the definition for HO +ρ (Definition 2.2.4) agrees with the one used in +[Goo69, p.61]. The existence of a dense set of strongly-entire vectors is well-understood for continuous unitary +representations of finite-dimensional Lie groups, yielding immediate necessary conditions for the existence +of strongly-entire representations in the infinite-dimensional setting. This will turn out to be quite restrictive. +Assume that dim(g) < ∞. Let us recall the definition used in [Goo69, p.61]. Let {eµ}d +µ=1 be a basis of g. +For v ∈ H∞ +ρ , we define +Es(v) := +∞ +� +n=0 +sn +n! +sup +1≤µk≤d +∥dρ(eµ1 · · · eµn)v∥ ∈ [0, ∞] +Set Hωt +ρ +:= +� +v ∈ H∞ +ρ +: Es(v) < ∞ for all 0 < s < t +� +for t > 0. Define HO′ +ρ +:= � +t>0 Hωt +ρ . Equip HO′ +ρ +with +the locally convex topology defined by the seminorms Es for s > 0. +Lemma 3.1.1. HO +ρ = HO′ +ρ +as an equality of locally convex vector spaces. +Proof. Define the compact subsets Bs := +� �d +µ=1 cµeµ : cµ ∈ C, |cµ| ≤ s +∀µ ∈ {1, · · · , d} +� +⊆ gC for s > 0. +Let s > 0. As seµ ∈ Bs for any µ ∈ {1, · · · , d}, it is immediate that Es(v) ≤ qBs(v). Conversely, take ξj ∈ Bs +for j ∈ {1, · · · , n}. Then ξj = �d +µj=1 cµjeµj ∈ Bs for some cµj ∈ C with |cµj| ≤ s. So +dρ(ξj1 · · · ξjn)v = +d +� +µ1,···µn=1 +cµ1 · · · cµndρ(eµ1 · · · eµn)v. +Consequently +∥dρ(ξj1 · · · ξjn)v∥ ≤ sn +d +� +µ1,···µn=1 +∥dρ(eµ1 · · · eµn)v∥ ≤ sndn +sup +1≤µk≤d +∥dρ(eµ1 · · · eµn)v∥. +Hence Es(v) ≤ qBs(v) ≤ Esd(v) for any s > 0. This shows that HO +ρ = HO′ +ρ +as locally convex vector spaces. +Following [AM66, p. 128], [Jen73, p. 115] and [Pen74], we define: +Definition 3.1.2. +— A finite-dimensional Lie group G is said to be of type R if Spec(Adg) ⊆ S1 for every g ∈ G, where +S1 ⊆ C is the unit-circle. +— A finite-dimensional Lie algebra g is said to be of type R if Spec(adξ) ⊆ iR for every ξ ∈ g. +Remark 3.1.3. Lie algebras of type R are by some authors also called weakly elliptic [Nee98, Def. II.1]. +Proposition 3.1.4 ([Jen73, Prop. 1.3]). +Let G be a finite-dimensional connected Lie group with Lie algebra g. Then G is of type R if and only if g is +of type R. +Proposition 3.1.5 ([Pen74, Lem. on p. 120]). +A finite-dimensional Lie algebra g is of type R if and only if it is the semi-direct product s ⋊ k of a compact +semisimple Lie algebra k and a type R solvable Lie algebra s. +Theorem 3.1.6 ([Pen74, Cor. II.5]). +Let G be a finite-dimensional Lie group and ρ a continuous unitary representation of G. Then HO +ρ is dense +if and only if ρ factors through a Lie group of type R. +In the setting where G is a possibly infinite-dimensional regular BCH Fr´echet-Lie group, this yields: +Corollary 3.1.7. Let G be a possibly infinite-dimensional regular BCH Fr´echet-Lie group. Suppose that +(ρ, Hρ) is a strongly-entire unitary representation of G. If ρ is injective, then any finite-dimensional Lie +subgroup of G is of type R. +Proof. Let H be a finite-dimensional Lie subgroup of G. +Then π := ρ|H is a continuous unitary H- +representation on Hπ := Hρ =: H. Since HO +ρ ⊆ HO +π , HO +π is dense in H. As ρ is injective, it follows by +Theorem 3.1.6 that H is of type R. +10 + +As an illustration: If ρ is injective and HO +ρ is dense, then G can not contain a single copy of the ax + b +group. On the other hand, Theorem 3.1.6 provides ample examples of continuous representations that admit +a dense set of strongly-entire vectors. Indeed, simply take any continuous unitary representation of a finite- +dimensional Lie group of type R. The following examples show that also infinite-dimensional Lie groups may +admit a dense set of strongly-entire vectors. +Example 3.1.8 (Norm-continuous representations). +Let G be a regular BCH Fr´echet-Lie group and let ρ : G → U(Hρ) a unitary representation of G which is +continuous w.r.t. norm-topology on U(Hρ). Equipped with the norm topology, U(Hρ) is a Banach-Lie group +with Lie algebra u(Hρ) := { T ∈ B(Hρ) : T ∗ = −T }, and the continuous homomorphism ρ : G → U(Hρ) is +automatically analytic by [Nee06, Thm. IV.1.18]. This implies that Hω +ρ = Hρ. Let us show that we even +have HO +ρ = Hρ. As the representation dρ : g → u(Hρ) is continuous, there exist a continuous seminorm p on +g s.t. ∥dρ(ξ)∥ ≤ p(ξ) for all ξ ∈ g [Tre67, Ch. I.7, Prop. 7.7]. So ∥dρ(ξ1) · · · dρ(ξn)ψ∥ ≤ p(ξ1) · · · p(ξn)∥ψ∥, +where ξj ∈ g for j ∈ N and ψ ∈ Hρ. So if B ⊆ g is bounded, then with M := sup p(B) < ∞ we get that +qB(ψ) := +∞ +� +n=0 +1 +n! sup +ξi∈B +∥dρ(ξ1) · · · dρ(ξn)ψ∥ ≤ +∞ +� +n=0 +M n +n! ∥ψ∥ = eM∥ψ∥ < ∞. +Using Lemma 2.2.5 this proves that HO +ρ = Hρ. +Example 3.1.9 (Positive energy representations of Heisenberg groups). +We recall the construction of positive energy representations of Heisenberg groups, and show that they admit +a dense set of strongly-entire vectors. Let V be a real Fr´echet space and ω a non-degenerate continuous skew +bilinear form V × V → R. Let G := Heis(V, ω) be the corresponding Heisenberg group, so its underlying set +is T × V and it has multiplication (z1, v1) · (z2, v2) := (z1z2e−iω(v1,v2), v1 + v2). As V is a Fr`echet space, it +is Mackey complete by [KM97, Thm. I.4.11]. Using [Nee06, Thm. V.1.8], this implies that G is regular. Let +GC := Heis(VC, ω) be the corresponding complexification. Let J be a compatible positive complex structure +on V , meaning that J ∗ω = ω and ω(v, J v) > 0 for any non-zero v ∈ V . The positive-definite sesquilinear +form ⟨v, w⟩ := ω(v, J w) + iω(v, w) makes V into a complex pre-Hilbert space, whose completion we denote +by VJ . Notice that the inclusion V → VJ is continuous. Equip the symmetric algebra S•(VJ ) with the inner +product satisfying +⟨v1 · · · vn, w1 · · · wn⟩ = +� +σ∈Sn +n +� +j=1 +⟨vj, wσj⟩, +for vj, wj ∈ VJ . +(3.1) +Let Hρ be the corresponding Hilbert space completion of S•(VJ ). Then Hρ contains and is generated by the +“coherent states” ev := �∞ +n=0 +1 +n!vn ∈ Hρ for v ∈ VJ , and there is a unitary representation ρ of Heis(V, ω) on +Hρ satisfying ρ(z, v)ew = ze− 1 +2 ∥v∥2−⟨v,w⟩ev+w [PS86, Sec. 9.5] for v, w ∈ V and z ∈ T. A direct computation +verifies the equation ρ(v1)ρ(v2) = e−iω(v1,v2)ρ(v1 + v2) for v1, v2 ∈ V . Let Ω ∈ Hρ be the vacuum vector. +The map +G → C, +(z, v) �→ ⟨Ω, ρ(z, v)Ω⟩ = ze− 1 +2 ∥v∥2 +is smooth, so it follows from [Nee10a, Thm. 7.2] that H∞ +ρ contains the cyclic vector Ω and is therefore dense +in Hρ. So ρ is smooth. The infinitesimal g-action dρ satisfies dρ(v)ψ = (c(v) − a(v))ψ for any v, w ∈ V and +ψ ∈ S•(VJ ), where c(v)ψ = vψ is the creation operator with core S•(VJ ) and a(v) := c(v)∗ is its adjoint, the +annihilation operator. From c(J v) = ic(v) and a(J v) = −ia(v) we obtain that the C-linear extension of dρ +to gC satisfies dρ(v + iw) = c(v + J w) − a(v − J w) for v, w ∈ V . +To see that HO +ρ is dense in Hρ, it suffices to show that it contains the cyclic vector Ω, because HO +ρ is G- +invariant. Let B be the open unit-ball in VJ . Let K be a compact subset of the real Fr´echet space V . +Then K is also compact as subspace of VJ , and is therefore contained in sB ⊆ VJ for some s > 0. If +v ∈ B, then ∥ c(v)|Sn(VJ ) ∥ = ∥ a(v)|Sn+1(VJ ) ∥ < √n + 1 [BR97, p. 9]. So if (vj)j∈N is a sequence in B, then +supv1,··· ,vn∈B ∥dρ(vn) · · · dρ(v1)Ω∥ < 2n√ +n! for any n ∈ N. Consequently, +qK(Ω) ≤ qsB(Ω) = +∞ +� +n=0 +sn +n! sup +vj∈B +∥dρ(vn) · · · dρ(v1)Ω∥ < +∞ +� +n=0 +(2s)n +√ +n! +< ∞, +∀s > 0. +It follows using Lemma 2.2.5 that Ω ∈ HO +ρ . Hence HO +ρ is dense in Hρ and ρ is strongly-entire. +3.2 +Properties of HO +ρ and holomorphic extensions +Let (ρ, Hρ) be a smooth unitary representation of G. In this section, we study some of the properties of the +locally convex space HO +ρ . These are summarized in Theorem 3.2.1 below: +11 + +Theorem 3.2.1. The locally convex space HO +ρ has the following properties: +1. The inclusion HO +ρ ֒→ H∞ +ρ +is continuous w.r.t. the weak topology on H∞ +ρ . +2. HO +ρ is Hausdorff and complete. +3. HO +ρ is both G- and g-invariant. +4. The map +gC × HO +ρ → HO +ρ , +(η, ψ) �→ +∞ +� +m=0 +f ψ +m(η) = +∞ +� +m=0 +1 +m!dρ(ηm)ψ +is entire and extends the map g × HO +ρ → HO +ρ , (η, ψ) �→ ρ(eξ)ψ. +5. The map G × HO +ρ → HO +ρ , (g, ψ) �→ ρ(g)ψ is real-analytic. +Before proceeding with the proof of Theorem 3.2.1, let us mention the following two important corollaries: +Corollary 3.2.2. Define the map +�ρC : gC → B(HO +ρ ), +�ρC(η)v := +∞ +� +n=0 +1 +n!dρ(ηn)v. +(3.2) +Let U ⊆ gC be open and convex. Assume that U ∩ g is non-empty and open in g. Suppose that the BCH +series defines a complex-analytic map ∗ : U × U → gC. Then �ρC(η ∗ ξ) = �ρC(η)�ρC(ξ) for any (η, ξ) ∈ U × U. +Proof. Define UR := U ∩g. Using Theorem 3.2.1(4) and the fact that compositions of analytic maps are again +analytic [BS71a, Thm. 6.4], it follows that the two maps (ξ, η, v) �→ �ρC(ξ)�ρC(η)v and (ξ, η, v) �→ �ρC(ξ ∗ η)v +are both complex-analytic U 2 × HO +ρ → HO +ρ . They agree on the real subspace UR × UR × HO +ρ , on which they +both equal (ξ, η, v) �→ ρ(eξ)ρ(eη)v = ρ(eξeη)v. It follows from Proposition 2.1.14 that they must be equal +everywhere, proving the assertion. +Corollary 3.2.3. Let (ρ, Hρ) be a continuous unitary G-representation and define �ρC : gC → B(HO +ρ ) by +equation (3.2). Let GC be a regular 1-connected complex BCH Fr´echet-Lie group with Lie(GC) = gC. Then +there is a representation ρC : GC → B(HO +ρ )× for which the corresponding action GC × HO +ρ → HO +ρ is complex- +analytic and such that ρC(eξ) = �ρC(ξ) holds true in a 0-neighborhood of gC. +Proof. As GC is a complex BCH Lie group, there are open symmetric convex 0-neighborhoods U, U ′ ⊆ gC such +that U ⊆ U ′, U ∩ g is open in g and the BCH series ∗ defines a complex-analytic map ∗ : U × U → U ′ ⊆ gC. +Shrinking U and U ′ if necessary, we may further assume that the restriction of expGC to U ′ is biholomorphic +onto some open 1-neighborhood V of GC. Define the function f : V → B(HO +ρ ) by f(eξ) := �ρC(ξ). In view of +Corollary 3.2.2, f satisfies +f(eξeη) = f(eξ∗η) = �ρC(ξ ∗ η) = �ρC(ξ)�ρC(η) = f(eξ)f(eη), +∀ξ, η ∈ U, +(3.3) +where the first equality follows from [Nee06, Thm. IV.2.8] and Proposition 2.1.15. In particular f(eξ) ∈ +B(HO +ρ )× and f(eξ)−1 = f(e−ξ) for any ξ ∈ U. As GC is a connected and simply connected topological +group, (3.3) further implies that there is a group homomorphism ρC : GC → B(HO +ρ )× extending f (cf. [GN, +Proposition C.2.1]). As expGC restricts to a biholomorphic map U ′ → V , it follows using Theorem 3.2.1(4) +that the action +V × HO +ρ → HO +ρ , +(eξ, v) �→ ρC(eξ)v = f(eξ)v = �ρC(ξ)v, +ξ ∈ U +is complex-analytic. As GC is a complex-analytic Lie group and V is an open 1-neighborhood in GC, this +implies that action GC × HO +ρ → HO +ρ , (g, v) �→ ρC(g)v is complex-analytic everywhere. +We proceed with the proof of Theorem 3.2.1. The inclusion HO +ρ ֒→ H∞ +ρ is continuous in the following sense: +Lemma 3.2.4. Let B ⊆ gC be compact and let ψ ∈ HO +ρ . +Then +1 +n!pn +B(ψ) ≤ qB(ψ) for any n ∈ N. +In +particular, the inclusion HO +ρ ֒→ H∞ +ρ +is continuous w.r.t. the weak topology on H∞ +ρ . +Proof. Let ψ ∈ HO +ρ . It is trivial that +1 +n!pn +B(ψ) ≤ qB(ψ). For the final statement, consider the continuous +seminorm pξ(ψ) := ∥dρ(ξ1 · · · ξn)ψ∥ on H∞ +ρ +for some ξ = (ξ1, · · · , ξn) ∈ gn. Taking for B the finite set +B := {ξ1, · · · , ξn} ⊆ gC, we obtain that +1 +n!pξ(ψ) ≤ +1 +n!pn +B(ψ) ≤ qB(ψ). +12 + +Lemma 3.2.5. HO +ρ is both Hausdorff and complete. +Proof. It is clear that HO +ρ is Hausdorff, because H∞ +ρ is so. Let us show that it is complete. Let (ψα)α∈I be +a Cauchy net in HO +ρ . Then it is also a Cauchy net in H∞ +ρ . The latter is complete [JN19, Prop. 3.19], where +we use that G is a regular Fr´echet-Lie group. Thus ψα → ψ in H∞ +ρ +for some ψ ∈ H∞ +ρ . We must show that +ψ ∈ HO +ρ and ψα → ψ in HO +ρ . Fix a compact set B ⊆ g. Let ǫ > 0. Choose ǫ0 > 0 such that ǫ0(1 + ǫ0) < ǫ. +Let t > 1 be such that +t +t−1 < 1 + ǫ0. As (ψα)α∈I is a Cauchy net in HO +ρ , there exists γ ∈ I such that +qtB(ψα − ψβ) < ǫ0 whenever α, β ≥ γ. In particular +1 +k!pk +B(ψα − ψβ) < ǫ0t−k for any α, β ≥ γ and k ∈ N≥0. +Consequently, for any ξi ∈ B with i ∈ {1, · · · , k} we have (using that ψα → ψ in H∞ +ρ ): +1 +k!∥dρ(ξ1 · · · ξk)(ψ − ψβ)∥ = 1 +k! lim +α ∥dρ(ξ1 · · · ξk)(ψα − ψβ)∥ ≤ ǫ0t−k +for β ≥ γ. +Thus +1 +k!pk +B(ψ − ψβ) ≤ ǫ0t−k for any β ≥ γ. Hence +qB(ψ − ψβ) = +∞ +� +k=0 +1 +k!pk +B(ψ − ψβ) ≤ ǫ0 +∞ +� +k=0 +t−k = +t +t − 1ǫ0 ≤ ǫ0(1 + ǫ0) < ǫ, +∀β ≥ γ +This shows that qB(ψ) ≤ qB(ψ − ψβ) + qB(ψβ) < ∞ and that qB(ψ − ψβ) < ǫ for all β ≥ γ. As B and ǫ were +arbitrary, we conclude (using the proof of Lemma 2.2.5) that ψ ∈ HO +ρ and ψα → ψ in HO +ρ . +Lemma 3.2.6. Let B, B0 ⊆ gC be compact subsets and let t > 1. Then there exists a compact subset B′ ⊆ gC +and some C > 0 such that B ⊆ B′ and +1 +m! +∞ +� +n=0 +1 +n! sup +ηj∈B0 +pn +B(dρ(η1 · · · ηm)ψ) < Ct−mqB′(ψ), +∀m ∈ N≥0, +∀ψ ∈ HO +ρ . +(3.4) +In particular, we have +1 +m!qB(dρ(ηm)ψ) ≤ Ct−mqB′(ψ) for any ψ ∈ HO +ρ , η ∈ B0 and m ∈ N≥0. +Proof. We may assume that B0 and B are both balanced. Define B′′ := B ∪ B0, which is again compact and +balanced in gC. For any η1, · · · , ηm ∈ B0 and ψ ∈ HO +ρ we have +pn +B(dρ(η1 · · · ηm)ψ) ≤ pn+m +B′′ (ψ) = t−(n+m)pn+m +tB′′ (ψ). +Thus supηj∈B0 pn +B(dρ(η1 · · · ηm)ψ) ≤ t−(n+m)pn+m +tB′′ (ψ). It follows that +∞ +� +n=0 +1 +n! sup +ηj∈B0 +pn +B(dρ(η1 · · · ηm)ψ) ≤ t−m +∞ +� +n=0 +t−n +n! pn+m +tB′′ (ψ) +≤ t−m +� ∞ +� +n=0 +t−n +�� ∞ +� +n=0 +1 +n!pn+m +tB′′ (ψ) +� += +t−m +1 − t−1 +∞ +� +n=0 +1 +n!pn+m +tB′′ (ψ) +Let s > 2. Notice that �∞ +n=0 +(n+m)! +n! +s−(n+m) < ∞, and +∞ +� +n=0 +1 +n!pn+m +tB′′ (ψ) = +∞ +� +n=0 +�(n + m)! +n! +s−(n+m) · +1 +(n + m)!pn+m +stB′′(ψ) +� +≤ +� ∞ +� +n=0 +(n + m)! +n! +s−(n+m) +� +· qstB′′(ψ). +Consequently, with Cm := �∞ +n=0 +(n+m)! +m!n! s−(n+m) and B′ := stB′′, we have: +1 +m! +∞ +� +n=0 +1 +n! sup +ηj∈B0 +pn +B(dρ(η1 · · · ηm)ψ) ≤ Cm +t−m +1 − t−1 qB′(ψ). +(3.5) +Using �N +k=0 +�N +k +� += 2N, notice that �∞ +m=0 Cm = �∞ +N=0 +� 2 +s +�N < ∞. Thus there exists C > 0 s.t. Cm ≤ C for +all m ∈ N≥0. Now simply observe using (3.5) that (3.4) holds for this C. Notice also that B ⊆ B′. +Lemma 3.2.7. HO +ρ is both G- and g-invariant. +13 + +Proof. Let ψ ∈ HO +ρ and let B ⊆ gC be compact. As the adjoint action of G on gC is continuous, Adg(B) is +again compact in gC. Since ρ(g) is unitary, we find that qB(ρ(g)ψ) = qAdg−1 (B)(ψ) < ∞. Thus ρ(g)ψ ∈ HO +ρ +and so HO +ρ is G-invariant. The g-invariance of HO +ρ is immediate from Lemma 3.2.6. +Lemma 3.2.8. Let E be a Fr´echet space and let X and Y be topological spaces. Let f : E × X → Y be +a function. Assume that f|B×X : B × X → Y is continuous for every compact subset B ⊆ E. Then f is +continuous. +Proof. Let U ⊆ Y be open. We write πE : E × X → E and πX : E × X → X for the canonical projections. +On the one hand, πX(f −1(U)) = � +e∈E πX(f|−1 +{e}×X (U)) is open in X. On the other hand, for any compact +B ⊆ E have B ∩ πE(f −1(U)) = πE ◦ f|−1 +B×X (U), which is open in B. As E is a Fr´echet space, it is first +countable and thus compactly generated. Therefore πE(f −1(U)) is open in E. So f −1(U) is open. +If B ⊆ gC is a compact subset, then the kernel of the seminorm qB on HO +ρ is trivial, ker(qB) = {0}, because +for ψ ∈ HO +ρ , qB(ψ) = 0 implies in particular that ∥ψ∥Hρ = 0. Let XB := HO +ρ +qB be the completion of +HO +ρ w.r.t. the norm qB on HO +ρ . We write ιB : HO +ρ ֒→ XB for the canonical continuous inclusion. The set +{ qB : B ⊆ gC compact } is directed and HO +ρ = lim +←−B XB is the corresponding projective limit of the Banach +spaces XB, as B runs over all compact subsets of gC. +Lemma 3.2.9. +1. The map fm : gC × HO +ρ → HO +ρ , fm(ξ, ψ) := +1 +m!dρ(ξm)ψ is continuous for every m ∈ N. +2. fm ∈ P m(gC × HO +ρ ; HO +ρ ) for every m ∈ N. +3. The series �∞ +m=0 fm(ξ, ψ) converges in HO +ρ for every (ξ, ψ) ∈ gC × HO +ρ and defines an entire map +f : gC × HO +ρ → HO +ρ , +f(ξ, ψ) := +∞ +� +m=0 +fm(ξ, ψ). +4. For any ξ ∈ g and ψ ∈ HO +ρ , we have f(ξ, ψ) = ρ(eξ)ψ. +Proof. +1. Let B0, B ⊆ gC be compact subsets and let t > 1. +By Lemma 3.2.6, there is a constant C > 0 +and a compact subset B′ ⊆ gC s.t. B ⊆ B′ and such that (3.4) holds true. In particular, we have +1 +m!qB(dρ(ηm)ψ) ≤ Ct−mqB′(ψ) for every η ∈ B0, ψ ∈ HO +ρ and m ∈ N≥0. +This implies for every +m ∈ N and η ∈ B0 that the linear map HO +ρ → XB, ψ �→ ιB(fm(η, ψ)) extends to a continuous linear +map Dm(η) : XB′ → XB whose operator norm satisfies ∥Dm(η)∥B(XB′ ;XB) ≤ Ct−m. Let m ∈ N. The +thus-obtained map +Dm : B0 × XB′ → XB, +Dm(η, ψ) := Dm(η)ψ +(3.6) +satisfies ιB ◦ fm|B0×HO +ρ += Dm ◦ (idB0 × ιB′) and is separately continuous in the XB′ variable by +construction. It is also continuous in the B0 argument. To see this, take ψ ∈ XB′ and let (ηn)n∈N be a +sequence in B0 with ηn → η for some η ∈ B0. We show that ∥Dm(ηn)ψ − Dm(η)ψ∥XB → 0 as n → ∞. +Suppose first that ψ ∈ HO +ρ . Consider the functions B0 → [0, ∞) defined by +τ(ξ) := qB +� +dρ(ξm − ηm)ψ +� +and +τN(ξ) := +N +� +k=0 +1 +k!pk +B +� +dρ(ξm − ηm)ψ +� +for N ∈ N. +The map τN is continuous for every N ∈ N, because +gm +C → H∞ +ρ , (η1, · · · , ηm) �→ dρ(η1 · · · ηm)ψ +is continuous w.r.t. the strong topology on H∞ +ρ +[JN19, Lem. 3.22], and because pk +B is a continuous +seminorm on H∞ +ρ w.r.t. to the strong topology. Moreover, for any ξ ∈ B0 we have +|τ(ξ) − τN(ξ)| = +∞ +� +k=N+1 +1 +k!pk +B(dρ(ξm − ηm)ψ) ≤ 2 +∞ +� +k=N+1 +1 +k! sup +ζ∈B0 +pk +B(dρ(ζm)ψ), +14 + +so it follows using (3.4) that τN → τ uniformly on B0. Hence τ is continuous and so τ(ηn) → τ(η) = 0 +as n → ∞. This means precisely that qB(dρ(ηm +n − ηm)ψ) → 0 as n → ∞. We thus obtain that +∥Dm(ηn)ψ − Dm(η)ψ∥XB = 1 +m!qB(dρ(ηm +n − ηm)ψ) +n→∞ +−−−−→ 0 +(3.7) +for every ψ ∈ HO +ρ ⊆ XB′, where we have suppressed ιB′ from the notation. Let us next consider general +ψ ∈ XB′. Let ǫ > 0 and ψ0 ∈ HO +ρ be s.t. ∥ψ − ψ0∥XB′ < ǫ. Using (3.7) we can find N ∈ N s.t. +∥Dm(ηn)ψ0 − Dm(η)ψ0∥XB < ǫ for all n ≥ N. Since ∥Dm(ξ)∥B(XB′;XB) ≤ Ct−m for all ξ ∈ B0, we +obtain for all n ≥ N that +∥Dm(ηn)ψ − Dm(η)ψ∥XB ≤ ∥Dm(ηn)(ψ − ψ0)∥XB + ∥Dm(ηn)ψ0 − Dm(η)ψ0∥XB + ∥Dm(η)(ψ0 − ψ)∥XB +< (2Ct−m + 1)ǫ +This shows that Dm(ηn)ψ → Dm(η)ψ in XB as n → ∞. So Dm is indeed separately continuous. +As B0 ⊆ gC is Fr´echet, XB′ and XB are both Banach and Dm(η) : XB′ → XB is linear for every +η ∈ B0, it follows using [Nee10a, Prop. 5.1] that Dm : B0 × XB′ → XB is jointly continuous. Since +ιB ◦ fm|B0×HO +ρ = Dm ◦ (idB0 × ιB′) we obtain that +ιB ◦ fm|B0×HO +ρ : B0 × HO +ρ → XB +is jointly continuous. This holds true for any compact subset B ⊆ gC, so using HO +ρ = lim +←−B XB we find +that fm|B0×HO +ρ : B0 × HO +ρ → HO +ρ is continuous. As the compact subset B0 ⊆ gC was arbitrary, it +follows by Lemma 3.2.8 that fm : gC × HO +ρ → HO +ρ is continuous. +2. Let m ∈ N. Define the symmetric m-linear map βm : (gC × HO +ρ )m → HO +ρ by +βm((η1, v1), · · · , (ηm, vm)) := 1 +m +m +� +i=1 +βvi +m(η1, · · · , ηm). +Notice that βm(∆m(η, v)) = βv +m(∆m(η)) = f v +m(η) = +1 +m!dρ(ηm)v, so fm is a homogeneous polynomial. +It is also continuous by the first item. So fm ∈ P m(gC × HO +ρ ; HO +ρ ). +3. Let B0, B ⊆ gC be compact subsets and let t > 1. By Lemma 3.2.6, there is a constant C ≥ 1 and +a compact subset B′ ⊆ gC s.t. (3.4) holds true and B ⊆ B′. We may assume that C ≥ 1. For every +m ∈ N, let Dm : B0 × XB′ → XB be defined by (3.6). As B ⊆ B′, we have qB ≤ qB′, so there is a +unique continuous linear contraction I : XB′ → XB extending ιB : HO +ρ → XB. Set D0(z, ψ) := z I(ψ) +for z ∈ C and ψ ∈ XB′. Then ∥Dm(η)ψ∥XB ≤ Ct−m∥ψ∥XB′ for any ψ ∈ XB′, η ∈ B0 and m ∈ N≥0. +Consequently, +∞ +� +m=0 +sup +η∈B0 +∥Dm(η)ψ∥XB ≤ C +� +∞ +� +m=0 +t−m +� +∥ψ∥XB′ = +C +1 − t−1 ∥ψ∥XB′, +∀ψ ∈ XB′. +(3.8) +As B and B0 were arbitrary, (3.8) in particular implies that the series �∞ +m=0 fm(η, ψ) converges in HO +ρ +for any η ∈ gC and ψ ∈ HO +ρ . It remains only to show that f is continuous. The function +D : B0 × XB′ → XB, +D(η, ψ) := +∞ +� +m=0 +Dm(η)ψ +satisfies ιB ◦ f|B0×HO +ρ = D ◦ (idB0 × ιB′), by the corresponding property of every Dm. It is moreover +separately continuous in the XB′ variable, by (3.8). Using that ∥Dm(η)∥B(XB′;XB) ≤ Ct−m for every +η ∈ B0 we know by a computation similar to (3.8) that for any compact subset K ⊆ XB′ we have +∞ +� +m=0 +sup +η∈B0 +sup +ψ∈K +∥Dm(η)ψ∥XB ≤ +C +1 − t−1 sup +ψ∈K +∥ψ∥XB′ < ∞. +This implies that the functions �N +m=0 Dm : B0 × XB′ → XB converge to D uniformly on compact +subsets. As �N +m=0 Dm is continuous for every N ∈ N and B0 × XB′ is Fr´echet, it follows that D is +continuous. Consequently, ιB ◦ f|B0×HO +ρ : B0 × HO +ρ → XB is continuous. This holds true for any +compact B ⊆ gC, which in turn implies that f|B0×HO +ρ : B0 × HO +ρ → HO +ρ is continuous. As the compact +subset B0 ⊆ gC was arbitrary, Lemma 3.2.8 implies that f : gC × HO +ρ → HO +ρ is continuous. +15 + +4. Let ψ ∈ HO +ρ . Then ψ is in particular a G-analytic vector, by Corollary 2.2.9. Consequently, the two +maps ξ �→ ρ(eξ)ψ and ξ �→ f(ξ, ψ) are both real analytic as maps g → Hρ. They moreover have the +same image under the jet-projection j∞ +0 : Cω(g; Hρ) → P(g; Hρ). Using Proposition 2.1.15, we conclude +that ρ(eξ)ψ = f(ξ, ψ) for all ξ ∈ g. +Lemma 3.2.10. The function G × HO +ρ → HO +ρ , (g, v) �→ ρ(g)v is real-analytic. +Proof. As the Lie group G is BCH, Lemma 3.2.9 implies that the map G × HO +ρ → HO +ρ , (g, v) �→ ρ(g)v is +analytic on U × HO +ρ for some 1-neighborhood U ⊆ G, which implies the assertion. +4 +A general approach to holomorphic induction +In this section, we define and study a notion of holomorphic induction of unitary representations of Lie groups. +The presented definition and results extend that of [Nee13], by removing the requirement of norm-continuity +of the representation that is induced from. We also no longer require the Lie group to be Banach, allowing +it to be Fr´echet instead. The precise setting we consider is as follows: +Let G be a connected regular BCH Fr´echet-Lie group with Lie algebra g. Let θ : gC → gC be the conjugation +defined by θ(ξ + iη) = ξ − iη for ξ, η ∈ g. We assume given a triangular decomposition gC = n− ⊕ hC ⊕ n+, +where n± and hC are closed Lie subalgebras of gC satisfying θ(n±) ⊆ n∓ and [hC, n±] ⊆ n±. Let H be a +connected Lie subgroup of G, in the sense that it is both a closed subgroup and an embedded submanifold +of G, with Lie algebra Lie(H) = h. Set b± := n± ⋊ hC. +The structure of this chapter is as follows. In Section 4.1 we establish some notation and preliminary def- +initions, in particular specifying a certain space of functions on G that takes the role usually taken by the +holomorphic sections of a complex homogeneous vector bundle over G/H. In Section 4.2 we present the +definition of holomorphically induced representations and establish an equivalent characterization. We then +proceed in Section 4.3, Section 4.4 and Section 4.5 to study the most important properties enjoyed by holo- +morphically induced representations. +As the theory of this section no longer has a clear interpretation in terms of holomorphic maps, we present in +Section 5 a stronger notion that involves complex geometry. The approach presented there depends crucially +on the availability of a dense set of strongly-entire vectors in the representation that is induced from. +4.1 +A substitute for holomorphic sections +Let (σ, Vσ) be an analytic unitary representation of H. +Let us establish some notation and preliminary +definitions. +Definition 4.1.1. For ξ ∈ g, define the differential operators Lv(ξ) and Lv(ξ)r on C∞(G; Vσ) by +(Lv(ξ)f)(g) := d +dt +���� +t=0 +f(getξ), +(Lv(ξ)rf)(g) := d +dt +���� +t=0 +f(e−tξg), +g ∈ G, f ∈ C∞(G; Vσ). +Extend both ξ �→ Lv(ξ) and ξ �→ Lv(ξ)r C-linearly to gC and further to algebra homomorphisms on U(gC), so +we have e.g. Lv(ξ1···ξn)r = Lv(ξ1)r · · · Lv(ξn)r for all ξk ∈ gC and k ∈ {1, · · · , n}. +Remark 4.1.2. We thus adopt the convention that for ξ ∈ g, v(ξ) denotes the left-invariant vector field on G +associated to ξ ∈ g whereas v(ξ)r denotes the right-invariant one. +Definition 4.1.3. Let D ⊆ V ω +σ be subspace that is dense in Vσ. +— An extension of dσ to b± with domain D is a Lie algebra homomorphism χ : b± → L(D) such that +χ(ξ) = dσ(ξ)|D for all ξ ∈ hC. We call (σ, χ) an (H, b−)-extension pair with domain D. +— The trivial extension of dσ to b± with domain D is defined by letting n± act trivially on D. +Definition 4.1.4. For k ∈ {1, 2}, let (σk, χk) be an (H, b−)-extension pair with domain Dk. +We say +that (σ1, χ) and (σ2, χ2) are unitarily equivalent if there is a unitary isomorphism U : Vσ1 → Vσ2 of H- +representation such that UD1 = D2 and Uχ1(ξ)v = χ2(ξ)Uv for all ξ ∈ b− and v ∈ D1. In this case we write +(σ1, χ1) ∼= (σ2, χ2). +16 + +Definition 4.1.5. For k ∈ {1, 2}, let (σk, χk) be an (H, b−)-extension pair with domain Dk. Define the +direct sum (σ1, χ1) ⊕ (σ2, χ2) := (σ1 ⊕ σ2, χ1 ⊕ χ2), where χ1 ⊕ χ2 is defined by +χ1 ⊕ χ2 : b− → L(D1 ⊕ D2), +(χ1 ⊕ χ2)(ξ)(v1, v2) = (χ1(ξ)v1, χ2(ξ)v2). +Definition 4.1.6. Let (σ, χ) be an (H, b−)-extension pair with domain D. We say that (σ, χ) is decomposable +if (σ, χ) ∼= (σ1, χ1) ⊕ (σ2, χ2) for some non-trivial (H, b−)-extension pairs (σ1, χ1) and (σ2, χ2). We say that +(σ, χ) is indecomposable if it is not decomposable. +Recall the definition of the involutions τ, θ and (−)∗ on U(gC), specified in Definition 2.2.1. Recalling that +θ(ξ + iη) = θ(ξ − iη) for ξ, η ∈ g, the involutions τ and (−)∗ satisfy τ(ξ) = −ξ and ξ∗ = −θ(ξ) for ξ ∈ gC. +Extensions are used to specify a suitable G-subrepresentation of Cω(G; Vσ)H: +Definition 4.1.7. Let (σ, χ) be an (H, b−)-extension pair with domain D. Define +Cω(G; Vσ)H := +� +f ∈ Cω(G, V ) : f(gh) = σ(h)−1f(g), +∀ g ∈ G, h ∈ H +� +Cω(G; Vσ)H,χ := +� +f ∈ Cω(G; V )H : ⟨v, Lv(ξ)f⟩ = −⟨χ(ξ∗)v, f⟩, +∀ξ ∈ b+, v ∈ D +� +. +Proposition 4.1.8. Let (σ, χ) be an (H, b−)-extension pair with domain D. Let f ∈ Cω(G; Vσ)H,χ. Then +f(g) ∈ dom(χ(x∗)∗) +and +(Lv(τ(x))f)(g) = χ(x∗)∗f(g), +∀x ∈ U(b+), ∀g ∈ G. +Proof. Let v ∈ D. Suppose that x = ξ1 · · · ξn for n ∈ N and ξi ∈ b+. Observe that f(g) ∈ dom(χ(η∗)∗) and +(Lv(η)f)(g) = −χ(η∗)∗f(g) for any g ∈ G and η ∈ b+, as a consequence of Definition 4.1.7. It follows by +induction on n ∈ N that ⟨v, Lv(ξ1···ξn)f⟩ = (−1)n⟨χ(ξ∗ +1 · · · ξ∗ +n)v, f⟩. This implies ⟨v, Lv(τ(x))f⟩ = ⟨χ(x∗)v, f⟩ +for any x ∈ U(b+) and v ∈ D. The assertion follows. +4.2 +Holomorphically induced representations +We now define holomorphically induced representations. Fix throughout the section an (H, b−)-extension +pair (σ, χ) with a domain Dχ ⊆ V ω +σ that is dense in Vσ. Let (ρ, Hρ) be a unitary representation of G. +Remark 4.2.1. The theory of holomorphic induction presented in the upcoming section makes use of repro- +ducing kernel Hilbert spaces. For more details thereon, one may refer e.g. to [Nee00, Ch. I-II]. The most +relevant properties are recalled in Section A below. +Definition 4.2.2. We say that (ρ, Hρ) is holomorphically induced from (σ, χ) if there exists a G-equivariant +injective linear map Φ : Hρ ֒→ Map(G; Vσ)H satisfying the following conditions: +1. The point evaluation Ex : Hρ → Vσ, Ex(ψ) := Φψ(x) is continuous for every x ∈ G. +2. ExE∗ +x = idVσ for every x ∈ G. +3. Dχ = +� +v ∈ Vσ : Φ(E∗ +e v) ∈ Cω(G; Vσ)H,χ � +. +Remark 4.2.3. The first condition entails that (ρ, Hρ) is unitarily equivalent to the natural G-representation +on the reproducing kernel Hilbert space HQ, where Q ∈ C(G × G, B(Vσ))H×H is the positive definite and +G-invariant kernel defined by Q(x, y) := ExE∗ +y, see also Proposition A.1.5 below. +We have the following equivalent characterization, whose proof comprises the remainder of the section: +Theorem 4.2.4. The following assertions are equivalent. +1. The G-representation (ρ, Hρ) is holomorphically induced from the (H, b−)-extension pair (σ, χ). +2. There is a closed H-invariant subspace V ⊆ Hρ with the following properties: +(a) V is cyclic for the G-representation Hρ. +(b) D�χ := V ∩ Hω +ρ satisfies dρ(n−)D�χ ⊆ D�χ. +(c) (σ, χ) is unitarily equivalent to (�σ, �χ), where (�σ, �χ) is the (H, b−)-extension pair defined by +�σ : H → U(V ), +�χ : b− → L(D�χ), +�σ(h) := ρ(h)|V , +�χ(ξ) := dρ(ξ)|D � +χ . +If these equivalent assertions are satisfied, then ρ is an analytic G-representation. +17 + +We proceed with the proof of Theorem 4.2.4. We have the following simple but important observation: +Lemma 4.2.5. Let Φ : Hρ ֒→ Map(G; Vσ)H be a G-equivariant injective linear map. Assume that the point +evaluation Ex(ψ) := Φψ(x) is continuous for every x ∈ G. Write fv := Φ(E∗ +e v) ∈ Map(G; Vσ)H for v ∈ Vσ. +Then: +1. Eg = Eeρ(g)−1 for any g ∈ G. +2. Φψ(g) = Eeρ(g)−1ψ for any ψ ∈ Hρ. In particular, fv(g) = Eeρ(g)−1E∗ +e v for v ∈ Vσ. +3. Let v ∈ Vσ. Then E∗ +e v ∈ Hω +ρ ⇐⇒ fv ∈ Cω(G; Vσ)H ⇐⇒ ⟨v, fv⟩ ∈ Cω(G; C). +Proof. +1. As Φ is G-equivariant, we have Egψ = Φψ(g) = Φρ(g)−1ψ(e) = Eeρ(g)−1ψ for any ψ ∈ Hρ. +2. This is immediate from the first assertion. +3. Let v ∈ Vσ. If E∗ +e v ∈ Hω +ρ , then the orbit map g �→ ρ(g)E∗ +e v is analytic G → Hρ. It follows that +fv(g) = Eeρ(g)−1E∗ +e v is analytic G → Vσ, which in turn implies that ⟨v, fv⟩ ∈ Cω(G; C). Assume that +⟨v, fv⟩ ∈ Cω(G; C). Then g �→ ⟨E∗ +e v, ρ(g)E∗ +e v⟩Hρ = ⟨v, fv(g−1)⟩V is analytic. As G is a BCH Fr´echet-Lie +group, this implies using [Nee11, Thm. 5.2] that E∗ +e v ∈ Hω +ρ . +We first prove that (1) =⇒ (2) in Theorem 4.2.4. Assume that ρ is holomorphically induced from (σ, χ). +Let the map Φ : Hρ → Map(G; Vσ)H satisfy the conditions in Definition 4.2.2. Let Ex := evx ◦Φ be the point +evaluation at x ∈ G. We write fv := Φ(E∗ +e v) ∈ Cω(G; Vσ)H,χ for v ∈ Dχ. +We show that the H-invariant subspace W := E∗ +e Vσ ⊆ Hρ satisfies the conditions in Theorem 4.2.4. Define +D�χ := E∗ +e Dχ ⊆ W. By Theorem A.1.3 we know that ρ(G)W = � +g∈G E∗ +g Vσ is total in Hρ, so that W is cyclic +for ρ. It is moreover immediate from Lemma 4.2.5 that D�χ ⊆ Hω +ρ . Because D�χ is dense in the cyclic subspace +W and Hω +ρ is G-invariant, we obtain that Hω +ρ is dense in Hρ. Hence ρ is analytic. +Lemma 4.2.6. Let v ∈ Dχ. The following assertions hold true: +1. Eeρ(g)E∗ +e v ∈ dom(χ(x∗)∗) and Eedρ(x)ρ(g)E∗ +e v = χ(x∗)∗Eeρ(g)E∗ +e v for any x ∈ U(b+) and g ∈ G. +2. dρ(b−)D�χ ⊆ D�χ and dρ(x)E∗ +e v = E∗ +e χ(x)v for any x ∈ U(b−). +Proof. +1. Let x ∈ U(b+). Since fv ∈ Cω(G; Vσ)H,χ, we obtain from Proposition 4.1.8 that fv(e) ⊆ dom(χ(x∗)∗) +and that Lv(τ(x))fv = χ(x∗)∗fv. On the other hand, notice using the formula fv(g) = Eeρ(g)−1E∗ +e v that +(Lv(τ(x))fv)(g) = Eedρ(x)ρ(g)−1E∗ +e v holds true for any g ∈ G, say by induction on the degree of x. We +thus obtain that Eedρ(x)ρ(g)−1E∗ +e v = χ(x∗)∗fv(g) = χ(x∗)∗Eeρ(g)−1E∗ +e v for any g ∈ G. +2. Let x ∈ U(b−). Recall from Lemma 4.2.5 that D�χ ⊆ Hω +ρ . Let ψ ∈ ρ(G)D�χ. Using the first assertion, +observe that ⟨E∗ +e χ(x)v, ψ⟩ = ⟨v, χ(x)∗Eeψ⟩ = ⟨v, Eedρ(x∗)ψ⟩ = ⟨dρ(x)E∗ +e v, ψ⟩. As D�χ is cyclic for G in +Hρ, it follows that E∗ +e χ(x)v = dρ(x)E∗ +e v. In particular dρ(b−)D�χ ⊆ D�χ. +Define the unitary H-action �σ on W by �σ(h) = ρ(h)|W . Consider the extension �χ(ξ) := dρ(ξ)|D � +χ of d�σ to +b−, whose domain is D�χ. By Lemma 4.2.6, E∗ +e defines a unitary equivalence between (σ, χ) and (�σ, �χ). +Lemma 4.2.7. D�χ = W ∩ Hω +ρ . +Proof. The inclusion D�χ ⊆ W ∩ Hω +ρ follows from Lemma 4.2.5. Let w ∈ W ∩ Hω +ρ . Then w = E∗ +e v for some +v ∈ Vσ. We must show that v ∈ Dχ. Lemma 4.2.5 implies that fv ∈ Cω(G, Vσ)H. Let v2 ∈ Dχ and ξ ∈ b+. +Using Lemma 4.2.6 and the formula fv(g) = Eeρ(g)−1E∗ +e v we obtain: +⟨v2, (Lv(ξ)fv)(g)⟩ = −⟨dρ(ξ∗)E∗ +e v2, ρ(g)−1E∗ +e v⟩ = −⟨χ(ξ∗)v2, Eeρ(g)−1E∗ +e v⟩ = −⟨χ(ξ∗)v2, fv(g)⟩. +It follows that fv ∈ Cω(G; Vσ)H,χ. By the third property in Definition 4.2.2, this means that v ∈ Dχ. +This completes the proof of (1) =⇒ (2) in Theorem 4.2.4. The converse is Lemma 4.2.8 below: +Lemma 4.2.8. Let (ρ, Hρ) be a unitary representation of G. Let V ⊆ Hρ be a closed H-invariant subspace. +Define a H-representation σ on V by σ(h) := ρ(h)|V . Set Dχ = V ∩ Hω +ρ . Assume that ρ(G)V is total in +Hρ, that Dχ is dense in Vσ and that dρ(n−)Dχ ⊆ Dχ. Define the extension χ(ξ)v := dρ(ξ)v of dσ to b− with +domain Dχ, where ξ ∈ b− and v ∈ Dχ. Then ρ is holomorphically induced from (σ, χ). +18 + +Proof. Let pV : Hρ → Vσ denote the orthogonal projection onto Vσ. For ψ ∈ Hρ, define Φψ(g) := pV ρ(g)−1ψ. +Consider the linear map Φ : Hρ → C(G; Vσ)H, ψ �→ Φψ. It is clear that Φ is G-equivariant and that the +point-evaluation Eg = pV ρ(g)−1 is continuous for any g ∈ G. +The map Φ injective because Φψ = 0 is +equivalent to ψ ⊥ ρ(G)V and ρ(G)V is total in Hρ, by assumption. Notice next that E∗ +g v = ρ(g)v for any +v ∈ V and so EgE∗ +g = idV . Write V 0 := {v ∈ V : Φv ∈ Cω(G; Vσ)H,χ}. It remains to show that Dχ = V 0. It +is immediate from the third assertion in Lemma 4.2.5 that V 0 ⊆ Dχ. Suppose conversely that v ∈ Dχ. Then +Φv ∈ Cω(G; Vσ)H by Lemma 4.2.5. Let ξ ∈ b+ and w ∈ Dχ. Using Lv(ξ)Φv(g) = −pV dρ(ξ)ρ(g)−1v, we find: +⟨w, Lv(ξ)Φv(g)⟩ = −⟨dρ(ξ∗)w, ρ(g)−1v⟩ = −⟨χ(ξ∗)w, ρ(g)−1v⟩ = −⟨χ(ξ∗)w, Φv(g)⟩. +Thus Φv ∈ Cω(G; Vσ)H,χ, which means that v ∈ V 0. Thus V 0 = Dχ. +4.3 +Uniqueness +In the following, we determine that there is up to unitary equivalence at most one unitary G-representation +that is holomorphically induced from a given (H, b−)-extension pair. Let (σ, χ) be such an (H, b−)-extension +pair, whose domain Dχ ⊆ V ω +σ is dense in Vσ. Let (ρ, Hρ) be a unitary representation of G. +Definition 4.3.1. We say that (σ, χ) is holomorphically inducible to G if there is a unitary G-representation +which is holomorphically induced from (σ, χ). +Proposition 4.3.2. Assume that ρ is holomorphically induced from (σ, χ). Let Φ : Hρ ֒→ Map(G; Vσ)H +satisfy the conditions in Definition 4.2.2 and write Ex := evx ◦Φ for the point evaluation at x ∈ G. Define: +F : G → B(Vσ), +F(g) := Eeρ(g)E∗ +e . +Then F satisfies the following properties: +1. F(e) = idVσ. +2. Q : G × G → B(Vσ), Q(x, y) := F(x−1y) is positive definite (c.f. Definition A.1.2). +3. Dχ = { v ∈ Vσ : g �→ ⟨v, F(g)v⟩ is real-analytic G → C }. +4. For all v, w ∈ Dχ and g ∈ G, ξ ∈ b+ we have: +� +Lv(ξ)r⟨w, Fv⟩ +� +(g) = −⟨χ(ξ)∗w, F(g)v⟩. +(4.1) +Finally, ρ is unitarily equivalent to the G-representation on the reproducing kernel Hilbert space HQ. +Proof. Define the �Q : G × G → B(Vσ) by �Q(x, y) := ExE∗ +y, which is positive-definite by Theorem A.1.3. In +view of the first assertion in Lemma 4.2.5, we have �Q(x, y)v = Eeρ(x−1y)E∗ +e v = F(x−1y)v = Q(x, y)v for any +v ∈ Vσ. Thus �Q = Q. In particular, Q is positive definite and F(e) = Q(e, e) = idVσ. Let v ∈ Vσ. Writing +fv := Φ(E∗ +e v), notice that fv(g) = F(g−1)v for g ∈ G. We find that E∗ +e v ∈ Hω +ρ +⇐⇒ ⟨v, Fv⟩ ∈ Cω(G; C), +using Lemma 4.2.5. Then Dχ = +� +v ∈ Vσ : E∗ +e v ∈ Hω +ρ +� += { v ∈ Vσ : ⟨v, Fv⟩ ∈ Cω(G; C) }, where we used +Lemma 4.2.7 in the first equality. Finally, notice that ⟨w, F(g)v⟩ = ⟨E∗ +e w, ρ(g)E∗ +e v⟩ for v, w ∈ Dχ and g ∈ G. +It thus follows from Lemma 4.2.6 that F satisfies (4.1) for all g ∈ G and ξ ∈ b+. The final statement is +immediate from Proposition A.1.5. +The next result, Theorem 4.3.3, gives a characterization of (σ, χ) being holomorphically inducible in terms +B(Vσ)-valued positive-definite functions on G. +Theorem 4.3.3. The following assertions are equivalent: +1. (σ, χ) is holomorphically inducible. +2. There is a function F : G → B(Vσ) satisfying the properties in Proposition 4.3.2. +Assume that these assertions are valid. Let F : G → B(Vσ) satisfy the conditions in Proposition 4.3.2. Then +F(g)∗ = F(g−1) for all g ∈ G. Moreover, for v ∈ Dχ and w ∈ Vσ we have: +� +Lv(x+)rLv(x−)⟨w, Fv⟩ +� +(g) = ⟨w, χ(τ(x+)∗)∗F(g)χ(x−)v⟩ +∀g ∈ G, x± ∈ U(b±) +(4.2) +� +Lv(x+x−)⟨w, Fv⟩ +� +(e) = ⟨w, χ(x∗ ++)∗χ(x−)v⟩, +∀ x± ∈ U(b±). +(4.3) +Finally, the function F : G → B(Vσ) is unique. +19 + +Proof. The implication (1) =⇒ (2) is immediate from Proposition 4.3.2. Conversely, let F : G → B(Vσ) be a +function satisfying the conditions in Proposition 4.3.2. Write Q(x, y) := F(x−1y) for x, y ∈ G. Let Hρ be the +corresponding reproducing kernel Hilbert space. Using Proposition A.1.5 we obtain a unitary representation +ρ of G on Hρ and a G-equivariant injective linear map Φ : Hρ → Map(G, Vσ)H for which the point evaluation +Ex := evx ◦Φ is continuous and satisfies Ex = Eeρ(x)−1 for every x ∈ G. From F(e) = idVσ it follows that +Q(x, x) = idVσ for every x ∈ G. Write fv := Φ(E∗ +e v) for v ∈ Vσ. +To see that (1) holds true, it remains only to show that Dχ = +� +v ∈ Vσ : fv ∈ Cω(G; Vσ)H,χ � +. Let x ∈ G +and v ∈ Vσ. From the equations fv(x) = ExE∗ +e v = Q(x, e)v = F(x−1)v and ExE∗ +e v = Eeρ(x)E∗ +e v, we conclude +that F(x)v = Eeρ(x)E∗ +e v = fv(x−1). It follows that +Dχ = { v ∈ Vσ : ⟨v, Fv⟩ ∈ Cω(G; C) } = +� +v ∈ Vσ : fv ∈ Cω(G; Vσ)H � +, +where Lemma 4.2.5 was used in the second equality. Assume that fv ∈ Cω(G; Vσ)H. Let w ∈ Dχ and ξ ∈ b+. +From the equation F(g)v = fv(g−1) we obtain that Lv(ξ)fv(g) = +� +Lv(ξ)rFv +� +(g−1) for any g ∈ G. Using +Equation (4.1) we find: +� +w, Lv(ξ)fv(g) +� += +� +Lv(ξ)r⟨w, Fv⟩ +� +(g−1) = −⟨χ(ξ∗)w, F(g−1)v⟩ = −⟨χ(ξ∗)w, fv(g)⟩ +∀g ∈ G. +Hence fv ∈ Cω(G; Vσ)H,χ. Thus Dχ = +� +v ∈ Vσ : fv ∈ Cω(G; Vσ)H,χ � +. We conclude that (ρ, Hρ) is holo- +morphically induced from (σ, χ). So (1) ⇐⇒ (2). +Assume these equivalent assertions are satisfied. From F(g) = Eeρ(g)E∗ +e it is immediate that F(g−1) = F(g)∗ +for all g ∈ G. We next show (4.2) and (4.3). Let v ∈ Dχ. Notice using F(g) = Eeρ(g)E∗ +e that for any +x, y ∈ U(gC) we have +� +Lv(y)rLv(x)Fv +� +(g) = Eedρ(τ(y))ρ(g)dρ(x)E∗ +e v, +∀g ∈ G. +(4.4) +Thus, for x± ∈ U(b±) we obtain using (4.4) and Lemma 4.2.6 that +� +Lv(x+)rLv(x−)Fv +� +(g) = Eedρ(τ(x+))ρ(g)dρ(x−)E∗ +e v = χ(τ(x+)∗)∗Eeρ(g)E∗ +e χ(x−)v, +(4.5) +� +Lv(x+x−)Fv +� +(e) = Eedρ(x+x−)E∗ +e v = χ(x∗ ++)∗χ(x−)v, . +(4.6) +From (4.5) we conclude that +� +Lv(x+)rLv(x−)Fv +� +(g) = χ(τ(x+)∗)∗F(g)χ(x−)v for all g ∈ G, which implies +(4.2). +On the other hand, (4.3) is implied by (4.6). +Finally, assume that F1 and F2 are two functions +satisfying the conditions in Proposition 4.3.2. Let v ∈ Dχ. The functions g �→ F1(g)v and g �→ F2(g)v are +both analytic and satisfy (4.6). As U(gC) is spanned by U(n+)U(b−) by the PBW Theorem, it follows that +j∞ +e (F1v) = j∞ +e (F2v). As G is connected, it follows from Proposition 2.1.15 that F1(g)v = F2(g)v for all g ∈ G +and v ∈ Dχ. For any fixed g ∈ G, the map v �→ (F1(g) − F2(g))v is continuous and vanishes on the dense +subset Dχ ⊆ Vσ. Hence F1 = F2. +Combining Proposition 4.3.2 with the uniqueness of F : G → B(Vσ) in Theorem 4.3.3, we obtain the desired +uniqueness of the holomorphically induced representation up to unitary equivalence: +Theorem 4.3.4. Let ρ1 and ρ2 be unitary G-representations which are both holomorphically induced from +(σ, χ). Then ρ1 ∼= ρ2 as unitary G-representations. +Finally, we focus our attention on the important special case where χ is a trivial extension. Using the PBW +Theorem, notice that we have the vector space decomposition U(gC) = U(hC) ⊕ (n+U(gC) + U(gC)n−). +Definition 4.3.5. Let E0 : U(gC) → U(hC) ∼= U(gC)/(n+U(gC) + U(gC)n−) be the quotient map. +Lemma 4.3.6. Assume that ρ is holomorphically induced from (σ, χ), where χ be the trivial extension of dσ +to b− with domain D ⊆ Vσ. Let v ∈ D and x ∈ U(gC). Then dσ(E0(x∗))v = dσ(E0(x))∗v. Moreover for all +w ∈ V ∞ +σ +we have ⟨w, dρ(x)v⟩ = ⟨w, dσ(E0(x))v⟩ and ⟨dρ(x∗)v, w⟩ = ⟨v, dσ(E0(x))w⟩. +Proof. By Theorem 4.2.4 we may assume that Vσ ⊆ Hρ is a closed subspace, Dχ = V ∩Hω +ρ , σ(h) = ρ(h)|Vσ and +χ(ξ) = dρ(ξ)|Dχ for every h ∈ H and ξ ∈ b−. Let pV : Hρ → Vσ be the orthogonal projection onto Vσ. Let +v ∈ Dχ, ξ+ ∈ n+, x ∈ U(gC) and ξ− ∈ n−. From Lemma 4.2.6 we obtain pV dρ(xξ−)v = pV dρ(x)χ(ξ−)v = 0 +and pV dρ(ξ+x)v = χ(ξ∗ ++)∗pV dρ(x)v = 0. Thus pV dρ(x)v = pV dρ(E0(x))v = dσ(E0(x))v for any x ∈ U(gC). +Let w ∈ Dχ. Recall from Lemma 4.2.7 that Dχ ⊆ Hω +ρ . We have: +⟨dσ(E0(x∗))v, w⟩ = ⟨dρ(x∗)v, w⟩ = ⟨v, dρ(x)w⟩ = ⟨v, dσ(E0(x))w⟩ = ⟨dσ(E0(x))∗v, w⟩. +As Dχ is dense in Vσ we conclude dσ(E0(x∗))v = dσ(E0(x))∗v. Consequently, if w ∈ V ∞ +ρ +then +⟨dρ(x∗)v, w⟩ = ⟨dσ(E0(x∗))v, w⟩ = ⟨dσ(E0(x))∗v, w⟩ = ⟨v, dσ(E0(x))w⟩. +20 + +We complement Theorem 4.3.3 with the following result, regarding the uniqueness of the domain: +Proposition 4.3.7. Let σ be an analytic unitary representation of H. Assume that there exists a subspace +Dχ ⊆ V ω +σ +dense in Vσ for which (σ, χ) is holomorphically inducible, where χ b− → L(Dχ) is the trivial +extension of dσ to b− with domain Dχ. Then Dχ is unique with this property. +Proof. Suppose that D1 and D2 are two such domains. For k ∈ {1, 2}, let χk denote the trivial extension +of dσ to b− with domain Dk. By assumption (σ, χk) is holomorphically inducible. Let Fk : G → B(Vσ) +satisfy the conditions in Proposition 4.3.2 for (σ, χk). Let vk ∈ Dk. Observe using Lemma 4.3.6 that for any +k ∈ {1, 2}, x ∈ U(b−) and v ∈ Dk we have χk(x)v = dσ(E0(x))v and χk(x)∗v = dσ(E0(x))∗v = dσ(E0(x∗))v. +Consider the functions a, b : G → C defined by a(g) := ⟨v1, F1(g)v2⟩ and b(g) := ⟨v1, F2(g)v2⟩. Notice that +both a and b are analytic, where we remark that a(g) = ⟨F1(g−1)v1, v2⟩. Let x± ∈ U(b±). Using (4.3) we +obtain: +(Lv(x+x−)b)(e) = ⟨v1, χ2(x∗ ++)∗χ2(x−)v2⟩ = ⟨v1, dσ(E0(x+))dσ(E0(x−))v2⟩ = ⟨dσ(E0(x∗ ++))v1, dσ(E0(x−))v2⟩. +We next compute (Lv(x+x−)a)(e). Let ι : G → G, g �→ g−1 denote the inversion on G and Σ : C → C, z �→ z +the conjugation on C. Define h : G → C by h(g) = ⟨v2, F1(g)v1⟩, so that a = Σ◦ h◦ ι. For any x ∈ U(gC) and +f ∈ C∞(G; C), we have +� +Lv(x)(f ◦ ι) +� +(e) = (Lv(τ(x))f)(e) and +� +Lv(x)(Σ ◦ f) +� +(e) = Σ +� +Lv(θ(x))f +� +(e). Using +these equations we obtain that (Lv(x)a)(e) = Σ +� +Lv(x∗)h +� +(e) for any x ∈ U(gC). By equation (4.3) we have +� +Lv(x∗ +−x∗ ++)h +� +(e) = ⟨v2, χ(x−)∗χ(x∗ ++)v1⟩ = ⟨v2, dσ(E0(x−))∗dσ(E0(x∗ ++))v1⟩ = ⟨dσ(E0(x−))v2, dσ(E0(x∗ ++))v1⟩. +Thus +(Lv(x+x−)a)(e) = Σ +� +Lv(x∗ +−x∗ ++)h +� +(e) = ⟨dσ(E0(x∗ ++))v1, dσ(E0(x−))v2⟩ = (Lv(x+x−)b)(e). +As U(gC) is spanned by elements in U(n+)U(b−) by the PBW Theorem, it follows that j∞ +e (a) = j∞ +e (b). Since +G is connected, it follows from Proposition 2.1.15 that a = b. Thus ⟨v1, F1(g)v2⟩ = ⟨v1, F2(g)v2⟩ for all g ∈ G, +v1 ∈ D1 and v2 ∈ D2. As both D1 and D2 are dense, it follows that F1 = F2 =: F. From the third property +in Proposition 4.3.2, we conclude that D1 = D2 = { v ∈ Vσ : g �→ ⟨v, F(g)v ∈ Cω(G; C) }. +Theorem 4.3.4 and Proposition 4.3.7 justify the following notation: +Definition 4.3.8. If ρ is holomorphically induced from (σ, χ), we write ρ = HolIndG +H(σ, χ). If addition- +ally χ is the trivial extension of dσ to b− on some necessarily unique domain Dχ ⊆ V ω +σ , we simply write +ρ = HolIndG +H(σ). +Remark 4.3.9. For k ∈ {1, 2}, let (σk, χk) be an (H, b−)-extension pair and let ρk be a unitary G-representation +with ρk = HolIndH(σk, χk). +In view of Theorem 4.3.4, one might wonder whether or not ρ1 ∼= ρ2 im- +plies (σ1, χ1) ∼= (σ2, χ2). This turns out to be false. For an explicit and simple counterexample, consider +G = SU(3). Let H ⊆ G be the subgroup consisting of diagonal matrices and let b− ⊆ sl(3, C) consist of +upper-triangular matrices. The defining representation ρ of G on C3 is holomorphically induced from the +two (H, b−)-extension pairs obtained by restricting ρ|H and dρ|b− to either Vσ1 := Ce1 or Vσ2 := Ce1 ⊕ Ce2, +as is quickly verified using Theorem 4.2.4. These are not unitary equivalent. +4.4 +Commutants +Suppose that ρ = HolIndG +H(σ, χ). +Definition 4.4.1. Let T ∈ T ∈ B(Vσ). We say that T commutes with (σ, χ) if T ∈ B(Vσ)H, T Dχ ⊆ Dχ and +T χ(ξ)v = χ(ξ)T v for every ξ ∈ b− and v ∈ Dχ. Define the ∗-closed commutant B(Vσ)H,χ of (σ, χ) by +B(Vσ)H,χ := +� +T ∈ B(Vσ)H : both T and T ∗ commute with (σ, χ) +� +. +Remark 4.4.2. Orthogonal projections in B(Vσ)H,χ correspond to direct sum decompositions of (σ, χ). To see +this, suppose p1 ∈ B(Vσ)H,χ is an orthogonal projection. Let p2 := 1 − p1. For k ∈ {1, 2}, write Vk := pkVσ +and Dk := pkDχ ⊆ Dχ. Define the (H, b−)-extension pair (σk, χk) by σk(h) := σ(h)|Vk and χk(ξ) := χ(ξ)|Dk, +where h ∈ H and ξ ∈ b−. Then (σ, χ) ∼= (σ1, χ1) ⊕ (σ2, χ2). +The main results of this section are Theorem 4.4.3 and Theorem 4.4.4 below: +21 + +Theorem 4.4.3. Suppose that ρ = HolIndG +H(σ, χ). Let Vσ be a closed subspace of Hρ satisfying the conditions +in Theorem 4.2.4.2. Let qV ∈ B(Hρ) be the orthogonal projection onto Vσ. Then +1. B(Vσ)H,χ is a von Neumann algebra. +2. Assume that qV ∈ ρ(G)′′. Then r : B(Hρ)G → B(Vσ)H,χ, r(T ) := T |Vσ defines a ∗-isomorphism of von +Neumann algebras. In particular, ρ is irreducible if and only if (σ, χ) is indecomposable. +Theorem 4.4.4. Consider the setting of Theorem 4.4.3 and assume that χ : b− → L(Dχ) is the trivial +extension of dσ to b− with domain Dχ. The following assertions are valid: +1. B(Vσ)H,χ = B(Vσ)H. +2. Assume that qV ∈ ρ(G)′′. Then B(Hρ)G ∼= B(Vσ)H. In particular, ρ is irreducible if and only if σ is. +Remark 4.4.5. In the context of positive energy representations, the case where χ is a trivial extension is +of central importance. In that setting we can typically guarantee that qV ∈ ρ(G)′′. The relation between +positive energy representations and holomorphic induction is considered Section 7. +Proof of Theorem 4.4.3 and Theorem 4.4.4 +Assume throughout the following that ρ is holomorphically induced from (σ, χ). In view of Theorem 4.2.4, +we may and do assume that Vσ ⊆ Hρ is a closed subspace, σ(h) = ρ(h)|Vσ for all h ∈ H, that Dχ = Vσ ∩ Hω +ρ , +dρ(b−)Dχ ⊆ Dχ and that χ(ξ)v = dρ(ξ)v for all ξ ∈ b− and v ∈ Dχ. We may further assume that the map +Φ : Hρ → Map(G; Vσ)H satisfying the conditions in Definition 4.2.2 is given by Φψ(g) = pV ρ(g)−1ψ. In +particular Ee = pV is the orthogonal projection pV : Hρ → Vσ and E∗ +e = ιV is the inclusion ι : Vσ ֒→ Hρ. We +also have qV = ιV pV . +Lemma 4.4.6. Let T ∈ B(Hρ)H,χ, x ∈ U(gC) and v, w ∈ Dχ. Then ⟨v, T dρ(x)w⟩ = ⟨v, dρ(x)T w⟩. +Proof. Using the PBW Theorem, it suffices to consider the case where x = x+x− for some x+ ∈ U(n+) and +x− ∈ U(b−). In that case we obtain using Lemma 4.2.6 and the fact that T ∈ B(Hρ)H,χ: +⟨v, T dρ(x)w⟩ = ⟨χ(ξ∗ ++)T ∗v, χ(x−)w⟩ = ⟨χ(ξ∗ ++)v, T χ(x−)w⟩ = ⟨χ(ξ∗ ++)v, χ(x−)T w⟩ = ⟨v, dρ(x)T w⟩. +Lemma 4.4.7. Let T ∈ B(Vσ). Assume that ⟨v, T ρ(eξ)w⟩ = ⟨v, ρ(eξ)T w⟩ for all v, w ∈ Dχ and all ξ in some +0-neighborhood in g. Then T Dχ ⊆ Dχ and +⟨w, T ρ(g)v⟩ = ⟨w, ρ(g)T v⟩, +∀g ∈ G, ∀v, w ∈ Vσ. +(4.7) +Proof. Let v, w ∈ Dχ. Both g �→ ⟨w, T ρ(g)v⟩ and g �→ ⟨w, ρ(g)T v⟩ are real-analytic G → C. As G is BCH, +so in particular locally exponential, these functions agree on some 1-neighborhood in G by assumption. As +G is connected, it follows from Proposition 2.1.14 that they are equal everywhere. We thus obtain that +⟨w, T ρ(g)v⟩ = ⟨w, ρ(g)T v⟩ for all g ∈ G. As Dχ is dense, equation (4.7) follows. Let v ∈ Dχ. Then using +(4.7) we find that ⟨T v, ρ(g)T v⟩ = ⟨T v, T ρ(g)v⟩ for all g ∈ G. The right-hand side defines a real-analytic +function G → C because v ∈ Hω +ρ . Thus also g �→ ⟨T v, ρ(g)T v⟩ is real-analytic. Recalling that G is a BCH +Fr´echet-Lie group, we conclude using [Nee11, Thm. 5.2] that T v ∈ Hω +ρ . Thus T v ∈ Hω +ρ ∩ Vσ = Dχ. +Lemma 4.4.8. B(Hρ)H,χ is a von Neumann algebra. Moreover we have +⟨w, T ρ(g)v⟩ = ⟨w, ρ(g)T v⟩, +∀T ∈ B(Hρ)H,χ, ∀g ∈ G, ∀v, w ∈ Vσ. +(4.8) +Proof. Let N ⊆ B(Vσ)H denote the von Neumann algebra in B(Vσ) generated by B(Hρ)H,χ. +We show +N = B(Hρ)H,χ. It only remains to show N ⊆ B(Hρ)H,χ. As N is ∗-closed, it suffices to show that T Dχ ⊆ Dχ +and that T χ(ξ)v = χ(ξ)T v for all T ∈ N, ξ ∈ b− and v ∈ Dχ. Let T ∈ N. Let (Tλ) be a net in B(Hρ)H,χ +such that Tλ → T strongly. Let v, w ∈ Dχ and x ∈ U(gC). Using Lemma 4.4.6 we have: +⟨v, T dρ(x)w⟩ = lim +λ ⟨v, Tλdρ(x)w⟩ = lim +λ ⟨v, dρ(x)Tλw⟩ = lim +λ ⟨dρ(x∗)v, Tλw⟩ = ⟨dρ(x∗)v, T w⟩ +(4.9) +As v, w ∈ Dχ ⊆ Hω +ρ , the orbit maps g �→ ρ(g)v and g �→ ρ(g)w are both real-analytic G → Hρ. We obtain +using (4.9) for all ξ ∈ g in a small-enough 0-neighborhood in g that: +⟨w, T ρ(eξ)v⟩ = +∞ +� +n=0 +1 +n!⟨w, T dρ(ξn)v⟩ = +∞ +� +n=0 +(−1)n +n! +⟨dρ(ξn)w, T v⟩ = ⟨ρ(e−ξ)w, T v⟩ = ⟨w, ρ(eξ)T v⟩. +(4.10) +22 + +It follows from Lemma 4.4.7 that T Dχ ⊆ Dχ and that equation (4.7) is valid for T . Thus NDχ ⊆ Dχ. +Differentiating (4.7) at the identity e ∈ G we find that ⟨w, T dρ(ξ)v⟩ = ⟨w, dρ(ξ)T v⟩ for all ξ ∈ gC and +w ∈ Dχ. Suppose ξ ∈ b−. Using that T Dχ ⊆ Dχ, we obtain +⟨w, T χ(ξ)v⟩ = ⟨w, T dρ(ξ)v⟩ = ⟨w, dρ(ξ)T v⟩ = ⟨w, χ(ξ)T v⟩, +∀w ∈ Dχ, +where Lemma 4.2.6 was used in the first and last equality. As Dχ is dense in Vσ, it follows for every ξ ∈ b− +and v ∈ Dχ that T χ(ξ)v = χ(ξ)T v. Thus T ∈ B(Hρ)H,χ. Hence N = B(Hρ)H,χ. +Combined with Lemma 4.4.8, Lemma 4.4.9 below completes the proof of Theorem 4.4.3. +Lemma 4.4.9. Assume that qV ∈ ρ(G)′′. Then the map +r : B(Hρ)G → B(Vσ)H,χ, +r(T ) := T |Vσ +defines an isomorphism of von Neumann algebras. +Proof. We know using Lemma 4.4.8 that B(Vσ)H,χ is a von Neumann algebra. Notice that the assumption +qV ∈ ρ(G)′′ is equivalent with T Vσ ⊆ Vσ for every T ∈ B(Hρ)G. Let T ∈ B(Hρ)G. Then T Hω +ρ ⊆ Hω +ρ and +T Vσ ⊆ Vσ. Recalling that Dχ = Vσ ∩ Hω +ρ , it follows that T Dχ ⊆ Dχ. Since both T and T ∗ are in B(Hρ)G, +it follows that r(T ) ∈ B(Vσ)H,χ, where we recall that ρ(h)|Vσ = σ(h) and dρ(ξ)|Dχ = χ(ξ) for h ∈ H and +ξ ∈ b−. It is clear that r is a norm-continuous, ∗-preserving and linear. It is also injective, because r(T ) = 0 +implies T ρ(G)Vσ = ρ(G)T Vσ = {0}, which in turn implies T = 0 because Vσ is cyclic for the G-representation +Hρ. Being an injective homomorphism of C∗-algebras, r is isometric and hence has closed range. Thus to +see r is surjective, it suffices to show that its image contains all orthogonal projections in B(Vσ)H,χ. Let +p1 ∈ B(Vσ)H,χ be an orthogonal projection and let p2 := 1 − p1. For k ∈ {1, 2}, define Vk, Dk, σk and χk as +in Remark 4.4.2, so that (σ, χ) ∼= (σ1, χ1) ⊕ (σ2, χ2). Let H1 and H2 be the closed G-invariant subspaces of +Hρ generated respectively by the subspaces V1 and V2 of Vσ. It suffices to show that H1 ⊥ H2. Let v1 ∈ V1 +and v2 ∈ V2. As p1 ∈ B(Vσ)H,χ, it follows from equation (4.8) that +⟨v1, ρ(g)v2⟩ = ⟨v1, p1ρ(g)v2⟩ = ⟨v1, ρ(g)p1v2⟩ = 0, +∀g ∈ G. +It follows that V1 ⊥ ρ(G)V2. Consequently H1 ⊥ H2. +Finally, it remains to prove Theorem 4.4.4: +Proof of Theorem 4.4.4: It remains only to prove the first point. The second will follow using Theorem 4.4.3. +It is clear that B(Vσ)H,χ ⊆ B(Vσ)H. Conversely, take T ∈ B(Vσ)H. Let v, w ∈ D. Then in particular the +orbit maps G → Hρ, g �→ ρ(g)v and g �→ ρ(g)w are real-analytic. Notice that T Dχ ⊆ V ∞ +σ +and similarly +T ∗Dχ ⊆ V ∞ +σ . Using Lemma 4.3.6, we obtain for all ξ in a small-enough 0-neighborhood that +⟨w, T ρ(eξ)v⟩ = +∞ +� +n=0 +1 +n!⟨w, T dρ(ξn)v⟩ = +∞ +� +n=0 +1 +n!⟨w, T dσ(E0(ξn))v⟩ = +∞ +� +n=0 +1 +n!⟨w, dσ(E0(ξn))T v⟩ += +∞ +� +n=0 +(−1n) +n! +⟨dρ(ξn)w, T v⟩ = ⟨ρ(e−ξ)w, T v⟩ = ⟨w, ρ(eξ)T v⟩, +From Lemma 4.4.7 it follows that T Dχ ⊆ Dχ. Hence B(Vσ)HDχ ⊆ Dχ and in particular T ∗Dχ ⊆ Dχ. Suppose +that v ∈ Dχ, ξ0 ∈ hC and ξ− ∈ n−. Then T χ(ξ0 + ξ−)v = T dσ(ξ0)v = dσ(ξ0)T v = χ(ξ0 + ξ−)T v. Hence +T χ(ξ)v = χ(ξ)T v for all ξ ∈ b− and v ∈ Dχ. We conclude that T ∈ B(Vσ)H,χ. Hence B(Vσ)H,χ = B(Vσ)H. +23 + +4.5 +Holomorphic induction in stages +Let us next consider holomorphic induction in stages. We specialize to the context of trivial extensions. Recall +from Section 4.5 that gC = n−⊕hC⊕n+ and that H ⊆ G is a connected Lie subgroup with Lie(H) = h. Assume +similarly that hC = a− ⊕ tC ⊕ a+, where a± and tC are closed subalgebras with θ(a±) ⊆ a∓ and [tC, a±] ⊆ a±. +Let T ⊆ H be a connected Lie subgroup integrating t ⊆ h. Using the notation of Definition 4.3.8: +Proposition 4.5.1 (Induction In Stages). Let (ρ, Hρ), (σ, Hσ) and (ν, Hν) be analytic unitary representations +of G, H and T , respectively. Then +1. ρ = HolIndG +T (ν) and σ = HolIndH +T (ν) +=⇒ +ρ = HolIndG +H(σ). +2. Suppose that σ = HolIndH +T (ν) and ρ = HolIndG +H(σ). +Assume w.l.o.g. +that Hν ⊆ Hσ ⊆ Hρ using +Theorem 4.2.4, the inclusions being T - and H-equivariant, respectively. If Hν ∩ Hω +ρ is dense in Hν, +then ρ = HolIndG +T (ν). +Proof. These observations follow from a repeated application of Theorem 4.2.4. +1. In view of Theorem 4.2.4, we may assume that Hν ⊆ Hρ as T -representations and that Hν ∩Hω +ρ is dense +in Hν and killed by dρ(n− ⊕ a−). Let (π, Hπ) denote the unitary H-representation in Hρ generated by +Hν ∩Hω +ρ ⊆ Hρ. Using Theorem 4.2.4 it follows that π = HolIndH +T (ν). By Theorem 4.3.4, it follows that +π ∼= σ as unitary H-representations. Thus we may assume Hσ = Hπ ⊆ Hρ, the last inclusion being +H-equivariant. The H-orbit of Hν ∩ Hω +ρ under ρ|H in Hσ is contained in Hσ ∩ Hω +ρ and is trivially total +for Hσ. Thus Hσ ∩ Hω +ρ is dense in Hσ. As Hν ∩ Hω +ρ is already cyclic for (ρ, Hρ), so is the larger space +Hσ ∩ Hω +ρ . To see that ρ = HolIndG +H(σ), it just remains to show that Hσ ∩ Hω +ρ is killed by dρ(n−). As +Hν ∩Hω +ρ is killed by dρ(n−) and AdH(n−) ⊆ n−, it follows that dρ(n−)ρ(H)ψ ⊆ ρ(H)dρ(n−)ψ = {0} for +any ψ ∈ Hν ∩ Hω +ρ . Thus dρ(n−) kills ρ(H)(Hν ∩ Hω +ρ ). As ρ(H)(Hν ∩ Hω +ρ ) is total in Hσ, it follows that +dρ(n−) kills Hσ ∩Hω +ρ . Having shown all conditions of Theorem 4.2.4, we conclude that ρ = HolIndG +H(σ). +2. As σ = HolIndT +H(ν) we may assume that Hν ⊆ Hσ as T -representations and that Hω +σ ∩ Hν is dense in +Hν, cyclic for the H-representation Hσ, and killed by dσ(a−). Similarly, as ρ = HolIndG +H(σ) we may +assume that Hσ ⊆ Hρ as H representations and moreover that Hω +ρ ∩ Hσ is dense in Hσ, cyclic for +the G-representation Hρ and killed by dρ(n−). Then Hν ⊆ Hσ ⊆ Hρ the inclusions being T - and H +equivariant, respectively. By assumption Hν ∩ Hω +ρ is dense in Hν. Since Hν is cyclic for (σ, Hσ) and +Hσ for (ρ, Hρ), it follows that Hν ∩ Hω +ρ is cyclic for (ρ, Hρ). For any ψ ∈ Hν ∩ Hω +ρ ⊆ Hσ ∩ Hω +ρ we have +dρ(a− ⊕ n−)ψ ⊆ dσ(a−)ψ + dρ(n−)ψ = {0}. Thus Hν ∩ Hω +ρ is killed by dρ(a− ⊕ n−). By Theorem 4.2.4 +it follows that ρ = HolIndG +T (ν). +5 +A geometric approach to holomorphic induction +In this section, a definition of holomorphically induced representations is presented which ensures that H∞ +ρ +embeds in a space of holomorphic mappings. Contrary to Section 4, this approach requires complex-geometry. +It is not as generally applicable, and in particular requires access to a dense set of strongly-entire vectors +in the representation that is to be induced, a condition that is well-understood for finite-dimensional Lie +groups but barely studied for infinite-dimensional ones. We first clarify the precise setting, after which the +homogeneous vector bundle G ×H V O +σ +is equipped with a suitable complex-analytic structure in Section 5.2. +We then proceed in Section 5.3 to define geometric holomorphic induction and compare the notion with the +one studied in Section 4. +Consider the setting of Section 4, so we have a decomposition gC = n− ⊕hC ⊕n+, where n± and hC are closed +Lie subalgebras of gC satisfying θ(n±) ⊆ n∓ and [hC, n±] ⊆ n±. Also, H ⊆ G is a connected Lie subgroup +integrating h ⊆ g. We assume further that GC is a complex regular BCH Fr´echet-Lie group with Lie algebra +Lie(GC) = gC as well as the existence of an embedding η : G ֒→ GC with Lie(η) : g ֒→ gC being the inclusion. +Observe in this setting that AdH(b±) ⊆ b±. We write p := (n− ⊕ n+) ∩ g, so that p ∼= g/h. Let M denote +the homogeneous space M := G/H. +Following [Nee14, Appendix C], we assume in addition that there exist open symmetric convex 0-neighborhoods +UC ⊆ gC, +Up ⊆ p ∩ UC, +Uh ⊆ h ∩ UC, +Un± ⊆ n± ∩ UC +and Ub− ⊆ b− ∩ UC +24 + +such that the following maps are analytic diffeomorphisms onto an open subset, where x ∗ y is defined by the +BCH series: +Up × Uh → g, +(x, y) �→ x ∗ y, +(A1) +Up × Ub− → gC, (x, y) �→ x ∗ y, +(A2) +Un+ × Ub− → gC, (x, y) �→ x ∗ y. +(A3) +Remark 5.1.1. +1. As mentioned in [Nee14, Appendix C], (A1) ensures that M carries the structure of a real-analytic +manifold for which the left G-action is analytic G × M → M, (A2) and (A3) ensure that M carries a +compatible G-invariant complex structure with TeHM ∼= gC/b− as complex vector spaces. Condition +(A3) is also needed to equip G ×H V O +σ +with the structure of a complex vector bundle over M, as we +shall see shortly. +2. In [Nee14, Example C.4], sufficient conditions are discussed that guarantee these assumptions are +satisfied. Using the Inverse Function Theorem, this is in particular the case if G is a simply connected +Banach-Lie group and M = G/H is a Banach homogeneous space. +Henceforth, we endow M = G/H with the G-invariant complex-analytic manifold structure mentioned in +Remark 5.1.1, for which TeHM ∼= gC/b− as complex vector spaces. +Lemma 5.1.2. Let f ∈ C∞(M; C) and let �f : G → C be its lift to G. Then f ∈ O(M) ⇐⇒ Lv(b−) �f = {0}. +Proof. Identify (TgG)C ∼= gC and TgHM ∼= gC/b− using the left G-action on M. By Proposition 2.1.11, f is +holomorphic if and only if T (f) is fiberwise C-linear, which in turn is equivalent to Lv(b−) �f = {0}. +5.2 +Complex structures on Eσ = G ×H V O +σ +Fix a unitary H-representations (σ, Vσ). Recall that V O +σ +denotes the space of strongly-entire vectors for the +H-representation σ on Vσ. Define the G-homogeneous vector bundle Eσ := G ×H V O +σ +with typical fiber V O +σ . +We adapt the proof of [Nee13, Thm. 1.6] to endow Eσ with a complex-analytic bundle structure using the +notion of entire extensions χ : b− → B(V O +σ ) of dσ to b−, see Definition 5.2.3 below. We write Lg : Eσ → Eσ +for the left G-action on Eσ. +Definition 5.2.1. Let W be a complete Hausdorff complex (resp. real) locally convex vector space and let +F : W → B(V O +σ ) be a function. +We say that F is complex-analytic (resp. real-analytic, smooth) if the +corresponding map W × V O +σ → V O +σ +is complex-analytic (resp. real-analytic, smooth). +Lemma 5.2.2. Consider the setting of Definition 5.2.1. If F is smooth then F is complex-analytic if and +only if the map Tx(F) : W → B(V O +σ ) is C-linear for every x ∈ W, where Tx(F)(w)v := +d +dt +�� +t=0 F(x + tw)v +Proof. By Proposition 2.1.11, the map F ∨ : W × V O +σ +→ V O +σ +is complex-analytic if and only if it is smooth +and T (F ∨) is fiber-wise C-linear. F ∨ is smooth by assumption and v �→ Tx(F ∨)(0, v) is trivially C-linear. +Thus F is complex-analytic if and only if w �→ Tx(F ∨)(w, 0) is C-linear for any x, w ∈ W, which is the +statement. +Definition 5.2.3. An entire extension χ of dσ : hC → B(V O +σ ) to b− is a homomorphism χ : b− → B(V O +σ ) of +Lie algebras such that: +1. χ|hC = dσ. +2. χ(Adh(ξ)) = σ(h)χ(ξ)σ(h)−1 for all h ∈ H and ξ ∈ b−. +3. The series �∞ +n=0 +1 +n!χ(ξ)nv converges in V O +σ +for all ξ ∈ b− and v ∈ V O +σ +and defines a holomorphic map +b− × V O +σ → V O +σ , +(ξ, v) �→ +∞ +� +n=0 +1 +n!χ(ξ)nv. +In this case, we write eχ(ξ)ψ := �∞ +n=0 +1 +n!χ(ξ)nψ. Then eχ : b− → B(V O +σ ) is complex-analytic. +25 + +Example 5.2.4. By Theorem 3.2.1, we know that the following map is entire: +hC × V O +σ → V O +σ → V O +σ , (η, v) �→ +∞ +� +n=0 +1 +n!dσ(ηn)v. +Consequently, the trivial extension χ : b− → B(V O +σ ) of dσ to b− with domain V O +σ +is an entire extension. +Using the notion of entire extensions, [Nee13, Thm. 1.6] adapts straightforwardly to the present setting: +Theorem 5.2.5. Let χ : b− → B(V O +σ ) be an entire extension of dσ to b−. Then Eσ = G ×H V O +σ +carries a +unique complex-analytic bundle structure satisfying the following properties: +1. The left G-action Lg is complex-analytic for any fixed g ∈ G. +2. The quotient map G × Vσ → Eσ is real-analytic. +3. Let U ⊆ G be a neighborhood of g ∈ G. A smooth function f ∈ C∞(UH, V O +σ )H corresponds to a local +holomorphic section of Eσ if and only if Lv(ξ)f = −χ(ξ)f for any ξ ∈ n−. +If the two entire extensions χ1 and χ2 of dσ to b− define the same complex-bundle structure, then χ1 = χ2. +Definition 5.2.6. Let χ : b− → B(V O +σ ) be an entire extension of dσ to b−. We denote by E(σ,χ) → M the +vector bundle Eσ → M equipped with the unique complex-analytic bundle structure satisfying the conditions +in Theorem 5.2.5. +Proof of Theorem 5.2.5: This proof essentially follows from trivial adaptations of [Nee13, Thm. 1.6]. Let us +indicate the required changes and recall the construction of the local charts, for later use. +Let qM : G → G/H denote the quotient map. Let Ug ⊆ U ∩ g and UG ⊆ G be neighborhoods of 0 ∈ g +and 1 ∈ G, respectively, s.t. expG|Ug : Ug → UG is an analytic diffeomorphism. Shrinking Ug if necessary, +there exists by (A1) some 0-neighborhoods Up ⊆ p and Uh ⊆ h s.t. the BCH series defines an analytic +diffeomorphism Up × Uh → Ug. Define UP := expG(Up) and UH := expG(Uh), so UG = UP UH. (Comparing +with the proof of [Nee13, Thm. 1.6], UP takes the role of UZ.) Define for any x ∈ G the open subsets +Ux := xqM(UP ) ⊆ M +and +�Ux := xUP H ⊆ G, +(5.1) +Using (A3), and replacing β : b− → B(Vσ)× in steps 2 − 4 of the proof of [Nee13, Thm. 1.6] by the entire +extension χ : b− → B(V O +σ ), we obtain after shrinking Ug if necessary for each x ∈ G a smooth function +Fx : �Ux → B(V O +σ )× satisfying the following properties: +1. Fx(gh) = σ(h)−1Fx(g) for all g ∈ �Ux and h ∈ H. +2. Lv(ξ)Fx = −χ(ξ)Fx for all ξ ∈ b−. +3. Fx(x) = idV O +σ . +Moreover, using (A3), that eχ : b− → B(V O +σ ) is complex-analytic and that the action H × V O +σ → V O +σ is real- +analytic by Theorem 3.2.1(5), observe from its construction that Fx is actually real-analytic. These functions +moreover satisfy Fyx(yg) = Fx(g) for any x, y ∈ G and g ∈ �Ux, as is immediate from their construction. +Following [Nee13, Thm. 1.6], we now define for each x ∈ G the trivialization +φx : Ux × V O +σ → Eσ|Ux , +(gH, v) �→ [g, Fx(g)v], +(5.2) +so that the transition function φxy := φ−1 +x +◦ φy on Ux ∩ Uy is given by +φxy : Ux ∩ Uy → B(V O +σ )×, +φxy(gH) = Fx(g)−1Fy(g). +Let us check as in [Nee13, Thm. 1.6] that these transition functions are complex-analytic. It suffices to show +that the lifted map �φxy : �Ux ∩ �Uy → B(V O +σ ) satisfies Lv(ξ) �φxy = 0 for all ξ ∈ b−. This follows from the three +properties of the functions Fx mentioned above: +Lv(ξ) �φxy = +� +Lv(ξ)F −1 +x +� +Fy + F −1 +x +� +Lv(ξ)Fy +� += −F −1 +x +� +Lv(ξ)Fx)F −1 +x Fy + F −1 +x +� +Lv(ξ)Fy +� += F −1 +x χ(ξ)Fy − F −1 +x χ(ξ)Fy += 0. +Thus the trivializations {φx}x∈G define a complex-analytic bundle structure on Eσ. Let E(σ,χ) denote the +thus-obtained complex-analytic bundle. We show that the properties 1 − 3 in Theorem 5.2.5 are satisfied: +26 + +1. Let x, g ∈ G. In the local charts defined by φx and φgx, Lg is represented by lg × idV O +σ , which is +complex-analytic from the corresponding property of lg : M → M. +2. Let x ∈ G. Consider the local coordinates of E(σ,χ) defined by φx. In these local coordinates, the +quotient map G × V O +σ → E(σ,χ) is represented by the real-analytic function +�Ux × V O +σ → Ux × V O +σ , +(g, v) �→ (gH, Fx(g)−1v). +3. Take f ∈ C∞(UH, V O +σ )H. The corresponding local section of E(σ,χ) is obtained by descending the +function �f : UH → Eσ, �f(g) := [g, f(g)] to the quotient qM(U). Let x ∈ U and define Wx := Ux ∩ U +and � +Wx := �Ux ∩ UH. Using the local chart φx, the map �f +���� +Wx +is represented by the smooth function +f : � +Wx → Ux × V O +σ , +f(g) = (gH, Fx(g)−1f(g)), +which is complex-analytic if and only if Lv(ξ)h = 0 for any ξ ∈ b−, where h is given by +h : � +Wx → V O +σ , +h(g) := Fx(g)−1f(g). +We compute that +Lv(ξ)h = +� +Lv(ξ)F −1 +x +� +f + F −1 +x +� +Lv(ξ)f +� += −F −1 +x +� +Lv(ξ)Fx)F −1 +x f + F −1 +x +� +Lv(ξ)f +� += F −1 +x χ(ξ)f + F −1 +x +� +Lv(ξ)f +� +. +Thus Lv(ξ)h = 0 if and only if Lv(ξ)f = −χ(ξ)f for any ξ ∈ b−. Consequently f corresponds to a +holomorphic local section of Eσ → M if and only if Lv(ξ)f = −χ(ξ)f for every ξ ∈ b−. The equation +is automatically satisfied for any ξ ∈ hC by the H-equivariance of f. The conclusion follows. +Step 5 in [Nee13, Thm. 1.6] shows that if the two entire extensions χ1 and χ2 define the same complex +bundle structure, then χ1 = χ2. To see that the complex-bundle structure is unique, we simply remark that +if E1 +σ and E2 +σ denote the vector bundle Eσ equipped `a priori with possibly different complex-analytic bundle +structures satisfying the properties 1 − 3 in Theorem 5.2.5, then by the third property they have the same +holomorphic local sections. This implies E1 +σ = E2 +σ as complex-analytic vector bundles over M. +5.3 +Geometric holomorphic induction +Having the complex-analytic G-homogeneous vector bundles E(σ,χ) at hand, we are now in a position to +define a stronger notion of holomorphic induction, which guarantees that H∞ +ρ +actually embeds into a space +of holomorphic mappings. +Let σ be a unitary representation of H on Vσ. +Consider an entire extension +χ : b− → B(V O +σ ) of dσ : hC → B(V O +σ ) to b−. Let (ρ, Hρ) be a unitary representation of G. +Definition 5.3.1. We say that (ρ, Hρ) is geometrically holomorphically induced from (σ, χ) if σ is strongly- +entire and there exists a G-equivariant injective linear map Φ : Hρ ֒→ Map(G; Vσ)H satisfying: +1. The point evaluation Ex : Hρ → Vσ, Ex(ψ) := Φψ(x) is continuous for every x ∈ G. +2. ExE∗ +x = idVσ for every x ∈ G. +3. For every w ∈ V O +σ , the following function is holomorphic: +fw : E(σ,χ) → C, +fw([g, v]) := ⟨E∗ +e w, ρ(g)E∗ +e v⟩. +We start with a lemma: +Lemma 5.3.2. Assume that Vσ ⊆ Hρ as unitary H-representations and that Vσ is cyclic for G in Hρ. +Assume further that σ is strongly-entire. Then the following assertions are equivalent: +1. V O +σ ⊆ H∞ +ρ +and dρ(ξ)v = χ(ξ)v for all ξ ∈ b− and v ∈ V O +σ . +2. fw ∈ O(E(σ,χ)) for every w ∈ V O +σ , where fw([g, v]) := ⟨w, ρ(g)v⟩. +If these assertions are satisfied, then we even have V O +σ +⊆ Hω +ρ . Moreover, fψ ∈ O(E(σ,χ)) for any ψ ∈ H∞ +ρ , +where fψ([g, v]) := ⟨ψ, ρ(g)v⟩. +27 + +Proof. Let ψ ∈ H∞ +ρ +and consider the function fψ : E(σ,χ) → C defined by fψ([g, v]) = ⟨ψ, ρ(g)v⟩. Consider +its lift to G × V O +σ , defined by �fψ : G × V O +σ +→ C, �fψ(g, v) = fψ([g, v]). Let x ∈ G. Define the open sets +�Ux ⊆ G and Ux ⊆ M as in (5.1), so Ux is an open neighborhood of xH ∈ M. Let Fx : �Ux → B(V O +σ )× be +defined as in the proof of Theorem 5.2.5. In particular, Fx satisfies Lv(ξ)Fx = −χ(ξ)Fx for any ξ ∈ b−. Let +φx : Ux × V O +σ +→ E(σ,χ) +�� +Ux be the corresponding chart of the holomorphic vector bundle E(σ,χ), defined in +(5.2). In these local coordinates, fψ and �fψ are represented by hψ,x and �hψ,x, respectively, where +hψ,x : Ux × V O +σ → C, +hψ,x(gH, v) = ⟨ρ(g)−1ψ, Fx(g)v⟩, +�hψ,x : �Ux × V O +σ → C, +�hψ,x(g, v) = ⟨ρ(g)−1ψ, Fx(g)v⟩. +As Fx is smooth, and because ψ ∈ H∞ +ρ , this shows in particular that fψ : E(σ,χ) → C is smooth for the +underlying real manifold structure. +Then hψ,x is complex-analytic if and only if Lv(ξ)�hψ,x = 0 for any +ξ ∈ b−. Let ξ ∈ b−. Using Lv(ξ)Fx = −χ(ξ)Fx, we compute for any (g, v) ∈ �Ux × V O +σ +that +(Lv(ξ)�hψ,x)(g, v) = ⟨dρ(ξ∗)ρ(g)−1ψ, Fx(g)v⟩ − ⟨ρ(g)−1ψ, χ(ξ)Fx(g)v⟩. +(5.3) +Thus if (1) holds true, then (5.3) shows that Lv(ξ)�hψ,x = 0 for any ξ ∈ b−, so that hψ,x is complex-analytic +for any x ∈ G. We then conclude that fψ ∈ O(E(σ,χ)) for any ψ ∈ H∞ +ρ . Since V O +σ ⊆ H∞ +ρ by assumption, we +in particular notice that (2) holds true. +Assume conversely that fw ∈ O(E(σ,χ)) for any w ∈ V O +σ . Let v ∈ V O +σ . Then g �→ ⟨v, ρ(g)v⟩ = fv([g, v]) is +real-analytic G → C, where we have used that the quotient map G × V O +σ → E(σ,χ) and the left-action of G +on itself are both real-analytic. As G is a BCH Fr´echet-Lie group, this implies by [Nee11, Thm. 5.2] that +v ∈ Hω +ρ . Hence V O +σ ⊆ Hω +ρ . We know from Corollary 2.2.9 that Hω +ρ ⊆ H∞ +ρ , so we also obtain that V O +σ ⊆ H∞ +ρ . +To see that (1) holds true, it remains to show that dρ(ξ)v = χ(ξ)v for all ξ ∈ b− and v ∈ V O +σ . Consider +the set D := +� +ψ ∈ H∞ +ρ +: fψ ∈ O(E(σ,χ)) +� +. The preceding shows that V O +σ +⊆ D. The set D is moreover +G-invariant. Indeed, if ψ ∈ D and g ∈ G, then fρ(g)ψ = fψ ◦ Lg−1 defines a holomorphic map on E(σ,χ), +because Lg−1 : E(σ,χ) → E(σ,χ) is holomorphic. As V O +σ +is dense in Vσ and Vσ is cyclic for G, it follows that +D is dense in Hρ. Let ξ ∈ b−, ψ ∈ D and v ∈ V O +σ . Recall that Fe(e) = idV O +σ . As fψ ∈ O(E(σ,χ)), we know +that (Lv(ξ)�hψ,e)(e) = 0. Using that v ∈ H∞ +ρ , it follows by evaluating (5.3) at (e, v) ∈ �Ue × V O +σ +that +⟨ψ, dρ(ξ)v⟩ = ⟨dρ(ξ∗)ψ, v⟩ = ⟨ψ, χ(ξ)v⟩. +As D is dense, it follows that dρ(ξ)v = χ(ξ)v for all ξ ∈ b− and v ∈ V O +σ , so that (1) holds true. We have also +shown that if these equivalent are satisfied, then V O +σ ⊆ Hω +ρ and fψ ∈ O(E(σ,χ)) for any ψ ∈ H∞ +ρ . +The following entails that H∞ +ρ +can be seen as a space of of holomorphic functions on the complex-analytic +bundle E(σ,χ) → M conjugate to E(σ,χ) → M: +Proposition 5.3.3. Assume that ρ is geometrically holomorphically induced from (σ, χ). Then there is an +injective G-equivariant C-linear map H∞ +ρ ֒→ O(E(σ,χ)) for which all point evaluations are continuous. +Proof. Assume that ρ is geometrically holomorphically induced from (σ, χ). In particular, this implies that +σ is strongly-entire. Let Φ : Hρ ֒→ Map(G; Vσ)H satisfy the conditions in Definition 5.3.1. We may consider +Vσ as a subspace of Hρ using the H-equivariant isometry E∗ +e . We know by Theorem A.1.3 that Vσ ⊆ Hρ +is cyclic. From Lemma 5.3.2 we obtain that fψ ∈ O(E(σ,χ)) for any ψ ∈ H∞ +ρ . The map ψ �→ fψ defines a +G-equivariant C-linear map H∞ +ρ → O(E(σ,χ)) that has continuous point evaluations, where O(E(σ,χ)) denotes +the vector space complex conjugate to O(E(σ,χ)), which may be identified with O(E(σ,χ)). This map is in- +jective because fψ = 0 implies that ψ ⊥ ρ(G)V O +σ , which in turn implies ψ = 0 because V O +σ is cyclic for Hρ. +Let us next compare the notion of geometric holomorphic induction with Definition 4.2.2: +Theorem 5.3.4. Assume that σ is strongly-entire. The following assertions are equivalent: +1. (ρ, Hρ) is geometrically holomorphically induced from (σ, χ). +2. There is a subspace D�χ ⊆ V ω +σ containing V O +σ +and an extension �χ : b− → L(D�χ) of dσ to b− such that +χ(ξ) = �χ(ξ)|V O +σ for every ξ ∈ b− and such that ρ = HolIndG +H(σ, �χ). +28 + +Suppose that χ is the trivial extension of dσ to b− with domain V O +σ . Then these assertions are equivalent to: +3. ρ = HolIndG +H(σ) and V O +σ ⊆ H∞ +ρ , where we considered Vσ as a subspace of Hρ using Theorem 4.2.4. +Proof. Assume that (ρ, Hρ) is geometrically holomorphically induced from (σ, χ), so in particular σ is +strongly-entire. Let Φ : Hρ ֒→ Map(G; Vσ)H satisfy the conditions in Definition 5.3.1. Identify Vσ with +a cyclic subspace of Hρ using E∗ +e . Define D�χ := Vσ ∩ Hω +ρ . From Lemma 5.3.2 we obtain that V O +σ +⊆ D�χ +and that dρ(ξ)v = χ(ξ)v for all ξ ∈ b− and v ∈ V O +σ . As V O +σ +is dense in Vσ, the latter in particular im- +plies that dρ(b−)D�χ ⊆ Vσ, which in turn implies dρ(b−)D�χ ⊆ D�χ. +From Theorem 4.2.4 it follows that +ρ = HolIndG +H(σ, �χ), where �χ : b− → L(D�χ) is the extension of dσ to b− with domain D�χ, defined by +�χ(ξ)v = dρ(ξ)v. This extension satisfies �χ(ξ)|V O +σ = χ(ξ) for any ξ ∈ b−, as required. +Conversely, let �χ : b− → L(D�χ) satisfy the conditions in (2), so in particular ρ = HolIndG +H(σ, �χ). +By +Theorem 4.2.4 we may assume that Vσ ⊆ Hρ as unitary H-representations, that D�χ = Vσ ∩ Hω +ρ and that +�χ(ξ)v = dρ(ξ)v for all ξ ∈ b− and v ∈ D�χ. As D�χ contains V O +σ +by assumption, it follows in particular that +V O +σ ⊆ Hω +ρ . From Lemma 5.3.2, we obtain that fw ∈ O(E(σ,χ)) for any w ∈ V O +σ , where fw([g, v]) = ⟨w, ρ(g)v⟩. +So the map +Φ : Hρ → Map(G; Vσ)H, +Φψ(g) := pV ρ(g)−1ψ +satisfies the conditions in Definition 5.3.1, where pV : Hρ → V O +σ +is the orthogonal projection. +Assume that χ is the trivial extension of dσ to b− with domain V O +σ . Assume that (2) holds true. Let the +subspace D�χ ⊆ Vσ and the extension �χ : b− → L(D�χ) satisfy the conditions in (2). We may consider Vσ as a +closed H-invariant linear subspace of Hρ satisfying the conditions in Theorem 4.2.4. In particular, we have +V O +σ ⊆ D�χ = Vσ ∩ Hω +ρ , so certainly V O +σ ⊆ V ∞ +σ . We also know that dρ(b−)V O +σ = {0}. As V O +σ +is dense in Vσ, +this further implies that dρ(b−)D�χ = {0}, so �χ is the trivial extension on D�χ. Hence (3) holds true. Assume +conversely that (3) is valid. Let �χ denote the trivial extension of dσ to b− on the domain D�χ := Vσ ∩ Hω +ρ . +By assumption ρ = HolIndG +H(σ, �χ) and V O +σ ⊆ H∞ +ρ . As D�χ is killed by dρ(n−) and dense in Vσ, it follows that +dρ(n−)V O +σ += {0}. Thus (1) in Lemma 5.3.2 is satisfied, from which we obtain that V O +σ ⊆ Hω +ρ . This means +that V O +σ ⊆ D�χ. So (2) is satisfied using the trivial extension �χ on the subspace D�χ ⊆ Vσ. +6 +Arveson spectral theory +In Section 7 below, we shall have need for a suitably general notion of Arveson spectral subspaces. As such, +we extend the already existing notion to a more general setting. Let V be a complete locally convex vector +space over C that is Hausdorff. We define Arveson spectral subspaces of V associated to a strongly continuous +R-representation α on V that satisfies a suitable condition, using the convolution algebra S(R) of C-valued +Schwartz functions on R. The results are adaptations of those in [Arv74, Sec. 2], [NSZ15, Sec. A.3], [Nee13, +Sec. A.2]. +6.1 +Certain classes of R-representations +Throughout the section, let α : R → B(V )× be a strongly continuous representation of R on V . In [NSZ15, Sec. +A.3], the R-action α is required to be polynomially bounded (see Definition 6.1.1 below). It will however be +convenient to define both a stronger and a weaker notion, that in turn are both still weaker than equicontinuity, +which is used in [Nee13, Sec. A.2]. +Definition 6.1.1. Let α : R → B(V )× be a representation of R on V . +— α is said to be equicontinuous if there is a basis of absolutely convex α-invariant 0-neighborhoods in V . +Equivalently, if the topology of V is defined by a family of α-invariant continuous seminorms. +— α is said to have polynomial growth if there is a basis B of absolutely convex 0-neighborhoods in V +such that for every U ∈ B there is a monic polynomial r ∈ R[t] such that αt(U) ⊆ r(|t|)U for all t ∈ R. +Equivalently, if there is a family P of defining seminorms on V such that for every p ∈ P there exists +a monic polynomial r ∈ R[t] such that p(αt(v)) ≤ r(|t|)p(v) for all t ∈ R and v ∈ V . +— α : R → B(V )× is called polynomially bounded if for every continuous seminorm p on V , there is a +0-neighborhood U ⊆ V and some N ∈ N such that +sup +v∈U +sup +t∈R +p(αt(v)) +1 + |t|N < ∞. +29 + +— α : R → B(V )× is said to be pointwise polynomially bounded if for every v ∈ V and continuous seminorm +p on V , there exists N ∈ N such that +sup +t∈R +p(αt(v)) +1 + |t|N < ∞. +Remark 6.1.2. Notice that we have the following implications: +α is equicontinuous +=⇒ +α has polynomial growth +=⇒ +α is polynomially bounded. +If V is a Banach space, then α has polynomial growth if and only if it is polynomially bounded. +Example 6.1.3. +1. The R-representations on both L2(R) and C∞(T) by translation are equicontinuous. +2. The R-action α on S(R) by translation is not equicontinuous but does have polynomial growth. Indeed, +one checks that the open set U := { f ∈ S(R) : supx∈R |xf(x)| < 1 } of S(R) satisfies � +t∈R αt(U) = {0}. +By [Nee13, Prop. A.1], this implies that α is not equicontinuous. It does have polynomial growth, +because the topology on S(R) is generated by the seminorms pn,m(f) := supx∈R(1 + |x|)n|(∂mf)(x)| +for n, m ∈ N≥0, which satisfy pn,m(αtf) ≤ +� �n +k=0 +�n +k +� +|t|n−k +� +pn,m(f) for all t ∈ R and f ∈ S(R). +3. The translation action α on C∞ +c (R) is pointwise polynomially bounded, since ∥αt(f)∥Cn +K(R) ≤ ∥f∥Cn(R) +for any f ∈ C∞ +c (R), t ∈ R, n ∈ N and compact K ⊆ R. +4. The action of R on C∞(R) by translations is not pointwise polynomially bounded. For example, the +smooth function f(x) = ex satisfies ∥αt(f)∥C([0,1]) = ∥f∥C[t,t+t] ≥ et for all t ∈ R. +Let P denote the set of continuous seminorms on V . For p ∈ P, let Np := { v ∈ V : p(v) = 0 } denote its +kernel. Let Vp := V/Np be the corresponding Banach space. If p, q ∈ P and p ≤ q, then Nq ⊆ Np and hence +there is a canonical contraction ηp,q : Vq → Vp. +Lemma 6.1.4. Assume that α is strongly continuous and has polynomial growth. Then α descends for each +p ∈ P to a representation of R on Vp with polynomial growth. Moreover V = lim +←− Vp as R-representations. +Proof. Let p ∈ P. Since α has polynomial growth, we have αt(Np) ⊆ Np for every t ∈ R. Consequently, α +descends to a strongly continuous R-representation α(p) on Vp that again has polynomial growth. If p, q ∈ P +and t ∈ R, then ηp,q ◦ α(q) +t += α(p) +t . We thus obtain an R-action on the projective limit lim +←− Vp for which the +canonical isomorphism V ∼= lim +←− Vp is R-equivariant. +Proposition 6.1.5. Assume that α is strongly continuous and has polynomial growth. +Then the action +α : R × V → V is continuous. +Proof. By Lemma 6.1.4 it follows that V = lim +←− Vp as R-representation on locally convex space. If p ∈ P, then +since Vp is a Banach space and the R-representation on Vp is strongly continuous, it follows from [Nee10a, +Prop. 5.1] that the R-action R × Vp → Vp is jointly continuous. +Using that V ∼= lim +←− Vp as topological +representations of R, it follows that the action α : R × V → V is jointly continuous. +6.2 +Arveson spectral subspaces +Let V be a complete locally convex vector space over V . Let α : R → B(V )× be a strongly continuous +representation of R on V . Assume that α is pointwise polynomially bounded. In the following, we define +the Arveson spectral subspaces of V associated to subsets E of R. We extend the results in [NSZ15, A.3] to +the case where α is only required to be pointwise polynomial bounded. We will use the convention that the +Fourier transform f �→ �f on S(R) is given by +�f(p) := +� +R +f(t)eitpdt. +(6.1) +Definition 6.2.1. +— If I ⊆ S(R) is an ideal, define its hull h(I) ⊆ R by +h(I) := +� +p ∈ R : �f(p) = 0 for all f ∈ I +� +. +30 + +— If E ⊆ R is a closed subset, define the ideal I0(E) of S(R) by +I0(E) := +� +f ∈ S(R) : supp( �f) ∩ E = ∅ +� +. +Lemma 6.2.2 ([NSZ15, Prop. A.8]). +1. If E ⊆ R is a closed subset, then h(I0(E)) = E. +2. If I ⊆ S(R) is a closed ideal, then I0(h(I)) ⊆ I. +Corollary 6.2.3. Let I ⊆ S(R) be a closed ideal with h(I) = ∅. Then I = S(R). +Proof. Since I0(∅) = S(R) it follows from Lemma 6.2.2 that S(R) = I0(∅) = I0(h(I)) ⊆ I. +We proceed by defining a representation of the convolution algebra (S(R), ∗) on V . +Lemma 6.2.4. Let f ∈ S(R) and v ∈ V . Then the weak integral +� +R f(t)αt(v)dt exists in V . +Proof. For any a > 0, the weak integral +� a +−a f(t)αt(v)dt exists in V because R → V, t �→ f(t)αt(v) is +continuous and V is complete (cf. [Mil84, p. 1021] or [GN, Prop. 1.1.15]). As α is pointwise polynomially +bounded and f ∈ S(R) is a Schwartz function, the limit v∗ := lima→∞ +� a +−a f(t)αt(v)dt exists in V , and we +have v∗ = +� +R f(t)αt(v)dt ∈ V . +Definition 6.2.5. For any Schwartz function f ∈ S(R), define the linear operator αf ∈ L(V ) by +αf(v) := +� +R +f(t)αt(v)dt. +Then f �→ αf defines a strongly continuous representation of the convolution algebra (S(R), ∗) on V . +Remark 6.2.6. If α is polynomially bounded, then αf ∈ B(V ) is a continuous operator for every f ∈ S(R). +Definition 6.2.7. +— Let Specα(V ) := h(ker α) ⊆ R be the hull of the closed ideal ker α in S(R). +— For v ∈ V , let S(R)v := { f ∈ S(R) : αf(v) = 0 } denote the annihilator of v in S(R), which is a closed +ideal in S(R), and let Specα(v) := h(S(R)v) ⊆ R be its hull. +— If E ⊆ R is a subset define Vα(E)0 := +� +v ∈ V : Specα(v) ⊆ E +� +and let Vα(E) := Vα(E)0 be its closure +in V . Define moreover V + +α (E) := � +N Vα(E + N), where N runs over all 0-neighborhoods in R. +If the action α is clear from the context, we drop α from the notation and simply write V (E)0, V (E) and +V +(E) instead of Vα(E)0, Vα(E) and V + +α (E). +Example 6.2.8. Let U : R → U(H) be a strongly continuous unitary representation of R. Then Ut = +etH for some self-adjoint operator H on H. Suppose that Ut = +� +R eitpdP(p) is the corresponding spectral +decomposition of U, for some projection-valued measure P on R. With the convention (6.1) we have Uf := +� +R f(t)Utdt = � +R �f(p)dP(p) for f ∈ S(R). The Arveson spectrum SpecU(H) coincides with Spec(H), the +spectrum of the self-adjoint operator H. Moreover, for a closed subset E ⊆ R, the corresponding spectral +subspace is given by HU(E) = P(E)H. +Remark 6.2.9. Let {Ei}i∈I be a family of closed subsets of R. Observe that � +i∈I V (Ei)0 = V +� � +i∈I Ei +� +0. +Remark 6.2.10. Notice for any v ∈ V that ker α ⊆ S(R)v, and so Specα(v) ⊆ Specα(V ). Thus +V = V (Specα(V ))0 = V +� +Specα(V ) +� += V +� +Specα(V ) +� +. +Combining this with Remark 6.2.9, we obtain for any closed subset E ⊆ R that V (E)0 = V (E ∩ Specα(V ))0. +Lemma 6.2.11. Let f ∈ S(R) and v ∈ V . Then Specα(αf(v)) ⊆ supp( �f). +Proof. Let p ∈ R\supp( �f) and choose g ∈ S(R) such that �g(p) ̸= 0 and �g|supp( � +f) = 0. Then g∗f = 0, because +�g �f = 0. It follows that αgαfv = αf∗gv = 0. Since we also have �g(p) ̸= 0 it follows that p /∈ Specα(αfv). +31 + +Proposition 6.2.12. Let v ∈ V . Then S(R)v = S(R) implies v = 0. Moreover v ̸= 0 implies Specα(v) ̸= ∅. +Proof. Assume that S(R)v = S(R). If λ ∈ V ′ is a continuous functional, it follows that +� +R f(t)λ(αtv)dt = 0 +for any f ∈ S(R). As t �→ λ(αtv) is continuous, this implies that λ(αtv) = 0 for all t ∈ R. In particular +λ(v) = 0. As V ′ separates the points of V by the Hahn-Banach Theorem [Rud91, Thm. I.3.4], it follow that +v = 0. Finally, if Specα(v) = ∅ then by Corollary 6.2.3 it follows that S(R)v = S(R) and hence v = 0. +Corollary 6.2.13. If E1, E2 ⊆ R are two disjoint closed subsets, then V (E1)0 ∩ V (E2)0 = {0}. +Proof. We have V (E1)0 ∩ V (E2)0 = V (E1 ∩ E2) = V (∅) = {0} by Remark 6.2.9 and Proposition 6.2.12. +If E ⊆ R is a subset, recall from Definition 6.2.1 that I0(E) ⊆ S(R) denotes the ideal of functions f ∈ S(R) +whose Fourier transform �f vanishes on a neighborhood of E ⊆ R. +Proposition 6.2.14 below provides a +convenient characterization of V (E)0 in terms of I0(E), which will be used repeatedly. +Proposition 6.2.14 ([NSZ15, Prop. A.8]). +For any subset E ⊆ R we have +V (E)0 = +� +v ∈ V : I0(E) ⊆ S(R)v +� += +� +v ∈ V : ∀f ∈ S(R) : supp( �f) ∩ E = ∅ =⇒ αf(v) = 0 +� +. +In particular V (E)0, V (E) and V +(E) are linear subspaces of V . +Proof. The proof of [NSZ15, Prop. A.8] continues to hold when α is only pointwise polynomially bounded. +Corollary 6.2.15. Assume that α is polynomially bounded. Then V (E)0 = V (E) = V +(E) for any E ⊆ R. +Proof. Let E ⊆ R be a subset. +By Remark 6.2.6 we know that αf is a continuous linear operator for +every f ∈ S(R). It then follows from Proposition 6.2.14 that V (E)0 is closed, so V (E)0 = V (E). Using +Remark 6.2.9, we further obtain that +V +(E) = +� +N +V (E + N) = +� +N +V (E + N)0 = V +� � +N +E + N +� +0 += V (E)0 = V (E)0. +The following will also be used frequently: +Corollary 6.2.16. Let E ⊆ R be a subset. The following assertions are equivalent: +1. Specα(V ) ⊆ E +. +2. V ⊆ V (E)0. +3. I0(E) ⊆ ker α. +Proof. Assume that Specα(V ) ⊆ E. +Then for any v ∈ V we have Specα(v) ⊆ Specα(V ) ⊆ E, by Re- +mark 6.2.10. This means that V ⊆ V (E)0. Assume next that V ⊆ V (E)0. By Proposition 6.2.14, this means +that I0(E) ⊆ S(R)v for all v ∈ V . So elements of I0(E) annihilate every v ∈ V . Thus I0(E) ⊆ ker α. If +I0(E) ⊆ ker α, then Specα(V ) = h(ker α) ⊆ h(I0(E)) = E, where the last equality uses Lemma 6.2.2. +Corollary 6.2.17. Specα(V ) = � +v∈V Specα(v). +Proof. Write E := � +v∈V Specα(v). By Remark 6.2.10 we have Specα(v) ⊆ Specα(V ) for any v ∈ V . As +Specα(V ) is closed, it follows that E ⊆ Specα(V ). Conversely, recall that V (E)0 = +� +v ∈ V : Specα(v) ∈ E +� +. +So from our definition of E, we trivially have V ⊆ V (E)0. Then Specα(V ) ⊆ E follows by Corollary 6.2.16. +Let us next record the behavior of spectral subspaces under continuous (multi-)linear maps: +Proposition 6.2.18. For j ∈ {1, 2}, let αj : R → B(Vj)× be a strongly continuous representation of R on +the complete and Hausdorff complex locally convex vector space Vj. Assume that αj is pointwise polynomially +bounded. Let T : V1 → V2 be a continuous R-equivariant linear map. Then for every subset E ⊆ R we have +T (V1(E)) ⊆ V2(E). If T is injective, then Specα1(V1) ⊆ Specα2(V2). +32 + +Proof. Let v ∈ V . As T is equivariant, we have S(R)v ⊆ S(R)T v. Hence h(S(R)T v) ⊆ h(S(R)v), which is to +say that Specα2(T v) ⊆ Specα1(v). Thus if E ⊆ R is a subset then T V1(E)0 ⊆ V2(E)0. As T is continuous, +it also follows that T V1(E) ⊆ V2(E). If T is injective then for any v ∈ V1 we have S(R)v = S(R)T v and con- +sequently Specα1(v) = Specα2(T v) ⊆ Specα2(V2). As Specα2(V2) is closed, it follows using Corollary 6.2.17 +that Specα1(V1) = � +v∈V1 Specα1(v) ⊆ Specα2(V2). +In the multi-linear context, we have the following analogue of [NSZ15, A.10]: +Proposition 6.2.19. For j ∈ {1, 2, 3}, let αj : R → B(Vj)× be a strongly continuous representation of R on +the complete and Hausdorff complex locally convex vector space Vj. Assume that αj is pointwise polynomially +bounded. Let β : V1 × V2 → V3 be a continuous R-equivariant bilinear map. Let E1, E2 ⊆ R be closed subsets. +Then +β(V1(E) × V2(E)) ⊆ V + +3 (E1 + E2). +In particular, if α3 is polynomially bounded then β(V1(E) × V2(E)) ⊆ V3(E1 + E2). +Before proceeding to to proof of Proposition 6.2.19, let us mention the following immediate consequence: +Corollary 6.2.20. Consider the setting of Proposition 6.2.19. Assume additionally that β has dense span +and that α3 is polynomially bounded. Then Specα3(V3) ⊆ Specα1(V1) + Specα2(V2). +Proof. We know by Proposition 6.2.19 that β(V1, V2) ⊆ V + +3 +� +Specα1(V1) + Specα2(V2) +� +. In view of Proposi- +tion 6.2.14 and Corollary 6.2.15, we further know that +V + +3 +� +Specα1(V1) + Specα2(V2) +� += V3 +� +Specα1(V1) + Specα2(V2) +� +0, +and this is a closed linear subspace of V3. As β(V1, V2) has dense linear span in V3, it follows that +V3 ⊆ V3 +� +Specα1(V1) + Specα2(V2) +� +0. +According to Corollary 6.2.16, this equivalent with Specα3(V3) ⊆ Specα1(V1) + Specα2(V2). +The proof of Proposition 6.2.19 requires some preparation. It closely follows that of [Arv74, Prop. 2.2] and +[Nee13, Prop. A.14]. We first introduce some additional notation: +Definition 6.2.21. For a subset E ⊆ R, define the ideal J(E) ⊆ S(R) and the subspace Rα(E)0 ⊆ V by +J(E) := +� +f ∈ S(R) : �f ∈ C∞ +c (R) and supp �f ⊆ E +� +, +Rα(E)0 := { αfv : f ∈ J(E), v ∈ V } ⊆ V +Let Rα(E) := Rα(E)0 be its closure. If α is clear from the context, we write simply R(E)0 and R(E) instead +of Rα(E)0 and Rα(E), respectively. +If E ⊆ R is a subset, recall from Definition 6.2.1 that I0(E) consists of all Schwartz functions f whose Fourier +transform �f vanishes on a neighborhood of E ⊆ R. On the other hand, J(E) is the ideal in S(R) generated +by those f ∈ S(R) for which �f has compact support contained in E. +Lemma 6.2.22. Let E ⊆ R be a closed subset and let N ⊆ R be a 0-neighborhood. Then +I0(E) + J(E + N) = S(R). +Proof. Let J2 := J(E + N)0 + I0(E) be the closed ideal of S(R) generated by J(E + N) and I0(E). Observe +that h(J(E + N)) ⊆ R \ E. +On the other hand, h(I0(E)) ⊆ E by Lemma 6.2.2. +We thus find that +h(J2) ⊆ h(I0(E))∩h(J(E+N)) ⊆ ∅ and hence h(J2) = ∅. It follows from Corollary 6.2.3 that J2 = S(R). +Lemma 6.2.23. Let v ∈ V and N ⊆ R be a 0-neighborhood. If J(Specα(V ) + N) ⊆ S(R)v, then v = 0. +Proof. Let E := Specα(V ). Assume that J(E + N) ⊆ S(R)v. Recall from Remark 6.2.10 that V = V (E)0. +By Proposition 6.2.14, this means that I0(E) ⊆ S(R)v. On the other hand, J(E + N) ⊆ S(R)v, by assump- +tion. Since S(R)v is closed we obtain using Lemma 6.2.22 that S(R) = I0(E) + J(E + N) ⊆ S(R)v. By +Proposition 6.2.12, this implies that v = 0. +33 + +Lemma 6.2.24. Let E ⊆ R be closed. Then V (E) ⊆ � +N R(E + N) ⊆ V +(E), where N runs over all open +0-neighborhoods in R. +Proof. This proof follows that of [Arv74, Prop. 2.2]. Lemma 6.2.11 entails that Specα(αfv) ⊆ supp( �f) for +any f ∈ S(R) and v ∈ V . If N ⊆ R is a 0-neighborhood and f ∈ J(E+N), then by definition supp �f ⊆ E+N +and hence Specα(αfv) ⊆ E + N for any v ∈ V . Recalling that R(E + N)0 is the subspace of V generated by +J(E +N), we obtain that R(E +N)0 ⊆ V (E +N)0. Consequently � +N R(E +N) ⊆ � +N V (E +N) = V +(E). +Next, take v ∈ V (E)0. We show that v ∈ � +N R(E + N). Let N be a 0-neighborhood in R. Let λ ∈ V ′ be a +continuous functional with λ(R(E+N)) = {0}. Trivially, αf(v) ∈ R(E+N)0 for any f ∈ J(E+N), and hence +λ(αfv) = 0. We further have I0(E) ⊆ S(R)v, by Proposition 6.2.14, and consequently λ(αgv) = 0 for any +g ∈ I0(E). Thus λ(αfv) = 0 for any f in the closed ideal J2 := I0(E) + J(E + N) of S(R) spanned by I0(E) +and J(E + N). By Lemma 6.2.22 this ideal equals S(R), so +� +R f(t)λ(αtv)dt = λ(αfv) = 0 for any f ∈ S(R). +As t �→ λ(αtv) is continuous, it follows that λ(αtv) = 0 for all t ∈ R. In particular λ(v) = 0. Using the +Hahn-Banach Theorem [Rud91, Thm. I.3.5], it follows that v ∈ � +N R(E + N). Thus V (E)0 ⊆ � +N R(E + N) +and consequently also V (E) ⊆ � +N R(E + N). +Proof of Proposition 6.2.19: Having Lemma 6.2.24 at hand, we proceed as in [Nee13, Prop. A.14]. Let N ⊆ R +be an open 0-neighborhood. Let N1, N2 ⊆ R be open 0-neighborhoods s.t. N1 + N2 ⊆ N. We show that +β +� +Rα1(E1 + N1)0 × Rα2(E2 + N2)0 +� +⊆ V +� +E1 + E2 + N +� +0. +(6.2) +As such, for k ∈ {1, 2}, take vk ∈ V and fk ∈ J(Ek + Nk), meaning that supp( �fk) ⊆ Ek + Nk. We show that +β(α1(f1)v1, α2(f2)v2) ∈ V +� +E1 + E2 + N +� +0. In view of Proposition 6.2.14, we must show that it is annihilated +by I0(E1 + E2 + N). Let f3 ∈ I0(E1 + E2 + N), so supp( �f3) ∩ E1 + E2 + N = ∅. Then +αf3β(αf1(v1), αf2(v2)) = +� +R +� +R +f1(t1)f2(t2)f3(t3)β(α1(t1 + t3)v1, α2(t2 + t3)v2)dt1dt2dt3, += +� +R +� +R +F(t1, t2)β(α1(t1)v1, α2(t2)v2)dt1dt2, +(6.3) +where F ∈ S(R2) is defined by +F(t1, t2) := +� +R +f3(t3)f1(t1 − t3)f2(t2 − t3)dt3. +The Fourier transform �F ∈ S(R2) of F is given by �F(p1, p2) = �f1(p1) �f2(p2) �f3(p1 + p2). +Observe that +supp( �f1) + supp( �f2) ⊆ (E1 + N1) + (E2 + N2) ⊆ E1 + E2 + N. Since �f3 vanishes on E1 + E2 + N, we +find that �F = 0. Hence F = 0. From Equation (6.3) we obtain that α3(f3)β(α1(f1)v1, α2(f2)v2) = 0. By +Proposition 6.2.14 we conclude that β(α1(f1)v1, α2(f2)v2) ∈ V +� +E1 + E2 + N +� +0. Thus (6.2) is valid. As β is +continuous, it follows that +β(V1(E) × V2(E)) ⊆ β +� +Rα1(E1 + N1) × Rα2(E2 + N2) +� +⊆ V +� +E1 + E2 + N +� +, +where the first inclusion uses Lemma 6.2.24. Thus +β(V1(E) × V2(E)) ⊆ +� +N +V +� +E1 + E2 + N +� += V +(E1 + E2). +Assume next that α3 is polynomially bounded. Then αf is continuous for every f ∈ S(R), by Remark 6.2.6. +By Corollary 6.2.15 it follows that V +(E1 + E2) = V (E1 + E2). +Let us next consider the behavior of spectra under tensor products and spaces of continuous linear maps: +Proposition 6.2.25. Let α and σ be R-representation on the complete and Hausdorff locally convex vector +spaces V and W over C, respectively. Assume that α and σ are strongly continuous and have polynomial +growth. Let n ∈ N. +1. The R-representation α�⊗σ on the completed projective tensor product V �⊗W has a continuous action +R × V �⊗W → V �⊗W, polynomial growth and satisfies +Specα�⊗σ(V �⊗W) ⊆ Specα(V ) + Specα(W). +(6.4) +34 + +2. Equip B(V ; W) either with the strong topology or that of uniform convergence on compact sets. The R- +representation γ on B(V ; W) defined by γtT = σt◦T ◦α−t is strongly continuous, pointwise polynomially +bounded and satisfies +Specγ(B(V ; W)) ⊆ Specσ(W) − Specα(V ). +(6.5) +Proof. Notice by Proposition 6.1.5 that the actions α : R × V → V and σ : R × W → W are continuous. +1. Write γt := αt �⊗σt for t ∈ R. We first show that the R-representation γ on V �⊗W has polynomial growth. +Let p and q be continuous seminorms on V and W respectively. Assume that p(αtv) ≤ rα(|t|)p(v) and +q(αtw) ≤ rσ(|t|)q(w) for all t ∈ R, v ∈ V and w ∈ W, where rα, rσ ∈ R[t] are monic polynomials. Using +this inequality, it follows from the definition of the seminorm p ⊗ q on V �⊗W (see Equation (2.1)) that +(p ⊗ q)(γtψ) ≤ rα(|t|)rσ(|t|)(p ⊗ q)(ψ) for all t ∈ R and ψ ∈ V �⊗W. Thus α�⊗σ has polynomial growth. +To see that α�⊗σ has a continuous action, it suffices by Proposition 6.1.5 to show it is strongly continuous. +Let ψ ∈ V �⊗W. It suffices to show that t �→ γtψ is continuous at t = 0. Assume first that ψ ∈ V ⊗W, so +that ψ = �n +k=1 vk ⊗ wk for some vk ∈ V and wk ∈ W. Let p and q be continuous seminorms on V and +W, respectively. Let rα, rσ ∈ R[t] be as above. Let ǫ > 0. As α and σ are strongly continuous, we can +find δ > 0 s.t. p(αtvk −vk)q(σtwk) < ǫ and p(vk)q(σtwk −wk) < ǫ for all t ∈ (−δ, δ) and k ∈ {1, · · · , n}. +Writing αtvk ⊗ σtwk − vk ⊗ wk = (αtvk − vk) ⊗ σtwk + vk ⊗ (σtwk − wk), we obtain +(p ⊗ q)(γtψ − ψ) ≤ +n +� +k=1 +p(αtvk − vk)q(σtwk) + p(vk)q(σtwk − wk) < 2kǫ, +∀t ∈ (−δ, δ). +This proves that γtψ → ψ as t → 0, for any ψ in the dense subspace V ⊗W. Let us next consider general +ψ ∈ V �⊗W. Let η ∈ V ⊗ W be s.t. (p ⊗ q)(ψ − η) < ǫ. For small enough δ > 0 we have rα(|t|)rσ(|t|) ≤ 2 +and (p⊗q)(γtη−η) < ǫ for all t ∈ (−δ, δ). Using that (p⊗q)(γt(ψ−η)) ≤ rα(|t|)rσ(|t|)(p⊗q)(ψ−η) < 2ǫ, +we find for all t ∈ (−δ, δ) that +(p ⊗ q)(γtψ − ψ) ≤ (p ⊗ q)(γt(ψ − η)) + (p ⊗ q)(ψ − η) + (p ⊗ q)(γtη − η) < 4ǫ +Thus R → V �⊗W, t �→ γtψ is continuous. +As the canonical bilinear map �⊗ : V × W → V �⊗W is continuous, R-equivariant and has dense span in +V �⊗W, the remaining assertion is immediate from Corollary 6.2.20. +2. It suffices to consider only the topology of uniform convergence on compact sets. Let T ∈ B(V ; W). +Let q be a continuous seminorm on W and let K ⊆ V be compact. Consider the continuous seminorm +on B(V ; W) defined by qK(T ) := supv∈K q(T v). As T is bounded, there is a continuous seminorm p on +V s.t. q(T v) ≤ p(v) for all v ∈ v. Let rσ, rα ∈ R[t] be monic polynomials s.t. q(σtw) ≤ rσ(|t|)q(w) and +p(αtv) ≤ rα(|t|)p(v) for all t ∈ R, v ∈ V and w ∈ W. Then +qK(γt(T )) = sup +v∈K +q(σtT α−tv) ≤ rσ(|t|)rα(|t|) sup p(K). +This implies that γ is pointwise polynomially bounded. +We next show that γ is strongly continuous. Let T ∈ B(V ; W), ǫ > 0 and O := q−1([0, ǫ)) ⊆ W. The +map Φ : R×V → W, (t, v) �→ γt(T )v −T v = σtT α−tv −T v is continuous, because the map T : V → W +and the actions α : R×V → V and σ : R×W → W are all continuous. Since {0}×K ⊆ Φ−1(O) and K +is compact, it follows from the Tube Lemma (cf. [Mun00, Lem. 26.8]) that there is an interval I ⊆ R con- +taining 0 s.t. Φ(I×K) ⊆ O. This means that qK(γt(T )−T ) < ǫ for all t ∈ I, so γ is strongly continuous. +It remains to show that (6.5) holds true. Write EV := Specα(V ) and EW := Specα(W). Let N ⊆ R +be a 0-neighborhood. Let T ∈ B(V ; W) be arbitrary. Let f3 ∈ I0(EW − EV + N), so f3 ∈ S(R) +is s.t. supp( �f3) ∩ EW − EV + N = ∅. We show that γf3(T ) = 0. Let N1, N2 ⊆ R be symmetric 0- +neighborhoods such that N1 + N2 ⊆ N. Let v ∈ V , f1 ∈ J(EV + N1) and f2 ∈ J(EW + N2). So �f1 and +�f2 have compact support contained in EV + N1 and EW + N2, respectively. One verifies that +σf2γf3(T )αf1v = +� +R +� +R +� +R +f1(t1)f2(t2)f3(t3)σt2+t3T αt1−t3v dt1dt2dt3 += +� +R +� +R +F(t1, t2)σt2T αt1dt1dt2, +(6.6) +35 + +where F ∈ S(R2) is given by +F(t1, t2) = +� +R +f1(t1 + t3)f2(t2 − t3)f3(t3)dt3. +The Fourier transform �F ∈ S(R2) of F is given by �F(p1, p2) = �f1(p1) �f2(p2) �f3(p2 − p1). Recalling that +N1 is symmetric, notice that +supp( �f2) − supp( �f1) ⊆ (EW + N2) − (EV + NV ) ⊆ EW − EV + (N1 + N2) ⊆ EW − EV + N. +As �f3 vanishes on EW − EV + N, it follows that �F = 0 and hence F = 0. From (6.6) we conclude +that σf2γf3(T )αf1v = 0 for all f2 ∈ J(EW + N2). This implies γf3(T )αf1v = 0, by Lemma 6.2.23. +Consequently, if λ ∈ W ′ is any continuous functional, then +� +R f1(t1)⟨λ, γf3(T )αt1v⟩dt = 0. As the +map t �→ ⟨λ, γf3(T )αt1v⟩ is continuous it follows that ⟨λ, γf3(T )αt1v⟩ = 0 for all t ∈ R. In particular +⟨λ, γf3(T )v⟩ = 0. As W ′ separates the points of W by the Hahn-Banach Theorem [Rud91, Thm. I.3.4], +it follows that γf3(T )v = 0. As v ∈ V was arbitrary we find that γf3(T ) = 0. We have thus shown that +I0(EW − EV + N) ⊆ ker γ. By Corollary 6.2.16, this is equivalent to Specγ(B(V ; W)) ⊆ EW − EV + N. +Hence Specγ(B(V ; W)) ⊆ � +N EW − EV + N = EW − EV . +Recall from Section 2.1 that P(V ; W) = �∞ +k=0 P k(V ; W) is equipped with the product topology, where each +P k(E; F) carries the topology of uniform convergence on compact sets. We will have need for the following +result in Section 7 below: +Corollary 6.2.26. Consider the setting of Proposition 6.2.25. Assume that V is Fr´echet. Define the rep- +resentation γ of R on P(V ; W) by γt(f)(v) := σt(f(α−t(v))). Then γ is strongly continuous and pointwise +polynomially bounded. Moreover, if Specα(V ) ⊆ (−∞, 0], then +inf Specγ(P(V ; W)) = inf Specσ(W) ∈ {−∞} ∪ R. +Proof. Let n ∈ N≥0. Notice that γ leaves the homogeneous component P n(V ; W) ⊆ P(V ; W) invariant. +Recall from Proposition 2.1.4 and Proposition 2.1.3 that +P n(V ; W) ∼= Symn(V, W) ⊆ Mult(V n; W) ∼= B(V �⊗n; W) +as locally convex vector spaces. The thus-obtained continuous linear embedding Φn : P n(V ; W) ֒→ B(V �⊗n; W) +is R-equivariant when B(V �⊗n; W) is equipped with the R-action defined by �γt(T ) := σtT α−t. By Propo- +sition 6.2.25, this action is strongly continuous and pointwise polynomially bounded. Consequently, also γ +is strongly continuous and pointwise polynomially bounded on P n(V ; W). As P(V ; W) carries the product +topology, the same holds for the R-action γ on P(V ; W). +For the final statement, notice that W = P 0(V ; W) ⊆ P(V ; W). +By Proposition 6.2.18 it follows that +Specσ(W) ⊆ Specγ(P(V ; W)), showing inf Specγ(P(V ; W)) ≤ inf Specσ(W). Conversely, let n ∈ N. As Φn +is continuous, injective and R-equivariant, we know that Specγ(P n(V ; W)) ⊆ Spec�γ +� +B(V �⊗n; W) +� +, by Propo- +sition 6.2.18. Furthermore, using Proposition 6.2.25 we notice that Specα⊗n(V �⊗n) ⊆ (−∞, 0] and therefore +also that Spec�γ +� +B(V �⊗n; W) +� +⊆ Specσ(W) + [0, ∞) =: E. Thus Specγ(P n(V ; W)) ⊆ E for any n ∈ N≥0. By +Corollary 6.2.16 this means that γfψn = 0 for any f ∈ I0(E), ψn ∈ P n(V ; W) and n ∈ N. Consequently, +γfψ = 0 for any f ∈ I0(E) and ψ ∈ P(V ; W). So I0(E) ⊆ ker γ. By Corollary 6.2.16, this is equivalent with +Specγ(P(V ; W)) ⊆ E. Hence inf Specσ(W) = inf E ≤ inf Specγ(P(V ; W)). +Finally, we record some useful facts regarding the space of smooth vectors of a unitary G-representation: +Proposition 6.2.27. Let G be a regular locally convex Fr´echet-Lie group. Let d ∈ g and assume that the +R-action ˙α : R → Aut(g) defined by ˙αt := Ad(exp(td)) is polynomially bounded. Let (ρ, Hρ) be a smooth +unitary representation of G. Let E ⊆ R be a closed subset. Then the following assertions hold: +1. The R-representation t �→ ρ(exp(td))|H∞ +ρ +on H∞ +ρ +is strongly continuous and pointwise polynomially +bounded, where H∞ +ρ +is equipped with the strong topology. +2. The operator π(f) := +� +R f(t)ρ(exp(td))dt on Hρ leaves H∞ +ρ +invariant for any f ∈ S(R). +3. H∞ +ρ (E) = Hρ(E) ∩ H∞ +ρ +36 + +4. For any open subset U ⊆ R, H∞ +ρ (U) is dense in Hρ(U). +5. If E1, E2 ⊆ R are closed subsets then dρ(gC(E1))H∞(E2) ⊆ H∞ +ρ (E1 + E2). +Proof. The second item follows from [NSZ15, Thm. 2.3], the fourth from [NSZ15, Prop. 3.2] and the fifth +from [NSZ15, Thm. 3.1]. We provide an alternative proof of the second assertion and prove the first and third. +Recall that the topology on H∞ +ρ is defined by the seminorms pB(ψ) := supξ∈B ∥dρ(ξ1 · · · ξn)ψ∥, where B ⊆ gn +is a bounded subset. By [JN19, Prop. 3.19], the locally convex space H∞ +ρ +is complete. Let ψ ∈ H∞ +ρ . By +[JN19, Lem. 3.24], the orbit map ρφ : G → H∞ +ρ , g �→ ρ(g)ψ is smooth. It follows in particular that the +R-representation t �→ ρ(exp(td)) on H∞ +ρ +is strongly continuous. It follows moreover that the multi-linear +map gn → H∞ +ρ , (ξ1, · · · , ξn) �→ dρ(ξ1 · · · ξn)ψ is continuous. Using Proposition 2.1.3, we find that there exist +a continuous seminorm p on g such that ∥dρ(ξ1 · · · ξn)ψ∥ ≤ �n +k=1 p(ξk) for every ξ ∈ gn. Let N ∈ N and +the 0-neighborhood U ⊆ g be s.t. C := supξ∈U supt∈R +1 +1+|t|N p( ˙αt(ξ)) < ∞. As B ⊆ gn is bounded, so is its +projection Bk ⊆ g onto the kth factor for every k ∈ {1, · · · , n}. Thus there exists s > 0 such that Bk ⊆ sU +for all 1 ≤ k ≤ n. We obtain that supξk∈Bk supt∈R +1 +1+|t|N p( ˙αt(ξk)) ≤ sC for every 1 ≤ k ≤ n. Using that ρ is +unitary we find that +pB(ρ(e−td)ψ) = sup +ξ∈B +∥dρ( ˙αt(ξ1) · · · ˙αt(ξn)ψ∥ ≤ sup +ξ∈B +n +� +k=1 +p( ˙αt(ξk)) ≤ Csn(1 + |t|N)n, +∀t ∈ R. +This implies that the R-action t �→ ρ(exp(td)) on H∞ +ρ +is pointwise polynomially bounded. As in Defini- +tion 6.2.5, we conclude that π∞(f)ψ := +� +R f(t)ρ(exp(td))ψdt defines a representation π : S(R) → L(H∞ +ρ ) of +S(R) on H∞ +ρ +by linear operators. It is clear that π∞(f) := π(f)|H∞ +ρ , so this proves that π(f) leaves H∞ +ρ +invariant for every f ∈ S(R). It is further immediate from Definition 6.2.7 that H∞ +ρ (E) = Hρ(E) ∩ H∞ +ρ . +7 +Positive energy representations and holomorphic induction +In this section we explore the connection between positive energy representations and holomorphic induction. +It is shown in Theorem 7.1.6 and Theorem 7.1.17 that these two are intimately related, as is to be expected +from similar known results in more restrictive settings, such as [PS86, Thm. 11.1.1], [Nee13, Sec. 3] and +[Nee14, Thm. 6.1]. This is used to transfer various results from holomorphic induction to the context of +positive energy representations, under suitable assumptions. Before proceeding to the main results, let us +clarify the setting and make some preliminary observations. +7.1.1 +Notation and preliminary observations +Let G be a connected regular BCH Fr´echet-Lie group with Lie algebra g. Let α : R → Aut(G) be a ho- +momorphism having a smooth action R × G → G and let ˙α be the corresponding R-representation on gC, +defined by ˙αs(ξ) := L(αs)ξ := +d +dt +�� +t=0 αs(etξ) for s ∈ R. Assume that ˙α has polynomial growth, in the sense +of Definition 6.1.1. Let Dξ := +d +ds +�� +s=0 ˙αs(ξ) be the corresponding derivation on gC. Define the Lie group +G♯ := G ⋊α R, which has Lie algebra g♯ := g ⋊D Rd, where we have written d := 1 ∈ R ⊆ g♯ for the standard +basis element. Then G♯ is again a connected regular Fr´echet-Lie group, using [Nee06, Thm. V.I.8], but not +necessarily BCH. +As ˙α is assumed to have polynomial growth, we can define the Arveson spectral subspaces of gC as in +Definition 6.2.7. If E ⊆ R is any subset, we write gC(E) for the spectral subspace of gC associated to E. +Define hC := ker D ⊆ gC({0}), h := hC ∩ g and +n− := +� +δ>0 +gC((−∞, −δ]), +n+ := +� +δ>0 +gC([δ, ∞)). +We assume that (gC, α) satisfies the so-called splitting condition, meaning that gC = n− ⊕ hC ⊕ n+. Define +b± := hC ⊕ n± ⊆ gC. Let H := (Gα)0 ⊆ G be the connected subgroup of α-fixed points in G. Let us first +establish that the assumptions on H, n± and hC made in Section 4.2 are presently satisfied. +37 + +Lemma 7.1.1. H is a closed embedded Lie subgroup of G with Lie algebra h. +Proof. Since G is locally exponential, we can find a 0-neighborhood Ug ⊆ g s.t. expG restricts to a diffeo- +morphism on Ug. Let ξ ∈ Ug arbitrary. Using the fact that αt(expG(ξ)) = expG( ˙αt(ξ)) for all t ∈ R, observe +that ξ ∈ ker D ⇐⇒ expG(ξ) ∈ Gα. This implies that expG(Ug ∩ h) = expG(Ug) ∩ H. We also obtain that +h = { ξ ∈ g : expG(Rξ) ⊆ H }. Using [Nee06, Thm. IV.3.3] we conclude that H is a Lie subgroup with Lie +algebra h. +Lemma 7.1.2. The subspaces n±, hC and b± are Lie subalgebras of gC and [hC, n±] ⊆ n±. +Moreover, +AdH(n±) ⊆ n±. Finally, θ(n±) ⊆ n∓ and θ(hC) ⊆ hC. +Proof. The Lie bracket [−, −] : gC × gC → gC is bilinear, continuous and R-equivariant, meaning that +˙αs([ξ, η]) = [ ˙αs(ξ), ˙αs(η)] for all s ∈ R and ξ, η ∈ gC. From Proposition 6.2.19 we obtain for any two closed +subsets E1, E2 ⊆ R that [gC(E1), gC(E2)] ⊆ gC(E1+E2). This implies that n±, hC and b± are Lie subalgebras +of gC and that [hC, n±] ⊆ n±. We next show that AdH(n±) ⊆ n±. Let h ∈ H. Then ˙αs and Adh commute +for any s ∈ R, so Adh : gC → gC is a continuous equivariant linear map. It follows using Proposition 6.2.18 +that Adh(gC(E)) ⊆ gC(E) for any closed subset E ⊆ R. Hence AdH(n±) ⊆ n±. Let us next consider the +conjugation θ. Using that ˙αt commutes with θ for any t ∈ R, observe that θ ˙αfθ = ˙αf for any f ∈ S(R). +Consequently, S(R)θ(ξ) = +� +f : f ∈ S(R)ξ +� +. Using that F(f)(p) = Ff(−p) for p ∈ R, we obtain for any +ξ ∈ gC that Spec ˙α(θ(ξ)) = h(S(R)θ(ξ)) = −h(S(R)ξ) = − Spec ˙α(ξ). So we have θ(gC(E)) = gC(−E) for any +closed E ⊆ R. This implies that θ(n±) ⊆ n∓ and θ(hC) ⊆ hC. +As the Lie group G♯ = G ⋊α R need not be analytic, we only have access to the analytic structure of G: +Definition 7.1.3. If (ρ, Hρ) is a unitary representation of G♯, we write HωG +ρ +for the space of G-analytic +vectors in Hρ. We further define H∞,n− +ρ +:= +� +ψ ∈ H∞ +ρ +: dρ(n−)ψ = {0} +� +and we write V (ρ) := H∞,n− +ρ +for +its closure. +Let us first clarify that V (ρ) can equivalently be defined as the closure of the set of G-smooth vectors in Hρ +that are killed by n−, as opposed to the G♯-smooth ones: +Lemma 7.1.4. Let ρ be a unitary G♯-representation. Let W(ρ) ⊆ Hρ be the closed linear subspace generated +by the set of G-smooth vectors in Hρ that are killed by dρ(n−). Then W(ρ) = V (ρ). +Proof. It is trivial that V (ρ) ⊆ W(ρ). Let ψ ∈ Hρ be a G-smooth vector s.t. dρ(n−)ψ = {0}. Let f ∈ C∞ +c (R) +and define πfψ := +� +R f(t)ρ(t)ψdt ∈ Hρ. Then πfψ is a smooth vector for G♯, e.g. using [NSZ15, Lem. A.4]. +Let ξ ∈ n−. Then ˙α−t(ξ) ∈ n− and hence dρ( ˙α−t(ξ))ψ = 0 for every t ∈ R. Using [NSZ15, Lem. A.4] to +differentiate under the integral, we obtain: +dρ(ξ)πfψ = +� +R +f(t)dρ(ξ)ρ(t)ψdt = +� +R +f(t)ρ(t)dρ( ˙α−t(ξ))ψdt = 0. +So πfψ ∈ H∞,n− +ρ +for any f ∈ C∞ +c (R). Approximating ψ by vectors of the form πfψ, we conclude that +ψ ∈ V (ρ). So V (ρ) = W(ρ). +To keep a uniform notation for G- and G♯-representations, we complement Definition 7.1.3 with: +Definition 7.1.5. If (ρ, Hρ) is a smooth unitary representation of G, we write HωG +ρ +:= Hω +ρ for the space of +G-analytic vectors in Hρ. Define H∞,n− +ρ +:= +� +ψ ∈ H∞ +ρ +: dρ(n−)ψ = {0} +� +and let V (ρ) := H∞,n− +ρ +denote its +closure. +Let us proceed with the task of relating the positive energy condition with holomorphic induction. Notice +that V (ρ) ⊆ Hρ is H × R-invariant for any smooth unitary G-representation ρ, because n− is invariant under +˙αt and Adh for any t ∈ R and h ∈ H. The following makes use of the notation specified in Definition 4.3.8: +Theorem 7.1.6. Let (ρ, Hρ) be a smooth unitary representation of G♯ and let σ be the unitary representation +of H × R on Vσ := V (ρ) defined by σ(h, t) := ρ(h, t)|V (ρ). The following assertions are equivalent: +1. ρ is of positive energy at d ∈ g♯, V (ρ) is cyclic for ρ and V (ρ) ∩ HωG +ρ +is dense in V (ρ). +2. σ is of positive energy at d ∈ g♯ and ρ|G = HolIndG +H(σ|H). +If these conditions are satisfied, then inf Spec(−idρ(d)) = inf Spec(−idσ(d)) ≥ 0. +38 + +We start the proof of Theorem 7.1.6 with two lemmas: +Lemma 7.1.7. Let W ⊆ V (ρ) be a H-invariant closed linear subspace that is cyclic for G and contains a +dense set of G-analytic vectors. Then W = V (ρ). +Proof. Let W ⊥ be the orthogonal complement of W in V (ρ), so V (ρ) = W ⊕W ⊥ as unitary H-representations. +It suffices to show that W ⊥ ⊥ ρ(G)W. Define W ωG := W ∩ HωG +ρ . Let w ∈ W ωG and v ∈ W ⊥ ⊆ V (ρ). +Consider the analytic function f : G → C, f(g) := ⟨v, ρ(g)w⟩. Let E0 : U(gC) → U(hC) be defined as in +Definition 4.3.5. As dρ(n−) kills both H∞,n− +ρ +and W ωG, observe that ⟨v, dρ(x)w⟩ = ⟨v, dσ(E0(x))w⟩ = 0 +for any x ∈ U(gC). It follows that j∞ +e (f) = 0. As G is connected and f is analytic, we conclude using +Proposition 2.1.14 that f = 0. Because W ωG is dense in W, it follows that W ⊥ ⊥ ρ(G)W. +Lemma 7.1.8. Let D ⊆ Hω +ρ be a linear subspace. Then dρ(U(gC))D is the closed G-invariant subspace of +Hρ generated by D. +Proof. Define F := dρ(U(gC))D and let F′ denote the closed G-invariant subspace generated by D. The +inclusion F ⊆ F′ is clear. It thus suffices to show that F is G-invariant. For any ψ ∈ D, define the analytic +map fψ : G → Hρ, fψ(g) := ρ(g)ψ. The set U := � +ψ∈D f −1 +ψ (F) contains 1 ∈ G and is closed, because each +fψ is continuous. As G is BCH and fψ(G) ⊆ Hω +ρ for any ψ ∈ D, the set U is also open. Since G is connected, +we obtain that U = G. Hence ρ(G)D ⊆ F. This implies that F is G-invariant. +Proof of Theorem 7.1.6: Define Dχ := V (ρ) ∩ HωG +ρ . Assume that (1) holds true. Then in particular, σ is of +positive energy at d. Let χ : b− → L(Dχ) be the trivial extension of dσ to b− with domain Dχ. By definition +of V (ρ), Dχ is killed by dρ(n−). The conditions for Vσ in Theorem 4.2.4 are satisfied for the (H, b−)-extension +pair (σ|H , χ), so (2) follows from Theorem 4.2.4. +Conversely, assume that ρ|G = HolIndG +H(σ|H) and that σ is of p.e. at d. It follows from Theorem 4.2.4 that +there is a H-invariant closed linear subspace W ⊆ Hρ s.t. W is cyclic for ρ and W ∩ HωG +ρ +is both dense in +W and killed by dρ(n−). The last condition implies using Lemma 7.1.4 that W ⊆ V (ρ). By Lemma 7.1.7 we +obtain that W = V (ρ). To see that (1) holds true, it only remains to show that ρ is of positive energy at d. +Define Φ : H∞ +ρ → C∞(G; Vσ)H by Φψ(g) := pV ρ(g)−1ψ for ψ ∈ H∞ +ρ , where pV : Hρ → V (ρ) is the orthogonal +projection. Using the exponential map as a local chart, identify J∞ +e C∞(G, Vσ) ∼= P(gC; Vσ) G-equivariantly. +Let A denote the composition +A : H∞ +ρ +Φ +−→ C∞(G; Vσ)H +j∞ +e +−−→ P(gC; Vσ) +restr +−−−→ P(n−; Vσ). +Observe that +Φρ(t)ψ(g) = pV ρ(g)−1ρ(t)ψ = pV ρ(t)ρ(α−t(g))−1ψ = σ(t)pV ρ(α−t(g))−1ψ = σ(t)Φψ(α−t(g)). +Consequently, A is R-equivariant if we equip P(n−; Vσ) with the R-action defined by (νtf)(ξ) := σ(t)f( ˙α−t(ξ)) +for t ∈ R and f ∈ P n(n−; Vσ). +Equip H∞ +ρ +with the strong topology (cf. Definition 2.2.2), with re- +spect to which it is complete because G is a regular Fr´echet-Lie group [JN19, Prop. 3.19]. +Recall that +P(n−; Vσ) = �∞ +n=0 P n(n−; Vσ) carries the product topology and each P n(n−; Vσ) carries the topology of +uniform convergence on compact sets. We show that A is continuous with respect to these topologies. For +any ψ ∈ H∞ +ρ , let fψ ∈ C∞(G; Hρ), f(g) := ρ(g)ψ denote the orbit map. Using that ρ is unitary, observe +that the linear map H∞ +ρ → C∞(G; Hρ), ψ �→ fψ is continuous w.r.t. the smooth compact-open topology on +C∞(G; Hρ). This implies that Φ is continuous. As j∞ +e +is continuous by Proposition 2.1.15, the continuity of +A follows. We remark further that the R-representation t �→ ρ(t) on H∞ +ρ +is strongly continuous and point- +wise polynomially bounded by Proposition 6.2.27, so that its Arveson spectrum can be defined according +to Definition 6.2.7. Similarly, because the R-actions on n− and Vσ both have polynomially growth and are +strongly continuous, it follows from Corollary 6.2.26 that the R-action ν on P(n−; Vσ) is strongly continu- +ous and pointwise polynomially bounded. Since n− and Vσ have non-positive and non-negative spectrum, +respectively (relative to the R-actions ˙αt and σ(t), respectively), we further obtain from Corollary 6.2.26 and +Example 6.2.8 that +inf Specν +� +P +� +n−; Vσ +�� += inf Spec(Vσ) = inf Spec(−idσ(d)) ≥ 0 +We show next that A is injective. Let ψ ∈ H∞ +ρ +and suppose that A(ψ) = 0. Then pV dρ(U(n−))ψ = {0}, +which implies ψ ⊥ dρ(U(n+))Dχ. Since Dχ is dρ(b−)-invariant, notice that dρ(U(n+))Dχ = dρ(U(gC))Dχ by +the PBW Theorem. By Lemma 7.1.8, this is the closed G-invariant subspace of Hρ generated by Dχ, which +equals all of Hρ because Dχ is dense in V (ρ) and V (ρ) is cyclic for ρ. Thus ψ ⊥ Hρ and so ψ = 0. Hence A is +39 + +injective, continuous and R-equivariant. It follows by Proposition 6.2.18 that Spec(H∞ +ρ ) ⊆ Specν +� +P +� +n−; Vσ +�� +, +where we consider the R-action t �→ ρ(t) on H∞ +ρ . Thus +inf Spec(H∞ +ρ ) ≥ inf Specν +� +P +� +n−; Vσ +�� += inf Spec(−idσ(d)), +Notice that Hρ and H∞ +ρ have the same spectrum, because H∞ +ρ is dense in Hρ. So +inf Spec(−idρ(d)) = inf Spec(Hρ) = inf Spec(H∞ +ρ ) ≥ inf Spec(−idσ(d)) ≥ 0. +Thus, ρ is of positive energy at d. So (2) holds true. Finally, the inclusion V (ρ) ⊆ Hρ is R-equivariant, so +by Proposition 6.2.18 we also have the reverse inequality inf Spec(−idρ(d)) ≤ inf Spec(−idσ(d)). +Let us state some important immediate consequences of Theorem 7.1.6. +Lemma 7.1.9. Let (ρ, Hρ) be a smooth unitary representation of G. Let qV ∈ B(Hρ) denote the orthogonal +projection onto V (ρ). Then qV ∈ ρ(G)′′. +Proof. Let T ∈ ρ(G)′ = B(Hρ)G. Then T H∞ +ρ +⊆ H∞ +ρ +and dρ(n−)T H∞,n− +ρ += T dρ(n−)H∞,n− +ρ +⊆ {0}. Thus +T H∞,n− +ρ +⊆ H∞,n− +ρ +. It follows that T V (ρ) ⊆ V (ρ), and so qV T = T qV . Hence qV ∈ ρ(G)′′. +Corollary 7.1.10. Suppose that the unitary G♯-representation ρ satisfies the equivalent conditions of Theo- +rem 7.1.6. Then T �→ T |V (ρ) defines isomorphisms of von Neumann algebras +B(Hρ)G ∼= B(V (ρ))H +and +B(Hρ)G♯ ∼= B(V (ρ))H×R. +Proof. That T �→ T |V (ρ) defines an isomorphism B(Hρ)G → B(V (ρ))H is immediate from Lemma 7.1.9 +and Theorem 4.4.4. Consequently, it suffices to show that any T ∈ B(Hρ)G with T |V (ρ) ∈ B(V (ρ))H×R +automatically commutes with the R-action t �→ ρ(t) on Hρ. Consider such T and let t ∈ R. Then +ρ(t)T ρ(g)v = ρ(t)ρ(g)T v = ρ(αt(g))ρ(t)T v = ρ(αt(g))T ρ(t)v = T ρ(αt(g))ρ(t)v = T ρ(t)ρ(g)v +(7.1) +for any g ∈ G and v ∈ V (ρ). As V (ρ) is cyclic for G, it follows that T ρ(t) = ρ(t)T for all t ∈ R. +Corollary 7.1.11. Suppose that the unitary G♯-representations ρ1 and ρ2 satisfy the equivalent conditions +of Theorem 7.1.6. Then the following assertions are valid: +1. If V (ρ1) ∼= V (ρ2) as unitary H-representations, then ρ1|G ∼= ρ2|G. +2. If V (ρ1) ∼= V (ρ2) as unitary H × R-representations, then ρ1 ∼= ρ2. +Proof. The first assertion is immediate from Theorem 4.3.4. Assume that the unitary u : V (ρ1) → V (ρ2) +intertwines the H ×R-actions. Consider the unitary G♯-representation ρ = ρ1 ⊕ρ2 on Hρ1 ⊕Hρ2. Notice that +V (ρ1 ⊕ ρ2) = V (ρ1) ⊕ V (ρ2) =: W and that ρ satisfies the equivalent conditions in Theorem 7.1.6. Define +S ∈ B(W)H×R by S(v1, v2) := (0, uv1). By Corollary 7.1.10, there is some T ∈ B(Hρ1 ⊕Hρ2)G♯ s.t. T |W = S. +As V (ρ1) and V (ρ2) are cyclic for G in Hρ1 and Hρ2, respectively, T is of the form T (ψ1, ψ2) = (0, Uψ1) for +some U : Hρ1 → Hρ2 intertwining the G♯-actions. Notice that S∗S and SS∗ are the orthogonal projections +onto V (ρ1) and V (ρ2), respectively. By Corollary 7.1.10 it follows that T ∗T and T T ∗ are the orthogonal +projections onto Hρ1 and Hρ2, respectively. This implies that U is unitary. +7.1.2 +The spectral gap condition +We will next assume that the so-called spectral gap condition is satisfied. We show that in this case, V (ρ) is +always cyclic for positive energy representations. +Definition 7.1.12. We say that the spectral gap (SG) condition is satisfied if there is some δ > 0 such that +gC = gC((−∞, −δ]) ⊕ hC ⊕ gC([δ, ∞)). +(7.2) +If ρ is a smooth unitary representation of G♯ and E ⊆ R is a subset, we write Hρ(E) and H∞ +ρ (E) for the +closed spectral subspaces associated to the R-representation t �→ ρ(t) on Hρ and H∞ +ρ , respectively, where we +recall that the R-action on H∞ +ρ +is pointwise polynomially bounded by Proposition 6.2.27. Recall also from +Proposition 6.2.27 that H∞ +ρ (E) = Hρ(E) ∩ H∞ +ρ . +40 + +Lemma 7.1.13. Assume that (SG) is satisfied. Let ρ be a smooth unitary representation of G♯ which is of +positive energy at d ∈ g♯. If Hρ ̸= {0} then V (ρ) ̸= {0}. +Proof. Let δ > 0 be such that (7.2) is satisfied. Set E0 := −i inf Spec(dρ(d)). Let 0 < ǫ < δ and define +U := [E0, E0 + ǫ). By definition of E0, the spectral subspace Hρ(U) is nonzero. Using Proposition 6.2.27(4), +we know that H∞ +ρ (U) is dense in Hρ(U) = Hρ((−ǫ, ǫ)). Since Hρ(U) is nonzero, so is H∞ +ρ (U). By the +last point in Proposition 6.2.27, we obtain that dρ(n−)H∞(U) ⊆ H∞ +ρ ((−∞, E0 + ǫ − δ]) = {0}. +Hence +H∞(U) ⊆ H∞,n− +ρ +⊆ V (ρ). It follows that V (ρ) ̸= {0}. +Proposition 7.1.14. Assume that (SG) is satisfied. Let ρ be a smooth unitary representation of G♯ which +is of positive energy at d ∈ g♯. Then V (ρ) is cyclic for ρ. +Proof. Let W be the closed G♯-invariant subspace of Hρ generated by V (ρ). Then W ⊥ carries a smooth +representation of G♯ that is of positive energy at d ∈ g♯. From (W ⊥)∞,n− ⊆ H∞,n− +ρ +⊆ V (ρ), we obtain that +(W ⊥)∞,n− ⊆ W ⊥ ∩ V (ρ) = {0}. Using Lemma 7.1.13 we conclude that W ⊥ = {0}, so W = Hρ. +7.1.3 +Ground-state representations +We now shift our attention to ground-state representations, where Theorem 7.1.6 simplifies somewhat. If ρ +is a smooth unitary representation of G♯ on Hρ, we define Hρ(0) = ker dρ(d), H∞ +ρ (0) := Hρ(0) ∩ H∞ +ρ +and +HωG +ρ (0) := Hρ(0) ∩ HωG +ρ . It will be convenient to make the following definition: +Definition 7.1.15. Let (ρ, Hρ) be a smooth unitary representation of G♯ that is ground-state at d ∈ g♯. We +say that ρ is analytically ground-state at d ∈ g♯ if HωG +ρ (0) is dense in Hρ(0). +Lemma 7.1.16. Let (ρ, Hρ) be a smooth unitary representation of G♯ that is of positive energy at d ∈ g♯. +Then H∞ +ρ (0) ⊆ H∞,n− +ρ +. If ρ is analytically ground-state at d, then V (ρ) = Hρ(0). +Proof. Using Proposition 6.2.19, we obtain that dρ(gC((−∞, −δ]))H∞ +ρ (0) ⊆ H∞ +ρ ((−∞, −δ]) = {0} for any +δ > 0. Hence H∞ +ρ (0) ⊆ V (ρ). If ρ is analytically ground-state at d, the preceding implies Hρ(0) ⊆ V (ρ). +Using Lemma 7.1.7 we conclude that Hρ(0) = V (ρ). +The following clarifies the tight relation between unitary representations of G ⋊α R that are analytically +ground-state at d ∈ g♯ and holomorphic induction: +Theorem 7.1.17. Consider the setting of Theorem 7.1.6. The following assertions are equivalent: +1. ρ is analytically ground-state at d ∈ g♯. +2. ρ|G = HolIndG +H(σ|H) and V (ρ) = Hρ(0). +Proof. Assume that (1) is valid. From Lemma 7.1.16 we obtain that V (ρ) = Hρ(0), so (2) follows from +Theorem 7.1.6. Suppose conversely that (2) holds true. Theorem 7.1.6 yields that ρ is of positive energy at +d, that Hρ(0) is cyclic for G and that HωG +ρ (0) is dense in Hρ(0). Thus (1) is valid. +Let us complement Theorem 7.1.17 with the following observation: +Proposition 7.1.18. Let ρ be a smooth unitary p.e. representation of G. Let ρ0 denote its minimal positive +extension to G♯. Assume that ρ0 satisfies the equivalent conditions of Theorem 7.1.6. If ρ is irreducible, then +it is analytically ground-state and V (ρ) = Hρ(0). +Proof. Define Vσ := V (ρ), σ0(h, t) := ρ0(h, t)|Vσ and σ(h) := ρ(h)|Vσ. Let M := ρ(G)′′ be the von Neumann +algebra generated by ρ(G). We obtain using Corollary 7.1.10 that B(Vσ)H = C idVσ, so σ is irreducible. It +follows that σ0(t) = eitp idVσ for some p ∈ R, because σ0(t) ∈ B(Vσ)H for any t ∈ R. As σ is of positive energy, +we have p ≥ 0. By Theorem 7.1.6 we know that inf Spec(−idρ(d)) = p. Consequently, ρ1(t) := ρ0(t)e−itp +defines a positive inner implementation of R → Aut(M), t �→ Ad(ρ0(t)). As ρ0(t) is minimal, it follows +that p ≤ 0. +Hence p = 0. +So V (ρ) ⊆ Hρ(0). +On the other hand, we know from Lemma 7.1.16 that +H∞ +ρ (0) ⊆ V (ρ). As σ is irreducible and H∞ +ρ (0) contains V (ρ) ∩ HωG +ρ , which is dense in V (ρ), it follows that +H∞ +ρ (0) = V (ρ) = Hρ(0). This implies that ρ is analytically ground-state. +41 + +7.1.4 +Strongly-entire ground-state representations for T-actions +The preceding results become particularly applicable for representations ρ which are both strongly-entire and +ground-state w.r.t. a T-action. In this case, we can always guarantee that they are analytically ground-state: +Lemma 7.1.19. Suppose that α descend to a T-action. Let ρ be a unitary p.e. representation of G ⋊α T. +We write HOG +ρ +for the vectors in Hρ that are strongly-entire for the G-action. Let P : Hρ → Hρ(0) denote +the orthogonal projection. Then PHO +ρ ⊆ HO +ρ . In particular, if ρ|G is strongly-entire then Hρ(0) ∩ HOG +ρ +is +dense in Hρ(0). +Proof. For a compact subset B ⊆ gC and ψ ∈ H∞ +ρ , we write pn +B(ψ) := supξj∈B ∥dρ(ξ1 · · · ξn)ψ∥ for n ∈ N≥0 +and set qB(ψ) := �∞ +n=0 +1 +n!pn +B(ψ). Let ψ ∈ HO +ρ and let B ⊆ gC be compact. Then B′ := α(T × B) ⊆ gC is +compact, T-invariant and satisfies B ⊆ B′. Observe that +pn +B(ρ(t)ψ) ≤ pn +B′(ρ(t)ψ) = pn +˙α−t(B′)(ψ) = pn +B′(ψ), +∀t ∈ T. +Identifying T ∼= R/2πZ, recall that P = +1 +2π +� 2π +0 +ρ(t)dt. Notice using e.g. [NSZ15, Lem. A.4] that PH∞ +ρ ⊆ H∞ +ρ , +and moreover that +pn +B(Pψ) ≤ 1 +2π +� 2π +0 +pn +B(ρ(t)ψ)dt ≤ pn +B′(ψ), +∀ψ ∈ H∞ +ρ , n ∈ N≥0. +We thus find that qB(Pψ) ≤ qB′(ψ). So PHO +ρ ⊆ HO +ρ . +Combining Theorem 7.1.17 and Lemma 7.1.19 we obtain: +Theorem 7.1.20. Assume that α is a T-action. Let (ρ, Hρ) be a unitary representation of G ⋊α T. Assume +that ρ|G is strongly-entire. Let σ be the unitary representation of H×T on V (ρ). The following are equivalent: +1. ρ is ground-state at d ∈ g♯. +2. ρ|G = HolIndG +H(σ|H) and V (ρ) = Hρ(0). +In this case, also σ is strongly-entire. +By Proposition 2.3.6(3), we know that any smooth unitary representation ρ of G which is of p.e. w.r.t. a +T-action α is automatically ground-state, and also that the minimal positive extension ρ0 of ρ to G♯ descends +to G ⋊α T. Combining Theorem 7.1.20, Corollary 7.1.10 and Corollary 7.1.11, we obtain: +Corollary 7.1.21. Assume α is a T-action and that every irreducible unitary representation of G that is of +positive energy w.r.t. α is strongly-entire. Then there is an injective map �Gpos(α) ֒→ �H, obtained by sending +ρ ∈ �Gpos(α) to the irreducible unitary H-representation on V (ρ). +Remark 7.1.22. Recall from Theorem 3.1.6 that if G is a finite-dimensional Lie group of type R, then every +continuous unitary G-representation is in fact strongly-entire. +It would be beneficial to obtain sufficient conditions for V (ρ) ∩ HωG +ρ +to be dense in V (ρ). We state the +following related open problem: +Problem 7.1.23. Assume there are 0-neighborhoods U ⊆ gC, U− ⊆ n−, U0 ⊆ hC and U+ ⊆ n+ for which +the map +U+ × U0 × U− → U, +(ξ+, ξ0, ξ−) �→ ξ+ ∗ ξ0 ∗ ξ− +is biholomorphic, where ∗ is defined by the BCH series. We write ξ �→ (ξ+, ξ0, ξ−) for its inverse. Let ρ +be a unitary representation of G that is of positive energy. Set Vσ := V (ρ), considered as a unitary H- +representation. Assume that Vσ is cyclic for ρ. Is it true that V ω +σ ⊆ HωG +ρ ? Taking v ∈ V ω +σ , the assumptions +imply that the map U → C, ξ �→ ⟨v, σ(eξ0)v⟩ is analytic on some 0-neighborhood. If it can be shown to locally +extend the map g → C, ξ �→ ⟨v, ρ(eξ)v⟩ on some 0-neighborhood in g, then it would follow from [Nee11, Thm. +5.2] that v ∈ HωG +ρ . +42 + +8 +Examples +Example 8.1.1 (Finite-dimensional Lie groups of type R). +Let G be a connected finite-dimensional Lie group of type R and let α be a T-action on G. Let H := (Gα)0 +be the connected subgroup of α-fixed points. In view of Theorem 3.1.6 and Theorem 7.1.20, any continuous +ground-state representation ρ of G is holomorphically induced from V (ρ). According to Corollary 7.1.21, this +defines an injection �Gpos(α) ֒→ �H. +Example 8.1.2 (Holomorphically induced, but not geometrically). +1. Consider G = SL(2, R) and let ρ be any non-trivial continuous unitary representation. Trivially, we +have ρ = HolIndG +G(ρ). However, as ρ admits no non-trivial strongly-entire vectors by Theorem 3.1.6, it +is not geometrically holomorphically induced from itself. +2. For a slightly less trivial example, consider the group G = K×SL(2, R), where K is a connected compact +simple Lie group. Let T ⊆ K be a maximal torus and set t := Lie(T ). Pick regular element H ∈ treg +and let ∆+ := { α ∈ ∆ : −iα(H) > 0 } be the corresponding system of positive roots, where ∆ ⊆ it∗ +denotes the set of all roots of k. Consider the T-action on G defined by αt(k, x) = (etHke−tH, x). Let +(ρ, Hρ) be a continuous irreducible unitary representation of G. Then ρ decomposes as Hρ = Hν ⊗ Hσ +for some irreducible unitary K- and SL(2, R)-representations (ν, Hν) and (σ, Hσ), respectively. Then +ρ is of positive energy w.r.t. α and V (ρ) = Cλ ⊗ Hσ, where Cλ ⊆ Hν is a lowest-weight subspace. +Since Hω +ρ = Hν ⊗ Hω +σ, Theorem 4.2.4 implies that ρ is holomorphically induced from the T × SL(2, R)- +representation on Cλ ⊗ Hσ. The latter admits no strongly-entire vectors by Theorem 3.1.6, so ρ is not +geometrically holomorphically induced from the T × SL(2, R)-representation on Cλ ⊗ Hσ. +Example 8.1.3 (Positive energy representations of Heisenberg groups). +Let V be a real Fr´echet space equipped with a non-degenerate continuous skew-symmetric bilinear form ω. +Let α : T → Sp(V, ω) be a homomorphism with smooth action T × V → V . Define Dv := +d +dt +�� +t=0 αtv and +consider the closed subspaces +V0 := ker D = { v ∈ V : αtv = v +∀t ∈ R } +and +Veff := Span { αtv − v : t ∈ R, v ∈ V } . +(8.1) +As α∗ +t ω = ω of all t ∈ R, we notice that V0 and Veff are symplectic complements, so (V, ω) ∼= (V0, ω0)⊕(Veff, ω1), +where ω0 and ω1 are the restrictions of ω to V0 and Veff, respectively. Consider G := Heis(V, ω). By Theo- +rem 7.1.17, we know for any unitary representation ρ of G ⋊α T that is analytically ground-state at d that +ρ|G is holomorphically induced by some analytic unitary representation of Heis(V0, ω0) =: H. +Let us consider a concrete example. Assume that ω1(v, Dv) > 0 for every nonzero v ∈ Veff. Assume that +(Veff)C decomposes as (Veff)C ∼= L+ ⊕ L− into the positive (L+) and negative (L−) Fourier modes of the T- +action α. Let J1 be the complex structure on V defined by J1(v+w) := iv−iw for v ∈ L+ and w ∈ L−. Then +J ∗ +1 ω1 = ω1 and ω1(v, J1v) > 0 for every v ∈ Veff, so J1 defines a compatible positive polarization on Veff. If +J0 is a compatible positive polarization on V0, then J = J0 ⊕ J1 defines one on V . As in Example 3.1.9, +we equip the (now complex) vector space V with the inner product ⟨v, w⟩J := ω(v, J w) + iω(v, w), making +V into a complex pre-Hilbert space, on which α acts unitarily. Let H be its Hilbert space completion and +let H0 and H1 be the closed subspaces of H spanned by V0 and Veff, respectively. Notice that as unitary +T-representations, we have (Veff, ⟨−, −⟩J1) ∼= (L+, ⟨−, −⟩L+), where ⟨v, w⟩L+ := 2iω(v, w) for v, w ∈ L+. So +the unitary T-representation α on H is of positive energy. Let F(H) be the Hilbert space completion of the +symmetric algebra S•(H) w.r.t. the inner product (3.1), and let ρ be the strongly-entire unitary representation +of G = Heis(V, ω) on F(H) constructed in Example 3.1.9. Similarly, we write ρ0 and ρ1 for the representations +of Heis(V0, ω0) and Heis(Veff, ω1) on F(H0) and F(H1), respectively. Letting T act on F(H) according to +the second quantization of α, we obtain an extension of ρ to a smooth representation of G ⋊α T on F(H), +which we denote again by ρ. This extension is ground-state w.r.t. α. The representation ρ of G ⋊α T on +F(H) decomposes as F(H) ∼= F(H0) ⊗ F(H1), and V (ρ) = F(H0) ⊗ Ω1 ⊆ F(H), where Ω1 ∈ F(H1) is the +vacuum vector. Theorem 7.1.20 implies that ρ|G is holomorphically induced from the H-representation ρ0 +on F(H0). Moreover, F(H0)∞ ⊗ Ω1 ⊆ H∞ +ρ . Indeed, the vacuum vector Ω1 is smooth for Heis(Veff, ω1), so +if ψ ∈ F(H0)∞ is a smooth vector for H then (z, v) �→ ρ(z, v)ψ = zρ0(v0)ψ0 ⊗ ρ1(v1)Ω1 is a smooth map +G → F(H). By Theorem 5.3.4 we conclude that ρ is geometrically holomorphically induced from ρ0. +43 + +Example 8.1.4 (Metaplectic representation). +We continue in the notation of Example 8.1.3. Let K be a connected regular BCH Fr´echet-Lie group acting +smoothly on V . Define G := V ⋊ K. Assume that α is a smooth T-action on G. Let H := Gα = V0 ⋊ Kα be +the (connected) subgroup of α-fixed points. Let HR be the real vector space underlying H. The symplectic +form ω on V extends to HR by setting ω(v, w) := Im⟨v, w⟩J for v, w ∈ HR. Define +Bres(HR) := { A ∈ B(HR) : [J , A] ∈ B2(H) } , +whose elements are ‘close’ to being C-linear. It is a real Banach algebra with norm ∥A∥res := ∥A∥ + ∥A∥2, +where B2(H) denotes the space of Hilbert-Schmidt operators on H. The restricted symplectic group is defined +by Spres(HR, ω) := Sp(HR, ω)∩Bres(HR), equipped with the subspace topology. Being an algebraic subgroup +of Bres(HR)×, we obtain using [Nee04, Prop. IV.14] that Spres(HR, ω) is a Banach-Lie group modeled on the +Banach-Lie algebra spres(HR, ω) := sp(HR, ω) ∩ Bres(HR). We assume that the canonical inclusion V ֒→ HR +extends to a continuous homomorphism +η : V ⋊ K → HR ⋊ Spres(HR, ω) =: HSpres(HR, ω) +of topological groups, which is then automatically analytic by [Nee06, Thm. IV.1.18]. By Example 8.1.3, +there is an irreducible projective unitary representation ρ of the Abelian Banach-Lie group (HR, +) on the +symmetric Fock space F(H), which is well-known to extend to HR ⋊ Spres(HR, ω) [Nee10b, Rem. 9.12]. We +denote this extension again by ρ. Then ρ ◦ η is a projective unitary G-representation. As in Example 8.1.3, +the T-action α is canonically implemented on F(H) by second quantization. Let �G, �H, � +HSpres(HR, ω) and +� +Spres(HR, ω) be the corresponding central T-extensions of G, H, HSp(HR, ω) and Spres(HR, ω), respectively, +obtained by pulling back the central T-extension U(F(H)) → PU(F(H)). Let ρ : � +HSpres(HR, ω) → U(F(H)) +be the lift of ρ to the central T-extension � +HSpres(HR, ω) to HSpres(HR, ω). Notice further that η lifts to a +homomorphism +η : �G → � +HSpres(HR, ω), +and that ρ ◦ η is the lift of ρ ◦ η. It is proven in [Nee10b, Thm. 9.3, Rem. 9.12] that � +Spres(HR, ω) is again +a Banach-Lie group and that the (cyclic) vacuum vector Ω ∈ F(H) is smooth for ρ, so that ρ is smooth. +By construction, the �G-representation ρ ◦ η extends to G ⋊α T and is of p.e. w.r.t. α. We show that it is in +fact holomorphically induced from the �H-representation on F(H0), for which it remains to show that F(H0) +contains a dense set of �G-analytic vectors, in view of Theorem 7.1.17. To see this, we simply remark that +equation (35) in the proof of [Nee10b, Thm. 9.3] shows not just that Ω is smooth, but even that it is analytic +for � +Spres(HR, ω). We know from Example 8.1.3 that Ω is analytic for Heis(HR, ω) (and even strongly-entire), +so +Heis(HR, ω) × � +Spres(HR, ω) → C, +(v, A) �→ ⟨Ω, ρ(v)ρ(A)Ω⟩ = ⟨ρ(v)−1Ω, ρ(A)Ω⟩ +is real-analytic. This implies using [Nee11, Thm. 5.2] that Ω is an analytic vector for the representation ρ +of � +HSpres(HR, ω). As Ω is cyclic for the action ρ0 of Heis((H0)R, ω) on F(H0), this implies that the set of +vectors in F(H0) ⊆ F(H) that are analytic for � +HSpres(HR, ω) is dense in F(H0). Thus ρ is holomorphically +induced from ρ0. As η is analytic, we obtain that F(H0) also has a dense set of �G-analytic vectors, so it +follows that ρ ◦ η = HolIndG +H(ρ0 ◦ η). +Let us give a concrete example, which is based on [Was98] and [PS86]. Consider V := C∞(S1; Cn), considered +as a real Fr´echet space. For K, we take the loop group LSU(n) := C∞(S1; SU(n)), so that G = V ⋊ LSU(n). +Let α : T → Aut(G) be the usual action of T on G by rotations. So V0 = ker D ∼= Cn is the subspace of +V consisting of constant loops in Cn, whereas Veff is spanned by the non-zero Fourier modes. Consider the +Hardy space H2 ++(S1; Cn) ⊆ L2(S1; Cn), spanned by the non-negative Fourier modes, and let H2 +−(S1; Cn) be +its orthogonal complement in L2(S1; Cn). Consider the Hilbert space H := H2 ++(S1; Cn) ⊕ H2 +−(S1; Cn), where +H2 +−(S1; Cn) denotes the Hilbert space complex conjugate to H2 +−(S1; Cn). Notice that the real vector space +HR underlying H is simply HR = L2(S1; Cn), which contains V = C∞(S1; Cn) as a dense subspace. Let +ω(v, w) := Im⟨v, w⟩H be the corresponding symplectic form on HR. Observe further that the T-action α on +V extends to a unitary p.e. representation of T on H, that α∗ +t ω = ω for all t ∈ T and that ω(v, Dv) > 0 for +all v ∈ V . It is known, and not hard to check, that LSU(n) embeds continuously into Spres(HR, ω) [PS86, +Sec. 6.3], defining an embedding +G = V ⋊ LSU(n) ֒→ HR ⋊ Spres(HR, ω) = HSpres(HR, ω). +So we are precisely in the setting described above. The preceding construction results in a projective unitary +representation of the BCH Fr´echet-Lie group G = V ⋊ LSU(n) on the Fock space +F(H) ∼= F(H2 ++(S1; Cn)) ⊗ F(H2 +−(S1; Cn)). +44 + +We have seen that the corresponding unitary positive energy representation of the central T-extension �G of G +is holomorphically induced from the representation of �H = Heis(Cn, ω0)⋊SU(n) on the energy-zero subspace +F(H)(0) ∼= F(Cn). +As a consequence, we further obtain that the representation ρ| � +K of �K = � +LSU(n) +is holomorphically induced from the SU(n)-representation on F(Cn), which decomposes into irreducible +components as F(Cn) ∼= �∞ +k=0 Sk(Cn). In particular, letting Fk denote the closed �K-invariant subspace of +F(H) generated by Sk(Cn) ⊆ F(H)(0) for k ∈ N≥0, we recover using Corollary 7.1.10 the known fact that +F(H) decomposes into irreducible components of ρ| � +K as F(H) ∼= +�∞ +k=0 Fk. +Example 8.1.5 (Groups of jets). +Let K be a 1-connected compact simple Lie group with Lie algebra k. Let V be a finite-dimensional real vector +space. We consider the Lie group Jn +0 (V ; K) of n-jets at 0 ∈ V of smooth maps V → K. Let γ : R → GL(V ) +be a continuous representation of R on V and let φ ∈ k. Assume that the R-action �αt(f) := etφf(α−t(x))e−tφ +on C∞ +c (V ; K) factors through T := R/Z. As γ fixes the origin, �α descends to a smooth T-action on Jn +0 (V ; K), +denoted α. Let D := +d +dt +�� +t=0 ˙αt be the corresponding derivation on Jn +0 (V ; k). Let G be a central T-extension +of Jn +0 (V ; K) ⋊α T, and let d ∈ g := Lie(G) cover (0, 1) ∈ Jn +0 (V ; k) ⋊D R. As usual, we write H := (Gα)0 ⊆ G +for the connected Lie subgroup of α-fixed points in G, whose Lie algebra is h = ker D. As G ∼= N ⋊ K for +some nilpotent Lie group N, it follows from Proposition 3.1.5 that G is of type R. By Example 8.1.1, we thus +obtain that any continuous unitary G-representation which is of positive energy w.r.t. α is holomorphically +by some unitary H-representation. A classification of �Gpos(α) amounts to determining the holomorphically +inducible elements in �H. Unitary positive energy representations of groups of jets are studied in more detail +in [Nie22]. +To make the preceding concrete, suppose that V = R2, n = 2k for some k ∈ N, that γ is the action of T +on R2 by rotations and that φ = 0. Then h ∼= R ⊕ω (Rk[x2 + y2] ⊗ k), where Rk[c] denotes the polynomial +ring in c truncated at the kth degree and where ω is a 2-cocycle on the Lie algebra Rk[x2 + y2] ⊗ k (which in +this case actually must be a coboundary). Every continuous unitary representation ρ of G that is of positive +energy w.r.t. α is holomorphically induced from the H-representation on V (ρ). +Example 8.1.6 (Gauge groups). +Let M be a compact manifold and let P → M be a principal bundle with structure group K, a simple compact +Lie group with Lie algebra k. Consider the group of gauge transformations Gau(P) = Γ(M; Ad(P)), where +Ad(P) = P ×Ad K is the adjoint bundle. This group is a regular BCH Fr´echet-Lie group with Lie algebra +gau(P) = Γ(M; P ×Ad k) [Nee06, Thm. IV.1.12]. Suppose that γ : T → Aut(P) is a smooth T-action on P by +automorphisms of P. Let η : T → Aut(Ad(P)) and γ : T → Diff(M) denote the induced T-actions on Ad(P) +and M, respectively. Explicitly, η is given by ηt([p, k]) := [γt(p), k] for p ∈ P, k ∈ K and t ∈ T. Then T acts +smoothly on Gau(P) by αt(s) := ηt ◦ s ◦ γ−t for s ∈ Gau(P) and t ∈ T. The paper [JN21] studies projective +unitary representations of Gau(P) which are smooth in the sense of admitting a dense set of smooth rays. +According to [JN19, Cor. 4.5, Thm. 7.3], these correspond to smooth unitary representations of a central +T-extension of Gau(P). +One of the main results of [JN19] is the full classification of smooth projective +unitary of the identity component Gau(P)0 which are of positive energy w.r.t α, provided that M has no +T-fixed points for γ [JN21, Thm. 8.10]. Let us consider a central T-extension G of the connected component +Gau(P)0 of the identity. Suppose that α lifts to a smooth T-action �α on G. In view of [JN21, Prop. 8.6], a +consequence of the classification [JN21, Thm. 8.10] is that every smooth unitary representation ρ of G which +is of positive energy w.r.t. �α is holomorphically induced from the corresponding representation of H := (G�α)0 +on V (ρ). The more general case where M is allowed to have T-fixed points is not yet fully understood. One +approach would be to determine, in specific cases, the irreducible unitary actor representations of H that are +holomorphically inducible to G, as an intermediate step towards the classification of the possibly larger class +of all p.e. factor representations of G. +Example 8.1.7 (Unitary groups of CIA’s). +An interesting class of examples to which the theory of Section 7 applies can be obtained using so-called +continuous inverse algebras (CIAs). +Suppose that A is a unital complex Fr´echet algebra that is a CIA, +meaning that its group of units A× is open in A and that the inversion a �→ a−1 is continuous A → A. Let +us suppose further that A carries a continuous conjugate-linear algebra involution A → A, a �→ a∗. In this +setting, A× is a complex regular BCH Fr´echet-Lie group modeled on A [Nee06, Thm. IV.1.11]. Moreover, +the unitary subgroup +U(A) := +� +a ∈ A× : a∗ = a−1 � +is a real Lie subgroup of A×, so that it is an embedded submanifold. It is modeled on the Lie algebra +u(A) := { a ∈ A : a∗ = −a } , +45 + +equipped with the commutator bracket. To see this, let U ⊆ A be a 0-neighborhood s.t. expA maps U +diffeomorphically onto its image in A×. We may assume that U = −U and that U ∗ = U, by shrinking U +if necessary. By [Gl¨o02a, Cor. 4.11] we know that expA(a) = �∞ +n=0 +1 +n!an for all a ∈ A. Using that both +a �→ a−1 and a �→ a∗ are continuous, it follows that expA(a)∗ = exp(a∗) and expA(a)−1 = expA(−a) for +all a ∈ U. This implies that expA(U ∩ u(A)) = expA(U) ∩ U(A). As U(A) is a closed subgroup of the +locally exponential Lie group A×, it follows from [Nee06, Thm. IV.3.3] that U(A) ⊆ A× is a Lie subgroup. It +is therefore a regular BCH Fr´echet-Lie group. Notice further that u(A)C = (A, [−, −]) as complex Lie algebras. +Suppose that α : R → Aut(A) is a homomorphism that has a smooth action R × A → A and that has +polynomial growth. Assume further that the splitting condition +A = A− ⊕ A0 ⊕ A+ +is satisfied. Setting G := U(A)0 and H := U(A0)0 = (Gα)0, all assumptions of both Section 4.2 and Section 7 +are satisfied. +Typically, such triples (A, R, α) can be obtained as the set of smooth points of a C∗-dynamical system +(B, G, γ), where B is a unital C∗-algebra, G is a Banach-Lie group and γ : G → Aut(B) is a strongly con- +tinuous G-action on B by automorphisms. By [Nee10a, Def. 4.1, Thm. 6.2], we know in this setting that the +set of smooth points A := B∞ is a G-invariant and ∗-closed subalgebra which naturally carries a Fr´echet +topology. Moreover, A is a CIA and the G-action γ : G × A → A is smooth w.r.t. this topology. If G is +finite-dimensional, then this topology coincides with the one obtained from the embedding A ֒→ C∞(G; B), +where C∞(G; B) carries the smooth compact-open topology [Nee10a, Prop. 4.6]. If ι : R ֒→ G is a one- +parameter subgroup of G for which the corresponding R-action α := γ ◦ ι on A has polynomial growth and +satisfies the splitting condition A = A−⊕A0⊕A+, then the triple (A, R, α) satisfies all the above assumptions. +As a concrete example, let Aθ := C∞ +θ (T2) be the smooth non-commutative 2-torus with parameter θ ∈ [0, 1 +2]: +Aθ := + + + +� +n,m∈Z +an,munvm : +� +n,m∈Z +(1 + |n| + |m|)k|an,m| < ∞ for all k ∈ N + + + , +where u and v are unitary operators satisfying uv = ei2πθvu, and where Aθ is equipped with the seminorms +pk(a) := � +n,m∈Z(1 + |n| + |m|)k|an,m| for k ∈ N≥0. This is a unital Fr´echet CIA carrying a continuous +involution, obtained as the smooth points of the natural T2-action on the ‘continuous’ non-commutative +2-torus Cθ(T2) with parameter θ. +Consider the smooth and equicontinuous T-action α on C∞ +θ (T2) that +satisfies αz(unvm) := zmunvm for all n, m ∈ Z and z ∈ T. Define G := U(Aθ)0. Then for any unitary +representation ρ of G⋊α T that is analytically ground-state w.r.t. α, we obtain from Theorem 7.1.17 that ρ|G +is holomorphically induced from the corresponding unitary representation of the connected Abelian group +H := (U(Aθ)α)0 ∼= C∞(T; T)0 on Hρ(0). In particular, if ρ(G)′′ is a factor, then as H is Abelian, we obtain +with Corollary 7.1.10 that ρ|G is holomorphically induced from a character of H. By Corollary 7.1.10 this +implies that ρ|G is irreducible. +A +Representations on reproducing kernel Hilbert spaces +In the following we summarize relevant properties concerning reproducing kernel Hilbert spaces in the context +of unitary group representations. Let H and V be Hilbert spaces and let G be a group. W write V G or +Map(G; V ) for the space of functions G → V and V (G) for the space of finitely-supported functions G → V . +Definition A.1.1. Suppose that H ⊆ V G. Then H is said to have continuous evaluation maps if for every +x ∈ G the linear map Ex : H → V, ψ �→ ψ(x) is bounded. +Definition A.1.2. A function Q : G × G → B(V ) is said to be positive definite if +∥v∥Q := +� +x,y∈supp(v) +⟨vx, Q(x, y)vy⟩V ≥ 0, +∀v ∈ V (G). +46 + +Theorem A.1.3 ([Nee00, Thm. I.1.4]). +Let Q : G × G → B(V ) be a function. The following assertions are equivalent: +1. Q is positive definite +2. There is a Hilbert space HQ ⊆ V G with continuous point-evaluations Ex : HQ → V s.t. Q(x, y) = ExE∗ +y +for all (x, y) ∈ G × G. +In this case HQ is unique up to unitary equivalence and { E∗ +xv : x ∈ G, v ∈ V } is total in HQ. +Definition A.1.4. A function Q : G × G → B(V ) is said to be a reproducing kernel for the Hilbert space H +if Q is positive definite and H ∼= HQ. +Proposition A.1.5. Let G be a topological group and let H ⊆ G be a closed subgroup. Let (σ, Vσ) be a strongly +continuous unitarily H-representation. Let Q ∈ C(G×G, B(Vσ))H×H, so Q(xh1, yh2) = σ(h1)−1Q(x, y)σ(h2) +for all x1, x2 ∈ G and h1, h2 ∈ H. Assume that Q is positive definite. +1. The left-regular action of G on V (G) +σ +induces a unitary G-action π on HQ if and only if Q is G-invariant. +In this case there is a function F : G → B(Vσ) such that Q(x, y) = F(x−1y). +2. Assume that Q is G-invariant. There is a G-equivariant linear map HQ ֒→ Map(G; Vσ)H with contin- +uous point-evaluations Ex for x ∈ G. These satisfy the equivariance condition Exπ(g) = Eg−1x for all +x, y ∈ G. +3. Assume that Q is G-invariant and strongly continuous as a map G × G → B(Vσ). Then the unitary +G-representation HQ is strongly continuous. +4. Suppose that (ρ, Hρ) is a unitary G-representation and that there is a G-equivariant injective linear +map Φ : Hρ ֒→ Map(G; Vσ)H having continuous point evaluations Ex := evx ◦Φ for x ∈ G. Then the +corresponding kernel Q is G-invariant, and Hρ ∼= HQ as unitary G-representations. +Proof. Let lg denote the left G-action on itself by left-multiplication. Recall that HQ = V (G) +σ +/NQ +⟨−,−⟩Q +, +where NQ := +� +f ∈ V (G) +σ +: ∥f∥Q = 0 +� +. For any x ∈ G we have a map δx : Vσ ֒→ V (G) +σ +defined by considering +elements of Vσ as functions on G with support {x}. Let qx : Vσ → HQ, v �→ [δx(v)] be its composition +with the quotient map V (G) +σ +→ HQ. We then have Ex = q∗ +x (cf. [Nee00, Thm. I.1.4] for more details). The +embedding HQ ֒→ V G +σ is defined by f �→ fψ, where fψ(x) = Ex(ψ). +1. For g ∈ G and f ∈ V (G) +σ +, we write g.f := f ◦ l−1 +g +for the left-regular action of G on V (G) +σ +. +Let +x, y ∈ G. Take v, w ∈ Vσ. Then g.δx(v) = δgx(v) and g.δy(w) = δgy(w) have support on {gx} and {gy}, +respectively. Thus ⟨g.δx(v), g.δy(w)⟩Q = ⟨v, Q(gx, gy)w⟩ whereas ⟨qx(v), qy(w)⟩Q = ⟨v, Q(x, y)w⟩. The +first assertion follows. If Q is G-invariant, then F(x) := Q(e, x) satisfies F(x−1y) = Q(x, y). +2. Let x ∈ G and h ∈ H. From Q(xh, y) = σ(h)−1Q(x, y) it follows that ExhE∗ +yv = σ(h)−1ExE∗ +yv for +any y ∈ G and v ∈ Vσ. As {E∗ +yv : y ∈ G, v ∈ Vσ} is total in HQ by Theorem A.1.3, it follows that +Exh = σ(h)−1Ex. Thus fψ ∈ Map(G; Vσ)H for any ψ ∈ HQ. We show that ψ �→ fψ is G-equivariant. +We have π(g)E∗ +xv = π(g)qx(v) = qgx(v) = E∗ +gx(v) for every x, g ∈ G and v ∈ Vσ. Hence π(g)E∗ +x = E∗ +gx +and Exπ(g) = Eg−1x for every x, g ∈ G. Thus for ψ ∈ HQ we obtain fψ(g−1x) = Eg−1xψ = Exπ(g)ψ = +fπ(g)ψ(x), so ψ �→ fψ is G-equivariant. +3. As G acts unitarily on HQ, it suffices to show that G → C g �→ ⟨ψ, π(g)ψ⟩Q is continuous for any ψ in +some total subspace. Consider ψ = E∗ +xv for arbitrary x ∈ G and v ∈ Vσ. Such vectors form a total set +in HQ by Theorem A.1.3. For g ∈ G, we have +⟨ψ, π(g)ψ⟩Q = ⟨v, Exπ(g)E∗ +xv⟩V = ⟨v, ExE∗ +gxv⟩V = ⟨v, Q(x, gx)v⟩V . +(A.1) +As Q : G×G → B(Vσ) is continuous w.r.t. the strong topology, the map g �→ ⟨ψ, π(g)ψ⟩Q is continuous. +4. As Φ is G-equivariant, we have Exρ(g) = Eg−1x for every x, g ∈ G. +As ρ is unitary this implies +that the corresponding kernel Q(x, y) := ExE∗ +y is G-invariant. +This kernel is also positive definite +by Theorem A.1.3, so HQ is a unitary G-representation by the first item. +We already know from +Theorem A.1.3 that HQ ∼= Hρ as Hilbert spaces. The unitary isomorphism U : HQ → Hρ is on the +dense subspace V (G) +σ +/NQ given by Uq(f) := � +x∈supp(f) E∗ +xf(x), where q : V (G) +σ +→ HQ denotes the +quotient map. Write π for the unitary G-action on HQ. Using qx = E∗ +x, qgx = π(g)qx and ρ(g)E∗ +x = E∗ +gx, +we obtain that +Uπ(g)qx(v) = Uqgx(v) = E∗ +gxv = ρ(g)E∗ +xv = ρ(g)Uqx(v). +47 + +References +[AM66] L. Auslander and C.C Moore. Unitary Representations of Solvable Lie Groups. Mem. Amer. Math. +Soc., 1966. +[Arv74] W. Arveson. On groups of automorphisms of operator algebras. J. Func. Anal., 15:217–243, 1974. +[Bel05] D. Beltit¸˘a. Integrability of analytic almost complex structures on Banach manifolds. Ann. Global +Anal. Geom., 28(1):59–73, 2005. +[BGN20] D. Beltita, H. Grundling, and K.-H. Neeb. Covariant representations for possibly singular actions +on C∗-algebras. Diss. Math., 549:1–94, 2020. +[Bor66] H.-J. Borchers. Energy and momentum as observables in quantum field theory. Comm. Math. +Phys., 2:49–54, 1966. +[Bor87] H.-J. Borchers. On the interplay between spectrum condition and locality in quantum field theory. +In Operator Algebras and Mathematical Physics, volume 62, pages 143–152. Amer. Math. Soc., +1987. +[BR87] O. Bratteli and D.W. Robinson. Operator Algebras and Quantum Statistical Mechanics 1. Springer- +Verlag, New York, second edition, 1987. C∗- and W ∗-algebras, symmetry groups, decomposition +of states. +[BR97] O. Bratteli and D.W. Robinson. Operator Algebras and Quantum Statistical Mechanics 2. Springer- +Verlag, Berlin, second edition, 1997. Equilibrium states. Models in quantum statistical mechanics. +[BS71a] J. Bochnak and J. Siciak. Analytic functions in topological vector spaces. Studia Math., 39:77–112, +1971. +[BS71b] J. Bochnak and J. Siciak. +Polynomials and multilinear mappings in topological vector spaces. +Studia Math., 39:59–76, 1971. +[DK00] J.J. Duistermaat and J.A.C. Kolk. Lie Groups. Springer-Verlag, Berlin, 2000. +[Gl¨o02a] H. Gl¨ockner. Algebras whose groups of units are Lie groups. Studia Math., 153(2):147–177, 2002. +[Gl¨o02b] H. Gl¨ockner. Infinite-dimensional Lie groups without completeness restrictions. In Geometry and +Analysis on Finite- and Infinite-dimensional Lie Groups, volume 55, pages 43–59. Polish Acad. Sci. +Inst. Math., 2002. +[GN] H. Gl¨ockner and K.-H. Neeb. Infinite Dimensional Lie Groups. Book in preparation. +[Goo69] R.W. Goodman. Analytic and entire vectors for representations of Lie groups. Trans. Amer. Math. +Soc., 143:55–76, 1969. +[Haa92] R. Haag. Local Quantum Physics. Springer-Verlag, Berlin, 1992. +[Jen73] J.W. Jenkins. Growth of connected locally compact groups. J. Func. Anal., 12:113–127, 1973. +[JN19] B. Janssens and K.-H. Neeb. Projective unitary representations of infinite-dimensional Lie groups. +Kyoto J. Math., 59(2):293–341, 2019. +[JN21] B. Janssens and K.-H. Neeb. Positive energy representations of gauge groups I: Localization, 2021. +In preparation. Available at arXiv:2108.03501. +[Kir76] A.A. Kirillov. Elements of the Theory of Representations. Springer-Verlag, Berlin-New York, 1976. +[KM97] A. Kriegl and P.W. Michor. The Convenient Setting of Global Analysis, volume 53. AMS, Provi- +dence, 1997. +[Lis95] W. Lisiecki. Coherent state representations. A survey. Rep. Math. Phys., 35(2-3):327–358, 1995. +[LM75] M. L¨uscher and G. Mack. Global conformal invariance in quantum field theory. Comm. Math. +Phys., 41:203–234, 1975. +[Mil84] J. Milnor. Remarks on infinite-dimensional Lie groups. In Relativity, Groups and Topology, pages +1007–1057. North-Holland, Amsterdam, 1984. +48 + +[Mun00] J.R. Munkres. Topology. Prentice Hall, Inc., 2000. +[Nee98] K.-H. Neeb. Some open problems in representation theory related to complex geometry. In Positivity +in Lie Theory: Open Problems, volume 26, pages 195–220. de Gruyter, Berlin, 1998. +[Nee00] K.-H. Neeb. Holomorphy and Convexity in Lie Theory, volume 28. Walter de Gruyter & Co., +Berlin, 2000. +[Nee01] K.-H. Neeb. Borel-Weil theory for loop groups. In Infinite Dimensional K¨ahler Manifolds, vol- +ume 31, pages 179–229. Birkh¨auser Verlag, 2001. +[Nee04] K.-H. Neeb. Infinite-dimensional groups and their representations. In Lie Theory, volume 228, +pages 213–328. Birkh¨auser Boston, 2004. +[Nee06] K.-H. Neeb. Towards a Lie theory of locally convex groups. Jpn. J. Math., 1(2):291–468, 2006. +[Nee10a] K.-H. Neeb. On differentiable vectors for representations of infinite dimensional Lie groups. J. +Funct. Anal., 259(11):2814–2855, 2010. +[Nee10b] K.-H. Neeb. Semibounded representations and invariant cones in infinite dimensional Lie algebras. +Confl. Math., 2(1):37–134, 2010. +[Nee11] K.-H. Neeb. On analytic vectors for unitary representations of infinite dimensional Lie groups. +Ann. Inst. Fourier, 61(5):1839–1874, 2011. +[Nee12] K.-H. Neeb. +Semibounded representations of Hermitian Lie groups. +Travaux math´ematiques., +21(5):29–109, 2012. +[Nee13] K.-H. Neeb. +Holomorphic realization of unitary representations of Banach-Lie groups. +In Lie +Groups: Structure, Actions, and Representations, volume 306, pages 185–223. Birkh¨auser/Springer, +New York, 2013. +[Nee14] K.-H. Neeb. +Semibounded unitary representations of double extensions of Hilbert-loop groups. +Ann. Inst. Fourier, 64(5):1823–1892, 2014. +[Nie22] M. Niestijl. Generalized positive energy representations of groups of jets, 2022. In preparation. +Available at arxiv:2211.10390. +[NR22] K.-H. Neeb and F.G. Russo. Ground state representations of topological groups. Math. Ann., pages +1–60, 2022. +[NS15] K.-H. Neeb and H. Salmasian. Classification of positive energy representations of the Virasoro +group. Int. Math. Res. Not., Volume 2015(18):8620–8656, 2015. +[NSZ15] K.-H. Neeb, H. Salmasian, and C. Zellner. On an invariance property of the space of smooth vectors. +Kyoto J. Math., 55(3):501–515, 2015. +[Ol’81] G.I. Ol’shanski˘ı. Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete +series. Funktsional. Anal. i Prilozhen., 15(4):53–66, 1981. +[Pen74] R. Penney. Entire vectors and holomorphic extension of representations. Trans. Amer. Math. Soc., +198:107–121, 1974. +[Per86] A. Perelomov. Generalized Coherent States and Their Applications. Springer-Verlag, Berlin, 1986. +[PS86] A. Pressley and G. Segal. Loop Groups. Oxford University Press, 1986. +[Rud91] W. Rudin. Functional Analysis. McGraw-Hill, Inc., New York, second edition, 1991. +[Sch90] K. Schm¨udgen. Unbounded Operator Algebras and Representation Theory, volume 37. Birkh¨auser +Verlag, Basel, 1990. +[Seg81] G. Segal. +Unitary representations of some infinite-dimensional groups. +Comm. Math. Phys., +80(3):301–342, 1981. +[SW64] R.F. Streater and A.S. Wightman. PCT, Spin and Statistics, and all That. Benjamin, New York, +1964. +49 + +[Tre67] F. Treves. +Topological Vector Spaces, Distributions and Kernels. +Academic Press, New York- +London, 1967. +[TW71] J.A. Tirao and J.A. Wolf. Homogeneous holomorphic vector bundles. Indiana Univ. Math. J., +20:15–31, 1970/71. +[Was98] A. Wassermann. +Operator algebras and conformal field theory. III. Fusion of positive energy +representations of LSU(N) using bounded operators. Invent. Math., 133(3):467–538, 1998. +50 + diff --git a/KNE4T4oBgHgl3EQfiA0C/content/tmp_files/load_file.txt b/KNE4T4oBgHgl3EQfiA0C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37d2c31a9ea5817122f5a8482261baedc7c2f866 --- /dev/null +++ b/KNE4T4oBgHgl3EQfiA0C/content/tmp_files/load_file.txt @@ -0,0 +1,5077 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf,len=5076 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='05129v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='RT] 12 Jan 2023 Holomorphic Induction Beyond the Norm-Continuous Setting, With Applications to Positive Energy Representations Milan Niestijl January 13, 2023 Abstract We extend the theory of holomorphic induction of unitary representations of a possibly infinite-dimensional Lie group G beyond the setting where the to-be-induced representation is required to be norm-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We allow the group G to be a connected regular BCH(Baker-Campbell-Hausdorff) Fr´echet-Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Given a smooth R-action α on G, we proceed to show that the corresponding class of so-called positive energy representations is intimately related with holomorphic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In particular, we show that if ρ is a unitary ground-state representation of G ⋊α R for which the energy-zero subspace Hρ(0) admits a dense set of G-analytic vectors, then ρ|G is holomorphically induced from the representation of the connected subgroup H := (Gα)0 of α-fixed points on Hρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As a consequence, we obtain an isomor- phism B(Hρ)G ∼= B(Hρ(0))H between the corresponding commutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We also find that any two such ground-state representations are necessarily unitary equivalent if their energy-zero subspaces are unitarily equivalent as H-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' These results were previously only available under the assumption of norm-continuity of the H-representation on Hρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Preliminaries 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1 Analytic functions on locally convex vector spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1 Homogeneous polynomials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2 Analytic functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 40 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3 Ground-state representations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 41 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4 Strongly-entire ground-state representations for T-actions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 42 8 Examples 43 A Representations on reproducing kernel Hilbert spaces 46 1 Introduction This paper is concerned with unitary representations a possibly infinite-dimensional connected Lie group G that is modeled on a locally convex vector space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' [Mil84, Nee06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let α : R → Aut(G) be a smooth action of R on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We consider those G-representations that extend to a unitary representation ρ of G ⋊α R which is smooth, in the sense that it admits a dense set of smooth vectors, and which is of positive energy, meaning that the self-adjoint generator −i d dt �� t=0 ρ(1G, t) of the unitary 1-parameter group t �→ ρ(1G, t) has non-negative spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For infinite-dimensional Lie groups, a full classification of all irreducible representations is typically not tractable, and even less so for factor representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The positive energy condition serves to isolate a class of representations that are more susceptible to systematic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It is also quite natural from a physical per- spective, because the Hamiltonian in quantum physics is nearly always required to be a positive self-adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It is then no surprise that positive energy representations of Lie groups are abundant in physics literature [SW64, Bor87, Bor66, Haa92, LM75, Ol’81, PS86, Seg81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Holomorphic induction has proven to be a particularly effective tool in the study of positive energy represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let us first describe the main idea of holomorphic induction in the case where G is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let H := (Gα)0 be the connected subgroup of α-fixed points in G, with Lie algebra h = Lie(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A unitary G-representation ρ is typically called holomorphically induced from the unitary H-representation σ on Vσ if the homogeneous Hermitian vector bundle V := G ×H Vσ over G/H can be equipped with a G-invariant complex-analytic bundle structure, with respect to which the Hilbert space Hρ can be G-equivariantly em- bedded into the space of holomorphic sections O(G/H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' V) of V, in such a way that the corresponding point evaluations Ex : Hρ → Vx are continuous and satisfy ExE∗ x = idVx for every x ∈ G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In particular, these conditions imply that Hρ is unitarily equivalent to the G-representation on a reproducing kernel Hilbert space, and that Hρ contains Vσ as H-subrepresentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' An important special case is obtained when Vσ is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If ρ is holomorphically induced from σ, we may identify Vσ with a cyclic ray [v0] in Hρ, whose G-orbit in the projective space P(Hρ) is a complex submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' This means that ρ is a so-called coherent state representation[Nee00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' XV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In this case, the G-homogeneous line bundle V is the pull-back of the tautological line bundle over P(Hρ) along the map G/H → P(Hρ), gH �→ [ρ(g)v0], and elements in the image of the corresponding map V → Hρ are usually called coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' This is also the setting of the well-known Borel-Weil Theorem [DK00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Such representations have been studied extensively [Per86, Nee00, Lis95], and are known to be tightly related to highest-weight representations [Nee00, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' XV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In particular, every unitary highest weight representation of G is a coherent state representation[Nee00, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' XV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The Schr¨odinger representation of the Heisenberg group Heis(R2, ω) provides a counterexample [Nee00, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' XV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Holomorphic induction, defined as above, was studied in [Nee13] in the context where G is a Banach-Lie group and where σ is bounded, meaning that it is continuous with respect to the norm-topology on B(Vσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Writing g for the Lie algebra of G and gC for its complexification, invariant complex structures on G/H correspond to closed Lie subalgebras b ⊆ gC satisfying b+b = gC, b∩b = hC and Adh(b) ⊆ b for all h ∈ H [Bel05, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 15] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' [Kir76, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 203] for the case where G is finite dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The corresponding G-invariant holomorphic bundle structures on V then turn out to be parametrized by extensions of dσ : h → B(Vσ) to a Lie algebra homomorphism χ : b → B(Vσ) satisfying χ(Adh(ξ)) = σ(h)χ(ξ)σ(h)−1 for all ξ ∈ b and h ∈ H [Nee13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6], as is to be expected from the finite-dimensional setting [TW71, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The holomorphic structure is used to relate various important properties of the G-representation ρ with those of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For example, [Nee13, 1 Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='12] entails that the commutants B(Hρ)G ∼= B(Vσ)H,χ are isomorphic as von Neumann algebras, which implies in particular that ρ is irreducible, multiplicity-free or of type I, II or III if and only if this is true for σ [Nee13, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Moreover, [Nee13, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='16] states that there is up to unitary equivalence at most one unitary G-representation ρ that is holomorphically induced from a given pair (σ, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The relation between holomorphically induced representations and the positive energy condition is then explained by [Nee13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='12, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='14], which essentially state that in the above context, and under suitable assumptions, holomorphi- cally induced representations correspond to so-called semibounded ones, the semiboundedness condition being a ‘stable’ and stronger version of the positive energy condition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' [Nee10b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' These observations suggest that the class of holomorphically induced representations may well admit a fruitful classification theory of its factor representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' This line of reasoning was pursued in [Nee14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='10] and [Nee12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1], resulting in a classification of the irreducible semibounded unitary representations of certain double extensions of Hilbert Loop groups and of hermitian Lie groups corresponding to infinite-dimensional irreducible symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In [Nee14, Appendix C], the theory of holomorphic induction was further developed, allowing G to be a connected regular BCH Fr´echet-Lie group, under certain additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Still, σ was required to be norm-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let us mention that a particular and well-known special case of such a situation had already appeared in the study of smooth positive energy representations of loop groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In fact, these had been completely classified using holomorphic induction [PS86] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' [Nee01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Still, the assumption of norm-continuity of σ is too restrictive in numerous examples, some of which we encounter in Section 8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It is typically only suitable for describing the class of semibounded unitary representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In order to obtain a theory that can be used to describe the possibly larger class of all positive energy representations, one necessarily needs to go beyond the norm-continuity of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The purpose of the present paper is to remove this assumption of norm-continuity of the representation σ that is induced from, whilst still allowing G to be a connected regular BCH Fr´echet-Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A main difficulty in this direction is that of equipping the homogeneous vector bundle G ×H Vσ with a G-invariant complex-analytic bundle structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The proof of [Nee13, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6] breaks down beyond the norm-continuous setting, so a new approach is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We provide two possible solutions to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As in [Nee14, Appendix C], we assume that gC admits a triangular decomposition of the form gC = n− ⊕ hC ⊕ n+, where n± and hC are closed Lie subalgebras of gC satisfying n± ⊆ n∓, and where b = hC ⊕ n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In the first, which we call the general approach, we avoid speci- fying a complex-analytic vector bundle altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Instead we replace the space of holomorphic sections by a suitable subspace Cω(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Vσ)H,χ of the space of real-analytic H-equivariant maps Cω(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Vσ)H, defined directly in terms of an extension χ : b → L(D) of dσ to b with some domain D ⊆ V ω σ consisting of analytic vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' This also avoids the need for a G-invariant complex structure on the homogeneous space G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In the second, which we call the geometric approach, we define a stronger notion of holomorphic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In this case, H∞ ρ actually embeds into a space of holomorphic mappings on a homogeneous vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It therefore requires complex geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A significant drawback of this approach is that it requires a dense set of so-called strongly-entire vectors, whose availability is usually not known, unless G happens to be finite-dimensional, in which case it is completely understood by the results of [Goo69] and [Pen74], see also Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let us also mention that this paper does not complete the story of holomorphic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The developed theory still excludes regular Fr´echet-Lie groups that are not BCH, such as the Virasoro group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Yet, it is known that holomorphic induction can be used to obtain a complete classification of the positive energy representations of the Virasoro group [NS15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Nevertheless, the present paper makes substantial progress towards a more complete understanding of holomorphic induction in the infinite-dimensional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In relation to positive energy representations, progress was made in a different direction in [NR22], where the class of ground-state representations is studied in the setting of topological groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Structure of the paper — In Section 2, we first recall some preliminaries regarding analytic functions on locally convex spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We proceed to define smooth, analytic and strongly-entire representations, which are increasingly regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We also recall some important results related to positive energy and ground-state representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — We proceed in Section 3 to define and study the space HO ρ of so-called strongly-entire vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We equip this space with a locally convex topology, and extend the results of [Goo69] from the setting 2 of finite-dimensional Lie groups to the present one, where G is allowed to be infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In particular, if GC is a complex 1-connected regular BCH Fr´echet-Lie group with Lie algebra gC, we obtain that HO ρ carries a representation of GC that has a holomorphic action GC × HO ρ → HO ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The space HO ρ plays an important role in the geometric approach to holomorphic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2 we present the general approach towards holomorphic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' After determining a useful equivalent formulation, we characterize the inducibility of pairs (σ, χ) in terms of positive definite functions on G, which leads to the uniqueness of the holomorphically induced representation up to unitary equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We then proceed to show that there is an isomorphism of von Neumann algebras B(hρ)G ∼= B(Vσ)H,χ between the commutants, provided that Vσ ⊆ Hρ is invariant under B(Hρ)G, in complete analogy with the previously described norm-continuous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We also briefly discuss holomorphic induction in stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — After equipping the G-homogeneous vector bundle Vσ := G ×H V O σ with a complex-analytic bundle structure, using an suitable extension χ of dσ with domain V O σ , we define in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3 the geometric notion of holomorphically induced representations, and compare it to the one presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — In relating holomorphic induction with the positive energy condition, we shall have need for a suitably general notion of Arveson spectral subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We therefore generalize in Section 6 the results of [NSZ15, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3] and [Nee13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2] to the level of generality needed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — In Section 7, we study the relation between holomorphic induction and the positive energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In particular, we show that if ρ is a unitary ground-state representation of G⋊αR for which the energy-zero subspace Hρ(0) admits a dense set of G-analytic vectors, then ρ|G is holomorphically induced from the H-representation on Hρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As a consequence, we obtain an isomorphism B(Hρ)G ∼= B(Hρ(0))H of von Neumann algebras between the corresponding commutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We also find that any two such ground-state representations are necessarily unitary equivalent if their energy-zero subspaces are unitarily equivalent as H-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — In Section 8, we consider numerous interesting examples of unitary representations that are holomor- phically induced from representations that are not norm-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Acknowledgments: This research is supported by the NWO grant 639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='734 “Cohomology and representation theory of infinite- dimensional Lie groups”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' I would like to thank my PhD supervisor Bas Janssens for his guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' I am more- over grateful to Karl-Hermann Neeb, who has carefully read an earlier version of this manuscript and given various suggestions for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The conversations with Karl-Hermann Neeb were also enlightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1 Analytic functions on locally convex vector spaces Let us recall some definitions and properties of analytic functions between locally convex vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The main references are [BS71b], [BS71a] and [Gl¨o02b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Throughout the following, fix locally convex vector spaces E and F over the field K that both are complete and Hausdorff, where K is either R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ∆k : E → Ek, ∆k(h) = (h, · · · , h) be the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1 Homogeneous polynomials Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose U ⊆ E is open and f ∈ C∞(U, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For any x ∈ U, define δ0 x(f) : E → F and δk x(f) : E → F by δ0 x(f)(v) := f(x) and δk x(f)(v) := dkf(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' v, · · · , v), where k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A map f : E → F is called a homogeneous polynomial of degree k if there exists a k-linear symmetric map �f : Ek → F such that f = �f ◦ ∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) denote the space of continuous homogeneous polynomials E → F of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For k = 0, we set P 0(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) := F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Set E0 := K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For k ∈ N≥0, we write Mult(Ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) for the space of continuous k-linear maps Ek → F, equipped with the topology of uniform convergence on products of compact sets in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For the case k = 1, we also write B(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) := Mult(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let Symk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) ⊆ Mult(Ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) denote the closed subspace of continuous symmetric k-linear maps Ek → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let E �⊗F denote the completed projective tensor product of E and F [Tre67, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Define E �⊗k := E �⊗ · · · �⊗E (k times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The topology on E �⊗k is defined by the 3 seminorms q1 ⊗ · · · ⊗ qk, where each qi is a continuous seminorms on E, see also [Tre67, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' On algebraic tensors t ∈ E⊗k, this seminorm is given by (q1 ⊗ · · · ⊗ qk)(t) := inf \uf8f1 \uf8f2 \uf8f3 � j k � i=1 qi(ξ(j) i ) : t = � j ξ(j) 1 ⊗ · · · ⊗ ξ(j) k , with ξ(j) i ∈ E \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1) On simple tensors we have (q1 ⊗ · · · ⊗ qk)(ξ1 ⊗ · · · ⊗ ξk) = �k i=1 qi(ξi), where ξi ∈ E [Tre67, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3 ([Tre67, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 465]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' There is a canonical linear isomorphism Mult(Ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) ∼= B(E �⊗k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It is a homeomorphism if E is Fr´echet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Equip P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) with the topology of uniform convergence on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If p is a continuous seminorm on F, B ⊆ E is a subset and f : E → F is a function, we write pB(f) := supx∈B p(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let k ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) ∼= Symk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) as locally convex vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If �f : Ek → F is a symmetric k-linear map and f = �f ◦ ∆k is the corresponding homogeneous polynomial, then �f can be recovered from f using the formula [BS71b, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A]: �f(x1, · · · xk) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 1 � ǫ1,··· ,ǫk=0 (−1)k−(ǫ1+···+ǫk)f(ǫ1x1 + · · · + ǫkxk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2) This formula moreover shows that �f is continuous if and only if f is so, and there is a linear isomorphism Symk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) → P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) given by �f �→ �f ◦ ∆k =: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It remains to show that this map is also a homeo- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose that f = �f ◦ ∆k for some �f ∈ Symk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If B ⊆ E is a compact subset and p is a continuous seminorm on F, then pB(f) ≤ pBk( �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hence Symk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) → P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F), �f �→ f is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For the continuity of the inverse, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2), from which it follows that if Bi ⊆ E are compact subsets for i ∈ N and p is a continuous seminorm on F, then sup xi∈Bi p( �f(x1, · · · , xk)) ≤ 2k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' pB(f), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3) where B = { ǫ1x1 + · · · + ǫkxk : ǫi ∈ {0, 1}, xi ∈ Bi for i ∈ {1, · · · , k} } , which is a compact subset of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Consequently the map f �→ �f is continuous P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) → Symk(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Define the locally convex space P(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) := �∞ k=0 P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F), equipped with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If F = K, we simply write P n(E) := P n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2 Analytic functions Let U ⊆ E be open and let f : U → F be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The function f : U → F is called complex-analytic or holomorphic if it is continuous, and for every x ∈ U there exists a 0 neighborhood V in E with x+V ⊆ U and functions fk ∈ P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) for k ∈ N≥0 such that: f(x + h) = ∞ � k=0 fk(h), ∀h ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — Suppose K = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The function f : U → F is called real-analytic if it extends to some complex-analytic map fC : UC → FC for some open neighborhood UC of U in EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The function f : U → F is called entire if it is continuous and there exist functions fk ∈ P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) for k ∈ N≥0 such that f(x) = �∞ k=0 fk(x) for all x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The above definition of a real-analytic map differs from the one used in [BS71a], where a function f : U → F is called real-analytic if it is continuous and for every x ∈ U there exists a 0-neighborhood V in U with x + V ⊆ U and homogeneous polynomials fk : E → F such that f(x + h) = �∞ k=0 fk(h) holds for all h ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The two notions are equivalent if E and F are Fr´echet spaces [Gl¨o02b, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9], [BS71a, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 4 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='7 ([BS71a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let fk ∈ P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) for every k ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let U ⊆ E be a 0-neighborhood s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' f(h) := � k fk(h) is convergent for every h ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that f : U → F is continuous at 0 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then, for every continuous seminorm p on F, there exists a 0-neighborhood V ⊆ U such that �∞ k=0 pV (fk) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let fn ∈ P n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) for every n ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Consider the following assertions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' f := �∞ n=0 fn defines an entire function E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' �∞ n=0 pB(fn) < ∞ for any compact subset B ⊆ E and continuous seminorm p on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We have that (1) =⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If E is a Fr´echet space, then also (2) =⇒ (1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that f = �∞ n=0 fn defines an entire function E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let B ⊆ E be a compact subset and let p be a continuous seminorm on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We may assume that B is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As f is continuous, f(2B) ⊆ F is compact and hence bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' So Mp := p2B(f) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As f is entire, we have f(zx) = �∞ n=0 fn(x)zn for any x ∈ E and z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let x ∈ 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then also zx ∈ 2B for any z ∈ C with |z| ≤ 1, as B is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Applying [BS71a, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2] to the holomorphic map g : C → F, g(z) := f(zx), we find that fn(x) = 1 2πi � |z|=1 g(z) zn+1 dz and moreover that p(fn(x)) ≤ sup |z|=1 p(g(z)) ≤ p2B(f) = Mp, ∀n ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hence p2B(fn) ≤ Mp, so that pB(fn) ≤ Mp2−n for all n ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus �∞ n=0 pB(fn) ≤ Mp �∞ n=0 2−n < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose that E is a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that (2) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then in particular the series �∞ n=0 fn(x) is convergent for any x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' So f := �∞ n=0 fn defines a function E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' To show f is entire, it remains only to show that it is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The condition (2) implies that sN → f uniformly on compact subsets, where sN := �N n=0 fn for any N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As sN is continuous for every N ∈ N and E is Fr´echet by assumption, this implies that f is continuous (by a standard 3ǫ argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9 ([Gl¨o02b, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Every real- or complex-analytic map is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='10 ([BS71a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If f : U → F is complex-analytic, then f(x + h) = �∞ k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='δk x(f)(h) for all h ∈ V , where V is the maximal balanced 0-neighborhood of E such that x + V ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='11 ([Gl¨o02b, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then f is complex-analytic if and only if f is smooth and δ1 x := df(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' −) : E → F is complex-linear for every x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='12 ([Gl¨o02b, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose K = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If f : U → F is complex-analytic, then so is df : U × E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' With these definitions, the chain rule holds for both real- and complex-analytic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' One proceeds to define real- and complex- analytic manifolds and Lie groups, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' [Mil84] and [Nee06] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If M is a real-analytic manifold and V is a locally convex vector space, we write Cω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' V ) for the set of analytic functions M → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If M is a complex-analytic manifold and V is complex, we write O(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' V ) for the space of complex-analytic mappings M → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='14 (Identity Theorems [BS71a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose that E and F are complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let f : U → F be complex-analytic and assume that U is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If f(x) = 0 for all x ∈ V for some open and non-empty V ⊆ U, then f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose that E is real and F is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let f : UC → F be complex-analytic, where UC ⊆ EC is open and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If UC contains a non-empty subset V ⊆ E that is open in E and f(x) = 0 holds for every x ∈ V , then f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The following linear map is continuous: j∞ x : C∞(U, F) → P(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F), f �→ ∞ � k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='δk x(f) If U is connected, then its restriction to Cω(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The map j∞ x is linear, as each δk x : C∞(U, F) → P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As P(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) = �∞ n=0 P n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) carries the product topology, to see j∞ x is continuous it suffices to show that δk x is continuous for every k ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' This is immediate from the definition of the compact-open C∞-topology on C∞(U, F) [Nee06, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1(d)], and the topology of uniform convergence on compact subsets carried by P k(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that U is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let f ∈ Cω(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' F) and suppose that j∞ x (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='10 it follows that f(x + h) = 0 for all h in some 0-neighborhood of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='14 this implies that f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2 Smooth, analytic and strongly-entire representations Let G be a BCH(Baker-Campbell-Hausdorff) Fr´echet-Lie group with Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We write gC for the complexification of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that G is regular in the sense of [Nee06, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We refer to [Nee06] and [Mil84] for an overview on locally convex Lie theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let us first clarify some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If D is a pre-Hilbert space, we write L(D) for the set of linear operators on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We further define L†(D) := { T ∈ L(D) : D ⊆ dom(T ∗) and T ∗D ⊆ D } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Set T † := T ∗|D for T ∈ L†(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then (−)† is an involution on L†(D) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' [Sch90, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We will also have need for various involutions on U(gC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let θ : gC → gC be defined by θ(ξ + iη) := ξ − iη for ξ, η ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Extend the conjugation θ on gC to a complex conjugate-linear automorphism of U(gC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let τ denote the involutive anti-automorphism of U(gC) extending ξ �→ −ξ on gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Define x∗ := τ(θ(x)) for x ∈ U(gC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Explicitly, θ, τ and (−)∗ satisfy the following relations, where ξj ∈ gC for j ∈ N: θ(ξ1 · · · ξn) = θ(ξ1) · · · θ(ξn), τ(ξ1 · · · ξn) = (−1)nξn · · · ξ1 and (ξ1 · · · ξn)∗ = (−1)nθ(ξn) · · · θ(ξ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If (ρ, Hρ) is a unitary G-representation, we say that it is continuous if it is so with respect to the strong operator topology on U(Hρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let (ρ, Hρ) be a continuous unitary representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' A vector ψ ∈ Hρ is called smooth, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' analytic, if the orbit map G → Hρ, g �→ ρ(g)v is smooth, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We write H∞ ρ and Hω ρ for the linear subspaces of smooth and analytic vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We say that the representation ρ is smooth if H∞ ρ is dense in Hρ and analytic if Hω ρ is dense in Hρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If ρ is a smooth unitary representation of G, then the derived representation dρ of gC on H∞ ρ extends to an algebra homomorphism dρ : U(gC) → L†(H∞ ρ ) satisfying dρ(x)† = dρ(x∗) for any x ∈ U(gC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let (ρ, Hρ) be a smooth unitary representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — Following [JN19, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9], we define two locally convex topologies on the space H∞ ρ : – The weak topology on H∞ ρ is defined by the seminorms pξ(ψ) := ∥dρ(ξ1 · · · ξn)ψ∥, where n ∈ N≥0 and ξ = (ξ1, · · · , ξn) ∈ gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' – The strong topology is defined by the seminorms pB(ψ) := supξ∈B ∥dρ(ξ1 · · · ξn)ψ∥, where B ⊆ gn is bounded and n ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The space H∞ ρ is complete w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' to either of these topologies [JN19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='19], where we used that G is a regular Fr´echet-Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — A vector ψ ∈ H∞ ρ is called entire if �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' supξ∈B ∥dρ(ξn)ψ∥ < ∞ for every compact B ⊆ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — If ψ ∈ H∞ ρ and B ⊆ gC, we define pn B(ψ) := supξ1,··· ,ξn∈B ∥dρ(ξ1 · · · ξn)ψ∥ and qB(ψ) := �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='pn B(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — A vector ψ ∈ H∞ ρ is called strongly-entire if qB(ψ) < ∞ for every compact subset B ⊆ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — We write HO ρ ⊆ H∞ ρ for the linear subspace of strongly-entire vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Equip HO ρ with the topology defined by the seminorms qB for compact subsets B ⊆ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' — We say that the representation ρ strongly-entire if HO ρ is dense in Hρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If ψ ∈ H∞ ρ , we write f ψ : G → Hρ for the orbit map f ψ(g) = ρ(g)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As f ψ is smooth, the homogeneous polynomial f ψ n (ξ) := 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ is continuous as a map gC → Hρ, so f ψ n ∈ P n(gC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Notice further that j∞ 0 (f ψ) = �∞ n=0 f ψ n ∈ P(gC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let βψ n be the unique element of Symn(gC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hρ) satisfying f ψ n = βψ n ◦ ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Explicitly, βψ n (ξ1, · · · , ξn) = 1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' )2 � σ∈Sn dρ(ξσ1 · · · ξσn)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 6 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ψ ∈ H∞ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that qB(ψ) < ∞ for every compact subset B ⊆ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then qB(ψ) < ∞ for every compact subset B ⊆ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let BC ⊆ gC be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Replacing BC by its balanced hull, we may assume that BC is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let B := � ξ + ξ : ξ ∈ BC � ⊆ g, which is compact in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then BC ⊆ B + iB and so qBC(ψ) ≤ q2B(ψ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let (ρ, Hρ) be a smooth unitary representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ψ ∈ H∞ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The following assertions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' ψ ∈ Hω ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' There exists a 0-neighborhood V ⊆ g such that �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ converges for every ξ ∈ V and the map V → Hρ, ξ �→ �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ converges for every ξ in a 0-neighborhood g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' There is a 0-neighborhood V ⊆ g such that � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='pn V (ψ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' There is a 0-neighborhood V ⊆ g such that �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='⟨ψ, dρ(ξn)ψ⟩ converges for all ξ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The map G → C, g �→ ⟨ψ, ρ(g)ψ⟩ is analytic at 1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that ψ ∈ Hω ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then the orbit map f ψ : G → Hρ is real-analytic, and hence so is f ψ ◦ exp : g → Hρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Notice that f ψ(eξ) = ρ(eξ)ψ, so that δn 0 (f ψ ◦ exp) = dρ(ξn)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='10, it follows that f ψ(eξ) = �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ on some balanced 0-neighborhood V ⊆ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' So (1) =⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We show that (2) =⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let V ⊆ g be a 0-neighborhood such that �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ converges for every ξ ∈ V and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' the map ξ �→ �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ is continuous on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Replacing V by some smaller balanced open set, we may assume that V is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Define hψ(ξ) := �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6, the assumptions imply that hψ is real-analytic on V , where it was used that g is Fr´echet and Hρ is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then hψ is smooth by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ξ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We show that hψ(ξ) = ρ(eξ)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let s ∈ I := [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then sξ ∈ V , because V is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Notice that d dt ���� t=s hψ(tξ)ψ = dρ(ξ)hψ(sξ), and d dt ���� t=s ρ(etξ)ψ = dρ(ξ)ρ(esξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let η ∈ H∞ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Using dρ(ξ)∗η = −dρ(ξ)η it follows that d dt �� t=s ⟨ρ(etξ)η, hψ(tξ)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As a consequence, ⟨η, ρ(e−tξ)hψ(tξ)⟩ = ⟨η, ψ⟩ for all t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As this is valid for any η in the dense set H∞ ρ it follows that ρ(e−tξ)hψ(tξ)ψ = ψ or equivalently that hψ(tξ)ψ = ρ(etξ)ψ for all t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In particular, taking t = 1 we conclude that hψ(ξ) = ρ(eξ)ψ for all ξ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As hψ is real-analytic on V , so is ξ �→ ρ(eξ)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Since G is BCH, this implies that g �→ ρ(g)ψ is analytic at 1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In turn, this implies that it is analytic everywhere, where we have used that G is a real-analytic Lie group and that the composition of real-analytic maps is again real-analytic [Gl¨o02b, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus ψ ∈ Hω ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The implication (2) =⇒ (3) is trivial whereas (3) =⇒ (4) follows from [BS71a, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2] because V is absorbing and g is a Baire space, as it is Fr´echet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' To see that (4) =⇒ (2), assume that V ⊆ g is a 0-neighborhood such that �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='pn V (ψ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' For ξ ∈ V , we write sN(ξ) := �N n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ and s(ξ) := �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It remains only to prove that s is continuous on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ξ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose that (ξk) is a sequence in V with ξk → ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let N ∈ N be such that �∞ n=N+1 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='pn V (ψ) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then for any η ∈ V we have ∥s(η) − sN(η)∥ ≤ �∞ n=N+1 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='pn V (ψ) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Using that sN is continuous, let N ′ ∈ N be s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' ∥sN(ξ) − sN(ξk)∥ < ǫ and ξk ∈ V for all k ≥ N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then ∥s(ξ) − s(ξk)∥ ≤ ∥s(ξ) − sN(ξ)∥ + ∥sN(ξ) − sN(ξk)∥ + ∥sN(ξk) − s(ξk)∥ < 3ǫ, ∀k ≥ N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus s(ξk) → s(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hence s is sequentially continuous at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As g is Fr´echet, this implies that s is continuous at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' It is trivial that (3) =⇒ (5) whereas (5) =⇒ (3) follows immediately from [Nee11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3] (by considering D := H∞ ρ and v := ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Finally, (6) ⇐⇒ (1) is precisely [Nee11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let us consider an analogous statements for entire vectors: 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let ψ ∈ H∞ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The following assertions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The series �∞ n=0 f ψ n (ξ) = �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ defines an entire function gC → Hρ, ξ �→ �∞ n=0 f ψ n (ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' ψ is an entire vector for ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=', �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' supξ∈B ∥dρ(ξn)ψ∥ < ∞ for every compact B ⊆ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The map g → Hρ, ξ �→ ρ(eξ)ψ extends to an entire function gC → Hρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' �∞ n=0 supξi∈B ∥βψ n (ξ1, · · · , ξn)∥ < ∞ for every compact B ⊆ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As gC is Fr´echet by assumption, we know using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='8 that the series �∞ n=0 f ψ n (ξ) = �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ defines an entire function on gC if and only if ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' sup ξ∈B ∥dρ(ξn)ψ∥ < ∞, ∀B ⊆ gC compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' That is, if and only if (2) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus (1) ⇐⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume next that (2) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As singletons are compact, it follows in particular that � n=0 f ψ n (ξ) converges for every ξ ∈ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6, this implies that ψ ∈ Hω ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Hence the orbit map f ψ : G → Hρ is real-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As G is BCH, the exponential map exp : g → G is real-analytic and hence ξ �→ f ψ(eξ) = ρ(eξ)ψ is a real-analytic map g → Hρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Since δn 0 (f ψ ◦ exp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' ξ) = dρ(ξn)ψ for every n ∈ N, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='10 implies that f ψ(eξ) = �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ on some 0-neighborhood in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As (2) and hence (1) hold by assumption, it follows that �∞ n=0 f ψ n is an entire function extending ξ �→ ρ(eξ)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus (3) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Suppose conversely that (3) is valid, so that f ψ ◦ exp extends to an entire function F : gC → Hρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='10 and using that δn 0 (f ψ ◦ exp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' ξ) = dρ(ξn)ψ for n ∈ N, we find that F(ξ) = �∞ n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξn)ψ for every ξ ∈ gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus (1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' We have shown (1) ⇐⇒ (2) ⇐⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Next we show (2) =⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let B ⊆ gC be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' As gC is complete, the closed convex hull of B is again compact [Tre67, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Thus we may assume that B is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Replacing B further by its balanced hull, we may assume that B is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then B + · · · + B (n times) ⊆ nB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' From equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3) it follows that sup ξi∈B ∥βψ n (ξ1, · · · , ξn)∥ ≤ 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' sup ξ∈nB ∥f ψ n (ξ)∥ = (2n)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' sup ξ∈B ∥f ψ n (ξ)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Choose some t > 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Since �∞ n=0 supξ∈B ∥f ψ n (ξ)∥ < ∞ for every compact B, it follows (by considering tB) that there exists some C > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' supξ∈B ∥f ψ n (ξ)∥ ≤ Ct−n for every n ∈ N≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then ∞ � n=0 sup ξi∈B ∥βψ n (ξ1, · · · , ξn)∥ ≤ C ∞ � n=0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' �2n t �n < ∞, The implication (4) =⇒ (2) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The characterization (4) of entire vectors in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='7 makes the difference between entire and strongly-entire vectors clear, namely whether one considers the symmetric n-linear maps βψ n or their non-symmetric analogues (ξ1, · · · , ξn) �→ 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='dρ(ξ1 · · · ξn)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Analogous to [Nee11, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='7], it is in general not known whether or not any entire vector is in fact strongly-entire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In the case where g is finite-dimensional, this follows immediately from [Pen74, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' HO ρ ⊆ Hω ρ ⊆ H∞ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Any strongly-entire vector is entire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Consequently, the first inclusion follows by combining Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='7 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The second one follows from the fact that if the orbit map f ψ : G → Hρ is real-analytic, then it is smooth by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' The space HO ρ of strongly-entire vectors will be considered in more detail in Section 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3 Positive energy and ground-state representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let G be a regular locally convex Lie group with Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' If H is a Hilbert space and S ⊆ H is a subset, we write �S� ⊆ H for the closed linear span of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='1 (Borchers-Arveson [BR87, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='46], [BGN20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let M ⊆ B(H) be a von Neumann algebra on the Hilbert space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Let (Ut)t∈R be a strongly continuous unitary one-parameter group satisfying UtMU −1 t ⊆ M for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Assume that Ut = eitH with H ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Define α : R → Aut(M) by αt(x) := AdUt(x) := UtxU −1 t for t ∈ R and x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Denote by Mα(S) ⊆ M the Arveson spectral subspace for S ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Then 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' There exists a strongly continuous unitary one-parameter group Vt = eitH0 in M with H0 ≥ 0 and AdVt = αt for every t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' � t>0�Mα[t, ∞)H� = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' Vt is uniquely determined by the additional requirement that for any other such V ′ t = eitH′ 0, we have H′ 0 ≥ H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE4T4oBgHgl3EQfiA0C/content/2301.05129v1.pdf'} +page_content=' In this case, the spectral projection P corresponding to Vt is determined uniquely by P[t, ∞)H = � s 2mψ/Ω, where mψ is the mass of +the emitted fermions. The emission rate at harmonic n is found using +dΓn = +� +s1,s2 +|Mn(s1, s2)|2(2π)δ(Ωn − ω1 − ω2) d3k1 +(2π)3ω1 +d3k2 +(2π)3ω2 +. +(2.7) +Here, k1 = (ω1, k1) and k2 = (ω2, k2) are the four-momenta of the fermion and anti-fermion +respectively, and s1(s2) is the spin of the fermion (anti-fermion). The microscopic physics enters +via Mn (s1, s2), which is the matrix element of the fermion-pair emission at harmonic n. At +leading order, this matrix element is obtained from the diagram in Fig. 1. In the diagram, ⊗ +denotes the classical source, which is given by the classical current, Jµ +cl(x), in the case of vector +mediator and by the density, ρcl(x), in the case of the scalar mediated radiation. The total +power loss via fermion-pair radiation is simply a sum of power losses over all harmonics +Ploss = +� +n +Pn, +Pn = +� +(ω1 + ω2) dΓn. +(2.8) +5 + +ψ, k1 +ψ, k2 +mediator +FIG. 1. Feynman diagram for a fermion pair emission by a classical current. +Here, Pn is the power loss of the nth harmonic. +In what follows, we consider two types of mediators: a massive gauge boson and a massive +scalar. We only consider s-channel exchange and remark on t-channel exchange at the end of +this subsection. +First, we consider a vector mediator, Aµ, that corresponds to a broken U(1)′ and has mass +mA. This gauge boson couples to a classical current Jµ +cl(x), which has charge Q under U(1)′. +The gauge boson Aµ is unstable and decays into a fermion pair. The terms in the effective +Lagrangian, relevant for the fermion-pair radiation via Aµ, are +Leff ⊃ gAµJµ +cl + gqψ ¯ψγµAµψ , +(2.9) +where qψ is the U(1)′ charge of the fermion ψ, g is a dimensionless coupling constant, and Jµ +cl(x) +is the classical current defined in Eq. (2.1). Both the vector boson and the fermions are assumed +to be massive with masses mA and mψ, respectively. The leading order matrix element for the +emission, at the n−th harmonic, is given by +Mn(s1, s2) = g2qψ ¯u(k1, s1)γµv(k2, s2) i(−ηµν + (k1 + k2)µ(k1 + k2)ν/m2 +A) +(k1 + k2)2 − m2 +A + imAΓA +Jν +cl(Ωn) , +(2.10) +where Jν +cl(Ωn) is the Fourier transform of Jν +cl(x), given by +Jν +cl(Ωn) = Ω +2π +� 2π/Ω +0 +dt +� +d3x ei(nΩt−p·x)Jν +cl(x) +(2.11) +with p = k1 + k2, ΓA is the decay width of the gauge boson, and 2π/Ω is the period. We +assume that the decay into a ¯ψψ pair is the only decay channel for the gauge boson Aµ, and +that the fermion mass mψ is negligible compared to the gauge boson mass mA. Under these +assumptions, the decay width of Aµ is given by +ΓA = g2q2 +ψmA +12π +. +(2.12) +The other case we consider is that of a scalar mediator, φ, for which the relevant terms in +the Lagrangian are +L ⊃ gφρcl + g′φ ¯ψψ, +(2.13) +6 + +where g is the dimensionless coupling between the scalar φ and the classical source, g′ is the +Yukawa coupling of the fermion ψ to the scalar φ, and ρcl(x) is the number density of relevant +particles in the classical source. Both the scalar and the fermions are assumed to be massive +with masses mφ and mψ, respectively. The matrix element in this case is given by +Mn(s1, s2) = gg′¯u(k1, s1)v(k2, s2) +iρcl(Ωn) +(k1 + k2)2 − m2 +φ + imφΓφ +, +(2.14) +where ρcl(Ωn) is the Fourier transform of ρcl(x), +ρcl(Ωn) = Ω +2π +� 2π/Ω +0 +dt +� +d3x ei(nΩt−p·x)ρcl(x), +(2.15) +and the decay width of the scalar is Γφ. As in the case of the vector mediator, we assume +that the fermionic decay mode is the only available mode, and the fermion mass mψ can be +neglected compared to the mass of a scalar mφ. Thus we have +Γφ = g′2mφ +8π +. +(2.16) +So far, we have only considered the s-channel contribution to the fermion pair radiation. +Fermion pair radiation via t−channel process mediated by a vector or scalar is also a possibility. +Such contributions, however, are highly suppressed for mS ≫ Ω, mM, where mS is the mass +of the particles in the source that couple to the fermion pairs ¯ψψ at the microscopic level, +and mM is the mediator mass. Since the emitted fermions have energy of the order of Ω, the +fundamental frequency of the system, the t-channel contribution to the momentum entering +the propagator is of the order of mS − Ω. Thus the t-channel propagator is schematically given +by +Π ∼ +1 +(mS − Ω)2 − m2 +M +. +(2.17) +In the case where mS is much larger than both Ω and mM, the propagator is dominated by the +mass of the source particles, and the process is heavily suppressed. In this paper, we assume +that the mass hierarchy mS ≫ Ω, mM and neglect the t−channel contributions to the fermion +pair radiation everywhere. +B. +Power loss formulae +Using Eqs. (2.7)–(2.14), we can calculate the power loss via fermion-pair radiation from +a point-like object moving in an elliptical orbit. The detailed derivations are shown in Ap- +pendix A, and here we only quote the final result. The power loss due to fermion-pair emission +7 + +in harmonic n > 2mψ/Ω, for the cases of the vector and scalar mediator, can be written as +P A +n = g4q2 +ψQ2 +12π3 +a2Ω4 BA +n (nA, nψ, nΓ), +(2.18) +P φ +n = g2g′2N 2 +12π3 +a2Ω4 Bφ +n(nφ, nψ, nΓ). +(2.19) +The functions BM +n (nA, nψ, nΓ), where M = A, φ, are given by +BM +n (nM, nψ, nΓ) ≡ +� +J′ +n(ne)2 + 1 − e2 +e2 +Jn(ne)2 +� � n−nψ +nψ +dx F M(x, n, nM, nψ, nΓ). +(2.20) +Here +nM ≡ mM/Ω, +nψ ≡ mψ/Ω, +nΓ ≡ ΓM/Ω, +(2.21) +and Jn(ne) is a Bessel function of order n with argument ne. +The integration variable in +Eq. (2.20) is defined by x ≡ ω1/Ω, where ω1 is the energy of one of the final-state fermion. In +what follows, for brevity, we use the notation +F M(x) ≡ F M(x, n, nM, nψ, nΓ), +BM +n ≡ BM +n (nM, nψ, nΓ). +(2.22) +The functions F M(x) have the general form +F M(x) = F M +0 (x) + F M +1 (x) +nMnΓ +� +tan−1 +�a(x) + b(x) +nMnΓ +� +− tan−1 +�a(x) − b(x) +nMnΓ +�� ++ F M +2 (x) tanh−1 +� +2a(x)b(x) +a(x)2 + b(x)2 + n2 +Mn2 +Γ +� +, +(2.23) +with a(x) and b(x) being universal for both gauge boson and scalar mediators, +a(x) = 2n2 +ψ − n2 +M + 2nx − 2x2 , +b(x) = 2 +� +x2 − n2 +ψ +� +(n − x)2 − n2 +ψ . +(2.24) +The functions F M +0 (x), F M +1 (x), and F M +2 (x) are different for the two cases. For a gauge boson +mediator, we obtain +F A +0 (x) = b(x)/2n , +F A +1 (x) = 1 +4n +� +n4 +A + 4n2n2 +ψ − n2 +An2 +Γ + 2n2 +An2 − 4nxn2 +A + 4x2n2 +A +� +, +F A +2 (x) = 1 +2n +� +n2 +A + n2 − 2nx + 2x2� +, +(2.25) +while for a scalar mediator, +F φ +0 (x) = −b(x)/2n , +F φ +1 (x) = 1 +4n +� +n2 +φn2 +Γ + (n2 − n2 +φ)(n2 +φ − 4n2 +ψ) +� +, +F φ +2 (x) = 1 +4n +� +n2 + 4n2 +ψ − 2n2 +φ +� +. +(2.26) +8 + +Eqs. (2.18)–(2.26) are the main results of our work. Analytical integration of F A(x) and +F φ(x) is challenging, but it still can be performed in certain limits. In Sec. III B, we consider +two limiting cases: the case of nM ≪ 1, which reproduces the Larmor formula, and nM ≫ 1, +which is relevant for the fermion pair radiation in the SM. In general, however, calculating the +power loss requires numerical analysis. We perform such an analysis in Sec. IV when we discuss +a particular phenomenological application of our result. +III. +DISCUSSION OF THE POWER-LOSS FORMULA +The power loss due to fermion-pair emission by a classical source on an elliptical orbit is +given by Eqs. (2.18)-(2.26). Below we discuss the main features and the asymptotic behavior +of this result. +A. +General features of the power-loss formula +We start with the general features that hold for both the vector and scalar cases. +• The radiation rate is proportional to the charge-squared; that is, the functions P A +n and P φ +n +depend on Q2 and N 2, respectively. This is a manifestation of the fact that the fermion-pair +radiation that we are considering is coherent. +• The form of F M(x), with M = A, φ, in Eq. (2.23) is somewhat general. We show in +Appendix A that the overall form of F M(x), at the tree level, is the same for any renormalizable +theory that couples fermions to a classical source moving in an elliptical orbit. Note that the +functions a(x) and b(x) defined in Eq. (2.24) are purely kinematic and thus have the same form +for any theory of fermion pair emission, while the form of F M +0 (x), F M +1 (x), and F M +2 (x) vary with +the theory considered. For instance, considering non-renormalizable interactions would lead to +a different momentum dependence of the matrix element that could, in principle, change the +form of F M(x). +• The power loss for both vector and scalar mediators behaves qualitatively the same way +despite the different functional forms of F A +i (x) vs. F φ +i (x), with i = 0, 1, 2. This is not surprising +since there is nothing fundamentally different between the matrix elements for the vector and +scalar cases. +• Energy conservation implies that the functions F M(x) are invariant under x → (n − x) +exchange. The reason is that the total energy radiated in fermion pairs in the n-th harmonic +is nΩ. The transformation x → (n − x) exchanges the energies of the emitted fermion and +anti-fermion, and the emission rate is the same regardless of the order in which the integrals +9 + +are carried out. This invariance results from the fact that the fermion-antifermion emission +from a classical system is essentially a 2-body decay. Note that this has nothing to do with the +details of the considered model. +• For nA < n, the power loss has a very weak dependence on nA. This is true for the +particular models that we chose here but is not expected to be true in general. For an example +when this is not the case, see the discussion of Proca fields in Ref. [6], where dependence on nA +appears due to the absence of gauge symmetry. +• There is an interplay of three energy scales: The mass of the mediator, mM, the mass of +the fermion, mψ, and the frequency of the harmonics, nΩ. The fermions cannot be produced +when 2mψ > nΩ. In the opposite limit, when 2mψ < nΩ, the production rate depends strongly +on the mediator mass. For mM < 2mψ < nΩ, fermion production is strongly suppressed since +the on-shell boson is kinematically forbidden from decaying into fermions. (Note that strictly +speaking, our result cannot be straightforwardly applied in this case as everywhere we assume +ΓM > 0.) For 2mψ < mM < nΩ, the fermions are produced via decay of the on-shell mediator. +Thus the power loss in the fermion-pair radiation is equal to that of the on-shell boson radiation. +The region of the parameter space where mM > nΩ > 2mψ is of the most interest to us, as in +this region the fermions are kinematically allowed, the mediator is off-shell, and therefore the +fermion pair emission is most significant. +• As an example that illustrates the qualitative features of the power loss, consider Fig. 2. +It shows BA +n , defined in Eq. (2.20), as a function of nA for massless fermions for the first four +harmonics. The most striking feature of the plots is a sharp drop at nA ∼ n. This behavior +follows from the fact that at nA ∼ n, the radiation regime switches from the radiation dominated +by on-shell boson production (nA < n), which is proportional to g2 to the off-shell production +(nA > n) proportional to g4. The power loss in the regime dominated by fermion-pair radiation +is thus suppressed by g2 compared to the power loss in the regime dominated by the on-shell +boson radiation. The power loss in the case of the scalar mediator exhibits the same behavior. +• Comparing our results to the cases of vector [3, 5, 6] and scalar radiation [3], we note that +from kinematic considerations alone, boson radiation drops to zero as soon as nM = n. This +is not what we observe for the fermion-pair emission. In the case of fermion-pair radiation, +off-shell boson production is possible, even though there is an extra suppression by g2 for +a vector and g′2 for a scalar compared to on-shell boson radiation. As a result, the regime +nM > n opens up new regions of the parameter space for each harmonic n and is of particular +phenomenological interest to us. +• Next, we remark on the dependence of the power loss on the eccentricity in the case of +orbits close to circular. +For that, we note that the eccentricity only enters the power loss +10 + +����� +����� +�� +���� +��-�� +��-�� +��-�� +�� +FIG. 2. BA +n vs nA for fixed eccentricity, e = 10−3, coupling constant g = 10−15, and massless final +state fermions, mψ = 0. See Eqs. (2.20)-(2.25) for the definition of BA +n . +through the Bessel function prefactor of BM +n in Eq. (2.20), which we denote as K(n, e), +K(n, e) = J′ +n(ne)2 + 1 − e2 +e2 +Jn(ne)2 . +(3.1) +We recall that Jn(z) and J′ +n(z) behave asymptotically, in the limit z ≪ 1, as +Jn(z) ≈ 1 +n! +�z +2 +�n +, +J′ +n(z) ≈ n +n! +1 +2 +�z +2 +�n−1 +≈ n +z Jn(z), +z ≪ 1. +(3.2) +Using Eq. (3.2), we find for the eccentricity dependent prefactor K(n, e), in the limit ne ≪ 1, +that +K(n, e) = J′ +n(ne)2 + n2 − (ne)2 +(ne)2 +Jn(z)2 ≈ J′ +n(ne)2 + +n2 +(ne)2Jn(z)2 += 2n2 +z2 Jn(ne)2 = (ne)2n−2 +22n−1 +n2 +(n!)2 = +(ne)2n−2 +22n−1 ((n − 1)!)2. +(3.3) +Thus we learn that in the limit ne ≪ 1, prefactor K(n, e) scales with the eccentricity as +K(n, e) ∝ (ne)2n−2 . +(3.4) +This shows that for small eccentricities (and thus orbits close to circular ones), the contributions +from higher harmonics die away very fast as n increases. For n = 1 and e ≪ 1, we have K(1, e) ≈ +1/2. For each subsequent harmonic power drops by a factor of order e2, until the factorial in +the denominator of K(n, e) (see Eq. (3.3)) starts to dominate. Then the contributions from the +higher harmonics start to decay away even faster. Fig. 2 illustrates the behavior of the power +loss for the first four harmonics in the case of small eccentricity e = 10−3. +• The case of highly eccentric orbits e ∼ 1 is significantly more involved. First, the contri- +butions from different modes do not follow the simple hierarchy of the low eccentricity case. +11 + +� +� +�� +�� +��-�� +��-�� +��-� +��-� +��� +� +� +� +�� +�� +��-�� +��-�� +��-�� +��-�� +��-�� +FIG. 3. Left: BA +n as a function of n in the regime where the radiation is dominated by on-shell boson +production. Different colors correspond to different values of eccentricity. The values of nψ, nA and g +are fixed. Right: BA +n as a function of n for a highly eccentric orbit with e = 0.6 in the regime where +the radiation is dominated by off-shell boson production. +The contributions from higher modes can be of the same order or even larger than the first +mode depending on the values of other parameters. See the left panel of Fig. 3 to compare +the n-dependence of BA +n for different eccentricity values. Second, as Fig. 3 demonstrates, the +hierarchy of modes in the on-shell dominated part of the parameter space does not carry into +the off-shell dominated region. Consider the green line corresponding to a highly eccentric orbit +with e = 0.6. For nA = 10−1 (left panel), the maximum contribution to the power loss comes +from the mode with n = 2 and the first 5 modes contribute at about the same order. The +situation is drastically different for nA = 50 (right panel). The maximum contribution to the +power loss comes from the n = 8 mode. We learn that for e ∼ 1, generally speaking, the power +loss per mode first increases as we increase n and then starts decreasing after reaching a certain +value of n. Where this maximum occurs depends on other parameters. +B. +Asymptotic behavior for the case of circular orbits +We now move to the discussion of the asymptotic behaviour of the power loss in two limiting +cases mM ≪ Ω and mM ≫ Ω, where mM is the mass of the mediator, M = A, φ. In this +subsection, for simplicity we consider the straightforward case of circular orbits (e = 0) and +massless fermions (mψ = 0). For the eccentricity dependent part of the power loss, K(n, e), we +have +lim +e → 0 K(n, e) = lim +e → 0 +� +J′ +n(ne)2 + 1 − e2 +e2 +Jn(ne)2 +� += 1 +2δn,1. +(3.5) +Thus the only mode that contributes to the power loss in the circular orbit limit is the mode +with n = 1. +12 + +First, let us consider the regime of light mediators, mM ≪ Ω, or equivalently nM ≪ 1. In +this limit, F M(x) defined in Eq. (2.23) is dominated by the second term. We thus neglect the +first and the third terms of F M(x) and take the second term’s limit nM → 0. After that, the +integral in (2.20) can be performed analytically, yielding the following asymptotic expressions +for the power radiated via vector and scalar, respectively: +P A(mA ≪ Ω) ≈ g2 +6πQ2a2Ω4, +(3.6) +P φ(mφ ≪ Ω) ≈ g2 +12πN 2a2Ω4. +(3.7) +The asymptotic behavior that we find for P A and P φ reproduces the known results for the +on-shell vector [3, 5, 6] and scalar [3] radiation. This is expected as, in the regime mM ≪ Ω, +the fermion pair radiation is dominated by on-shell boson production. Additionally, Eq. (3.6) +also reproduces the Larmor formula for the power of the electromagnetic wave radiation given +in Eq. (1.1). To see this, recall that the acceleration on a circular orbit is equal to aΩ2, where +a is the radius of the orbit and Ω is the frequency of revolution. +Next, we study the regime when on-shell boson production is kinematically forbidden, and +the fermion pair radiation takes place through the off-shell mediator. This is the limit of heavy +mediators, mM ≫ Ω, or equivalently nM ≫ 1. As in the case of the light mediators, we take the +nM → ∞ limit of F M(x) and find that the resulting expression can be integrated analytically. +Upon performing the integration, we find that the vector and scalar-mediated radiation behave +as +P A(mA ≫ Ω) ≈ g4q2 +ψQ2 +210π3 +a2Ω8 +m4 +A += +1 +35π2 +g2q2 +ψΩ4 +m4 +A +× P A(mA ≪ Ω), +(3.8) +P φ(mφ ≫ Ω) ≈ g2g′2N 2 +840π3 +a2Ω8 +m4 +φ += +1 +70π2 +g′2Ω4 +m4 +φ +× P φ(mφ ≪ Ω). +(3.9) +We learn that in the limit of heavy mediators, the fermion pair radiation is suppressed compared +to on-shell boson radiation by the following factors: +1. A factor of g2q2 +ψ or g′2, which, at the amplitude level, comes from the coupling of the +mediator to the fermion pair. +2. A factor of Ω4/m4 +φ, which comes from the propagator of the mediator. +3. A phase space factor of 1/35π2 or 1/70π2, which arises from the fact that there are more +particles in the final state in the case of the off-shell pair production than in the case of +the on-shell boson production. +Note that Eqs. (3.8) and (3.9) can be interpreted as integrating out the heavy mediator, +resulting in an effective 4-Fermi interaction with a coefficient proportional to g2/m2 +A or gg′/m2 +φ. +Thus, it is also valid for t-channel and u-channel interactions. +13 + +Last, we compare the results of the vector to that of the scalar mediators. Consider mA = mφ, +Q2 = N 2 and g′ = gqψ. In this case, the power radiated via the vector mediator is greater than +the power radiated via the scalar mediator in both radiation regimes. In particular, we have +P A(mA ≪ Ω) +P φ(mφ ≪ Ω) ≈ 2, +P A(mA ≫ Ω) +P φ(mφ ≫ Ω) ≈ 4. +(3.10) +These factors are related to the different number of degrees of freedom between the vector and +scalar cases. There are two polarization states for an on-shell massless vector, while the scalar +has only one. For the deeply off-shell mediator, the correspondence is not so clear, but it seems +to us that it is related to the fact that off shell gauge boson, Aµ, has four degrees of freedom +C. +Fermion-pair radiation in the SM +The expression in Eq. (3.8) can be used to estimate the power loss due to fermion pair +radiation by classical sources within the SM. In this subsection, we consider neutrino pair +radiation mediated by Z-boson. The contribution due to W-boson mediated pair emission is +qualitatively the same as the Z-boson contribution and is expected to be of the same order. +The main difference between the two contributions is due to the fact that W-boson mediated +radiation is only relevant for leptons in the source while Z-boson contribution is present for all +types of fermions. +Consider a source made of NΨ fermions of type Ψ with the total weak charge Q = NΨqΨ. +To apply Eq. (3.8) to the neutrino pair radiation in the SM, we need to recall that Eq. (3.8) +was derived under the assumption of vectorial couplings, while the SM is a chiral theory. The +relevant parts of the SM Lagrangian are different from the Lagrangian in Eq. (2.9); in particular, +in the SM we have +LSM ⊃ −i +g +2 cos θW +� +¯Ψγµ(cΨ +V − cψ +A)Ψ + ¯νγµ(cν +V − cν +A)ν +� +Zµ. +(3.11) +Thus Eq. (3.8) yields the following expression for the Z-boson mediated power loss due to the +neutrino pair radiation in the SM +P Z(mZ ≫ Ω) ≈ +1 +210π3 +g4q2 +νq2 +ΨN 2 +Ψ +16 cos4 θW +a2Ω8 +m4 +Z +, +(3.12) +where we perform the replacement g → g/(2 cos θW) in Eq. (3.8) and define +q2 +ψ = q2 +ν = (cν +V )2 + (cν +A)2, +qΨ = cΨ +V , +mA = mZ . +(3.13) +Note that, for the source, only vectorial coupling cΨ +V enters the power loss. This is because we +consider coherent radiation. +14 + +The expression in Eq. (3.12) can be rewritten as +P Z(mZ ≫ Ω) ≈ G2 +effq2 +Ψq2 +νN 2 +Ψ +a2Ω8 +210π3 , +(3.14) +where Geff = +√ +2GF and GF is the Fermi constant. When the power loss is written in the form +of Eq. (3.14), it becomes clear that it is the same as what one would obtain by performing the +calculation for the effective Fermi theory with the effective Lagrangian given by +LZ +eff ⊃ Geff[¯Ψγµ(cΨ +V − cΨ +Aγ5)Ψ][¯νγµ(cν +V − cν +Aγ5)ν]. +(3.15) +This, of course, is not surprising as we consider radiation at the energy Ω, which is much less +than the electroweak scale, Ω ≪ mZ. In fact, the result in Eq. (3.14) applies to any effective +4-Fermi interaction. While we derive our results for s-channel exchange, in the limit where the +mediator is much heavier than the orbit frequency, we do not need to distinguish between s- +channel and t-channel. Thus, Eqs. (3.12) and (3.14) can also be used for t-channel W-exchange +in the SM. +Finally, we discuss the situation when there are several different types of fermions in the +source. In this case, we need to first add all the amplitudes that correspond to the radiation +by different fermions Ψ (for leptons, we add both Z-boson and W-boson contributions). Then, +we square the sum of the relevant amplitudes to obtain the total emission rate. +We end this subsection with the following remark. The power loss due to neutrino pair +radiation in the SM was estimated in Ref. [4] to be P Z +SM ∼ G2 +FΩ6. Using the explicit calculation, +however, we find that P Z +SM ∼ G2 +Fa2Ω8. That is, there is an extra factor of a2Ω2 compared to +the estimation of Ref. [4]. In fact, our result includes the semi-major axis a as an additional +energy scale of the system. +IV. +FERMION PAIR RADIATION BY PULSAR BINARIES +We now move to discuss the phenomenological applications of our results to astrophysical +systems. We focus on the neutrino-pair emission from pulsar binaries [23–33]. A pulsar binary +is a binary system of a pulsar and companion. This choice is motivated by the availability of +extensive period decay data for such systems. In particular, we apply our results to two binaries: +Hulse-Taylor binary PSR B1913+16 [34–36] (a system of a pulsar and a neutron star) and PSR +J1738+0333 [29, 37] (a system of a pulsar and a white dwarf). The parameters characterizing +the two systems are summarized in Table I. +In what follows, we first discuss the applicability of our results of Section II B to pulsar +binaries in general. Then we estimate the contribution to the power loss due to neutrino pair +15 + +Binary system +PSR B1913+16 [36] +PSR J1738+0333 [29] +Eccentricity e +0.6171340(4) +3.4(11) × 10−7 +Pulsar mass m1 (M⊙) +1.438(1) +1.46(6) +Companion mass m2 (M⊙) +1.390(1) +0.181(8) +Binary period Tb (GeV−1) +4.240 × 1028 +4.657 × 1028 +Intrinsic period decay ˙Tb +−2.398(4) × 10−12 +−2.59(32) × 10−14 +Predicted period decay due to GW ˙TGW +−2.40263(5) × 10−12 +−2.77(19) × 10−14 +Ratio of period decays R = ˙Tb/ ˙TGW +0.9983(16) +0.94(13) +Orbital frequency Ω = 2π/Tb (GeV) +1.482 × 10−28 +1.349 × 10−28 +Semi-major axis a (GeV−1) +9.878 × 1024 +8.77 × 1024 +TABLE I. The relevant parameters for the PSR B1913+16 and PSR J1738+0333 binary systems. +Figures in parenthesis are the 1σ uncertainties in the last quoted digit, where all the uncertainties +are symmetrized. M⊙ is the mass of the sun. The relative experimental error of the binary period +Tb is ∼ 10−12 for PSR B1913+16, and ∼ 10−11 for PSR J1738+0333. +The double line separates +binary parameters quoted in Ref. [29, 36] and the ones we derive. Values of the semi-major axis a are +calculated using Eq. (4.5). +emission in the SM and show that it is negligible compared to the gravitational wave radiation. +We then consider neutrino pair radiation in two BSM scenarios via ultralight vector and scalar +mediators and apply our results to the pulsar binaries with the parameters in Table I. +A. +Pulsar binaries as a classical source +The results for the fermion pair radiation, summarized in Eqs. (2.18)-(2.26), were derived for +the case of classical current describing non-relativistic point-like object following an elliptical +orbit. To justify the application of our results to pulsar binaries, we note the following: +1. A pulsar binary can be treated as a classical source. The typical size of a pulsar binary +can be estimated as the size of the semi-major axis which varies between 106 and 108 km, +that is, a ∼ 1024 − 1026 GeV−1. The wavelength of the radiation is determined by the +fundamental frequency of the orbit, and for a typical pulsar binary with periods in the +range of 10−1 − 103 days, the wavelength is λ ∼ 1028 − 1032 GeV−1. Thus, λ ≫ a and we +conclude that pulsar binaries can be treated as classical radiation sources. +2. Stars of the pulsar binary can be treated as point-like objects. Typical sizes of stars +in a binary vary from r ∼ 10 km ∼ 1019 GeV−1, for neutron stars, and r ∼ 103 km ∼ +16 + +1021 GeV−1, for white dwarfs. Thus r ≪ a, λ and both pulsar and its companion can be +treated as point-like objects. Moreover, r ≪ λ implies the coherence of the radiation. +3. The motion of the pulsar and its companion in the binary system is non-relativistic. We +can roughly estimate the orbital velocity of the stars in a binary as v ∼ aΩ, which for +characteristic values quoted above implies v ≲ 10−2. +4. For a wide range of pulsar binary systems, the observed power loss is such that it has no +significant effect on the eccentricity of the orbit. Thus we can treat the orbit as elliptical +over the time of observation. +For example, the Hulse-Taylor binary has e ∼ 1, with +Tb(de/dt) ≲ 10−11, where Tb is the binary period and de/dt is the time derivative of the +eccentricity [36]. +Now that we have established that the results of Section II B can be applied to pulsar +binaries, we proceed in two steps. First, we modify our expressions for the classical current and +number density in Eqs. (2.1) and (2.2) to the case of two point-like objects on an elliptical orbit. +Second, we perform the standard reduction of the two-body problem to a one-body problem. +We write the classical current and number density as +Jµ +cl(x) = +� +b=1,2 +Qb δ3(x − xb(t))uµ +b , +(4.1) +and +ρcl(x) = +� +b=1,2 +Nb δ3(x − xb(t) , +(4.2) +respectively. Here, b = 1, 2 is the index that labels the stars of the binary system, xb(t) is the +position of the b-th star at time t, and uµ +b is its four-velocity. +Next, we move to the binary system’s Center-of-Mass (CoM) frame. For that, we define R, +the coordinate of center of mass, and r, the distance between the two stars, +R = +m1 +m1 + m2 +x1 + +m2 +m1 + m2 +x2, +r = x1 − x2 , +(4.3) +where m1 and m2 are the masses of the two stars. +As we are not concerned with the translational motion of the system as a whole, which is +described by R, we can solely focus on r. This is the standard two-body to one-body problem +reduction for central force motion. The non-relativistic classical trajectory of the stars in the +CoM frame can thus be described by the vector r = (x, y, 0) and is given by elliptical orbits as +in Eq. (2.3): +x = a(cos ξ − e), +y = a +√ +1 − e2 sin ξ, +Ωt = ξ − e sin ξ, +(4.4) +17 + +where e is the eccentricity, a is the semi-major axis of the elliptical orbit, and the fundamental +frequency of revolution is given by +Ω = +� +GN(m1 + m2) +a3 +. +(4.5) +The results of Eqs. (2.18)-(2.26) generalize to the case of binary systems via the following +replacements that follow from the 2-body to 1-body reduction procedure: +Q2 → M 2 +�Q1 +m1 +− Q2 +m2 +�2 +, +N 2 → M 2 +�N1 +m1 +− N2 +m2 +�2 +, +(4.6) +where +M = +m1m2 +m1 + m2 +(4.7) +is the reduced mass of the binary system. As a result we obtain the following expressions for +the power loss in n-th harmonic for a vector and scalar mediators respectively: +P A +n = g4q2 +ψ +12π3M 2 +�Q1 +m1 +− Q2 +m2 +�2 +a2Ω4 BA +n (nA, nψ, nΓ), +(4.8) +P φ +n = g2g′2 +12π3 M 2 +�N1 +m1 +− N2 +m2 +�2 +a2Ω4 Bφ +n(nφ, nψ, nΓ), +(4.9) +where the functions BA +n and Bφ +n are defined in Eqs. (2.20)-(2.26). +B. +Neutrino pair radiation by pulsar binaries in the SM +In the SM, for the pulsar binary, the power loss via electroweak mediators is discussed in +Sec. III C. Here, we simply generalize it to the case of 2-body motion using Eq. (4.6). We obtain +the following expression for the power loss in neutrino pair radiation via Z-exchange in the SM +PSM ≈ G2 +F +� +cν +V +2 + cν +A +2� +105π3 cos2 θW +M 2a2Ω8 +� +1 +m1 +� +i=n,p,e,... +ci +V N1iQ1i − 1 +m2 +� +i=n,p,e,... +ci +V N2iQ2i +�2 +(4.10) +where the sum goes over all microscopic constituents of binary stars, such as neutrons (n), +protons (p), electrons (e), etc. To perform a numerical estimate, we consider a pulsar binary +with a neutron star companion and assume that all of the neutron star mass is in the form of +neutrons. We consider a typical pulsar-neutron star binary with +m1,2 ∼ M⊙ ∼ 1057GeV, +a ∼ 1025 GeV−1, +Ω ∼ 10−28 GeV, +(4.11) +and non-zero dipole moment +M 2 +�Q1 +m1 +− Q2 +m2 +�2 +∼ Q2 +1,2 ∼ 10114, +(4.12) +18 + +where Qb = Nb(n)−Nb(¯n) ≈ Nb(n) ≈ M⊙/mn ≈ 1057, with b = 1, 2, are the neutron charges of +the neutron stars, Nb(n) and Nb(¯n) are the numbers of neutrons and anti-neutrons respectively, +mn is the neutron mass. Using cν +V = cν +A = 1/2, cn +V = −1/2, and the measured values of mn, +GF, and θW, we find the following numerical estimate for the radiated power +PSM ∼ 10−56eV2. +(4.13) +To see if the above result is significant, we compare it to the power loss in the form of +gravitational wave (GW) radiation. Using the quadrupole formula for the GW radiation [38] +for the case of circular orbit (e = 0) we have +PGW = 32 +5 GNM 2a4Ω6 ∼ 108 GeV2 +(4.14) +where GN is Newton’s gravitational constant. The rough estimates in Eqs. (4.13) and (4.14) +show that, in the SM, the fermion-pair radiation by astrophysical objects is completely negligible +compared to the gravitational wave radiation. +We close the subsection with one remark. Within the SM, neutron stars also emit syn- +chrotron radiation of fermion-antifermion pairs in their self-produced magnetic fields, as shown +in Ref. [39]. This phenomenon is different from the one we consider here. Synchrotron radiation +is an incoherent effect. Thus, the power loss, in this case, scales as N, the number of neutrons +in the star. In the case we are considering, the radiation is coherent and comes from the star’s +acceleration as a whole. Then, the net power that is radiated is proportional to N 2. +C. +New physics constraints from the neutrino pair radiation by pulsar binaries +Since extra radiation in the SM is negligible, any observed deviation from the gravitational +wave radiation would be strong evidence for the physics beyond the SM. In particular, fermion- +pair radiation can be enhanced in BSM models with light vector or scalar mediators, with +mA,φ ≪ mZ. To explain why such light bosonic states have evaded detection so far, we must +require that they have small couplings, thus evading all the available constraints. The smallness +of couplings, however, still can be compensated in cases where the object has a large charge +under the new symmetries. This can be the case for astrophysical objects. Thus, such objects +are our prime focus in the rest of this work. +In particular, in this subsection, we demonstrate how our results can be used to derive new +physics bounds from the neutrino pair radiation by pulsar binaries. As we mentioned above, +we use two distinct pulsar binary systems, the Hulse-Taylor binary PSR B1913+16 and PSR +J1738+0333. +The relevant properties of the two systems are summarized in Table. I. The +19 + +Hulse-Taylor binary is a pulsar binary with a neutron star companion, it is highly eccentric, +and the mass ratio of the two stars is close to 1. The PSR J1738+0333, on the other hand, +is a pulsar-white dwarf binary with an almost circular orbit and a high pulsar-to-companion +mass ratio. For both systems, the data on the orbital period decay is shown in Table I. Both +binaries lie within 1σ of the general relativity prediction. +In our analysis, we exploit the fact that typical neutron stars contain a very large number of +muons, N(µ) ∼ 1055 [40–43]. Thus, the effects of muonophilic new physics can be significantly +enhanced. The presence of the large muon number in neutron stars is attributed to the fact +that when the electron chemical potential, µe, is larger than the muon mass µe > mµ, it +becomes energetically favorable for relativistic electrons at the Fermi surface to decay into +muons via e− → µ− + ¯νµ + νe. Moreover, the muonic beta-decay n → p + µ− + ¯νµ and inverse +beta-decay p + µ− → n + νµ reactions become energetically favorable, while the muon decay +µ− → e− + ¯νe + νµ is forbidden by Fermi statistics. +Being motivated by the neutron star muonic content, we consider neutrino pair emission by +pulsar binaries via the following two types of BSM mediators: +• U(1)Lµ−Lτ massive gauge boson with +L ⊃ gAα (¯µγαµ − ¯τγατ + ¯νµγανµ − ¯ντγαντ) , +(4.15) +• Massive muonophilic scalar with +L ⊃ gφ¯µµ + g′φ¯νµνµ . +(4.16) +It is known that at least two of the SM neutrinos are massive, while the third neutrino can +be very light or massless. This means that only one neutrino mass eigenstate can be radiated +in the two scenarios we consider here. A realistic treatment of neutrino emission would include +insertions of the corresponding PMNS matrix elements [44], resulting in an additional factor of +order one. Since we already neglecting an O(1) factor coming from the estimate of the muon +number density in the neutron stars, we also ignore any PMNS factors in the rest of this section. +Note also that in a theory with general couplings to the left and right-handed neutrinos, +i.e., gAα¯νγα(cV − cAγ5)ν, the results for the power loss are qualitatively similar. Moreover, in +the case of massless neutrinos, the power loss for the case of the general coupling is the same +as the power loss for the case of purely vectorial coupling up to g2 → g2(c2 +A + c2 +V ) replacement. +This is why in what follows, for simplicity, we consider the case of the vectorial coupling only. +These two BSM models imply the possibility for the neutrino pair radiation at rates enhanced +compared to the SM. Our results from Eqs. (4.8) and (4.9) thus can be used to set bounds on +the coupling constants and masses of the new bosons. +20 + +The presence of the muonophilic new physics, however, not only alters the radiation patterns +of pulsar binaries, but it also has important implications for the neutron star’s equation of state. +In particular, the presence of a repulsive (vector) or attractive (scalar) interaction between +muons could affect the muon number, which depends on the coupling g to the new physics. In +the following, we write the muon number as N(µ, g) to keep the dependence on g explicit. +The number of muons becomes g-dependent as the interactions change the muon chemical +potential. The muon interaction due to the Lµ − Lτ vector boson is repulsive, and thus the +chemical potential is increased compared to its SM value by ε ∼ g2N(µ, g)/R, where R is +the radius of the neutron star the boson mass is neglected. When the coupling g is small, +such that ε ≪ mµ, the effect of the new interaction is insignificant, and the number of muons +is approximately given by its value in the limit of no interaction N(µ, g = 0). +When the +interaction is strong, such that ε ≫ mµ, it becomes energetically less favorable to have muons +inside the neutron star and thus N(µ, g) < N(µ, g = 0). +Similar reasoning applies to the case of the scalar mediator. The only difference is the sign +of the interaction. In the scalar case, the interaction between muons is attractive. Thus the +muon chemical potential is decreased by ε. This leads to the increase of the muon number +for larger couplings N(µ, g) > N(µ, g = 0). In both cases, the change from the regime when +N(µ, g) ≈ N(µ, g = 0) to the situation when the interaction starts to affect the muon number +happens for couplings such that ε ∼ mµ, or numerically g ∼ 10−18 for a typical neutron star [5]. +However, in what follows, we ignore the effect of the new physics on the muon number. +Everywhere in our analysis, we use the muon number in the limit of no new physics interaction, +that is we set N(µ) = N(µ, g = 0) ∼ 1055 [40–43]. In principle, g-independence of muon +number can be achieved in models with both vector and scalar mediators with fine-tuned +coupling constants such that the repulsive and attractive interactions cancel each other. +To apply Eqs. (4.8) and (4.9), we define Nb(µ) and Nb(¯µ) as the number of muons and +antimuons respectively in neutron star labeled by b = 1, 2. Then, as there are almost no tau +leptons in neutron stars, Qb = Nb(µ) − Nb(¯µ) is the total charge of the neutron star under the +Lµ−Lτ gauge symmetry, and Nb = Nb(µ)+Nb(¯µ) is the total number of muons and anti-muons +in the star. Additionally, since Nb(¯µ) ≈ 0, we have Qb ≈ Nb. +The energy lost through radiation in a binary star system can be directly probed by measur- +ing the decay of the orbital period. Assuming that the attractive gravitational force between +the two stars is such that their orbits stay Keplerian, the decay rate of the period of revolution +Tb is related directly to the energy lost via radiation [6]: +˙Tb = −6πa5/2G−3/2 +N +(m1m2)−1(m1 + m2)−1/2 × Ploss, +(4.17) +21 + +where ˙Tb is the time derivative of the binary period, GN is the gravitational constant, m1 and m2 +are the masses of the stars in the binary system, a is the semi-major axis of the elliptical orbit, +and Ploss is the total power radiated. The decay of the period per unit of time is dimensionless +and is measured experimentally. +GW emission is the dominant source of power loss in a binary star system. Assuming that +the GW emission and neutrino pair emission are the only sources of energy loss, we have +Ploss = PGW + P¯νν, +(4.18) +where P¯νν is the power loss due to the neutrino pair radiation and PGW is the power loss +due to GW emission, which, to the leading order, is given by the GW quadrupole radiation +formula [38], +P GW +loss = 32 +5 GΩ6M 2a4(1 − e2)−7/2 +� +1 + 73 +24e2 + 37 +96e4 +� +, +(4.19) +where M is the reduced mass of the system, as defined in Eq. (4.7). The binary period decay +˙Tb thus can be written as a sum of two contributions, +˙Tb = ˙TGW + ˙T¯νν . +(4.20) +We next introduce the period decay ratio R as the ratio of the measured period decay to +the theoretical prediction of the period decay due to GW radiation, +R = +˙Tb +˙TGW += 1 + +˙T¯νν +˙TGW +. +(4.21) +We use the measured value of R to set 2σ limits on the masses and couplings of the BSM +mediators of neutrino pair radiation as +˙T¯νν +˙TGW +≤ (R − 1) + 2σ . +(4.22) +The resulting constraints on the parameter space (g, mA) and (g, mφ) that we derive from +the period decay data for the Hulse-Taylor binary and PSR J1738+033 are shown in Fig. 4. +When deriving the constraints, we use Qb = Nb = 1055 with b = 1, 2 and qν = 1. For the gauge +boson mediator (left panel), we calculate the period decay due to neutrino pair emission, ˙T¯νν, +using Eqs. (4.8) and (4.17). As we take all three neutrinos to be massless, and as Lµ −Lτ boson +couples to two neutrino types, there is an extra factor of 2 in Eq. (4.8). Similarly, for the case +of the scalar mediator (right panel), we use Eqs. (4.9) and (4.17). As there is no symmetry +that requires equality of g and g′ in the case of the scalar mediator, we present our results for +the scalar case in the (g, mφ) plane for four different values of g′ that vary from 10−7 to 10−1. +First, let us discuss the left panel of Fig. 4, which shows constraints on the mass and coupling +of the gauge boson. For the PSR J1738+0333 (red line), whose orbit is very close to circular, the +22 + +��� �����+�� +��� �����+���� +��-�� +��-�� +��-�� +��-�� +��-�� +��-�� +��-�� +��-�� +��-�� +��-�� +��-� +��-� +PSR J1738+0333 +10-22 +10-20 +10-18 +10-16 +10-20 +10-18 +10-16 +10-14 +10-12 +10-10 +10-8 +10-6 +FIG. 4. Left: Constraints on g vs mA from the highly eccentric PSR B1913+16 (Hulse-Taylor) Bounds +from the neutrino pair radiation (solid) and vector boson radiation (dashed) are shown such that the +region above the curves is excluded by the measurements of the period decay. The system parameters +are taken from Table I. Right: Constraints on g vs mφ from PSR J1738+033. The dashed gray line +corresponds to the bound set by the emission of the scalar boson only, while the solid lines show the +bounds from including a coupling g′ to the neutrinos. +effect of neutrino pair radiation becomes significant for the mediator masses greater than the +second harmonic frequency, mA > 2Ω. For the highly eccentric Hulse-Taylor binary, off-shell +radiation dominates for mA > 85Ω. In the region mA > 2Ω (mA > 85Ω) for PSR J1738+0333 +(Hulse-Taylor binary), the boundary of the excluded region is approximately quadratic in the +mediator mass. This is in stark contrast with the case of the on-shell boson emission discussed +in Ref. [3, 5, 6], where the boundary of the excluded region jumps in steps at mA = nΩ, with +n being an integer. For comparison, the dashed lines in Fig. 4 show the bounds due to the +on-shell boson radiation. +Finally, we comment on the right panel of Fig. 4, which shows the constraints on the mass +mφ and coupling g for different values of g′ in the case of the scalar mediated radiation. We +only demonstrate the constraints for PSR J1738+0333; the results for the Hulse-Taylor binary +are qualitatively the same. Depending on the value of g′ the off-shell scalar radiation starts +to dominate for mφ > Ω (g′ ≳ 10−4) or mφ > 2Ω (g′ ≲ 10−8). As one can see from the plot, +g′ = 10−1 provides the strongest bound. +We conclude this section by noting that we do not perform a detailed analysis of the bounds +on muonophilic light states. We only remark that very strong bounds on light states are derived +23 + +from fifth force searches. Most of these bounds do not apply in our case as these experiments +are done using materials made out of protons, neutrons, and electrons. +V. +CONCLUSION +It is well known that fermion pairs can behave as bosons in several circumstances. In this +work, we show that fermion pairs can also constitute classical radiation just like bosonic states +do. We use this understanding to derive the generalization of the Larmor formula for the case +of the fermion pair emission. +Being motivated by the potential of applying fermion pair radiation to astrophysical objects, +we consider the case of classical sources following elliptical orbits. The most interesting regime +of fermion pair radiation is when the mediator is off-shell, which takes place when the mass of +the mediator is much smaller than the frequency of the periodic motion of the source. In this +regime, the fermion pair emission takes over from on-shell boson production. This opens up a +window into a broader region of parameter space for various models that allow for the fermion +pair radiation by classical sources. +Subsequently, we apply our results to neutrino-antineutrino emission by two pulsar binary +systems PSR B1913+16 and PSR J1738+0333. Neutrino pair emission by binary systems is +highly suppressed in the SM compared to GW radiation, but can be significantly enhanced in +various BSM scenarios. In particular, we consider two possibilities: light muonophilic vector +and scalar mediators that couple to the SM neutrinos. Using period decay data for the two +binary systems, we derive bounds on the parameters of the two models. While we did not +perform a comprehensive study of the relevance of these bounds, the key point is that they +provide a demonstration of the fact that fermion pair radiation can be used to enhance BSM +probes using astrophysical data. +There are several future directions to go from here. Here are a few that we find particularly +interesting: +• A thorough and detailed study of the bounds that we find on specific models is called for. +This, however, is complicated by the large uncertainties that come from the estimates on +the neutron star constituents. In particular, new physics interactions alter the equation +of state of a neutron star and, currently, there is no precise quantitative understanding +of how this affects its content. +• It also would be interesting to see if we can find more systems to which our results can +be applied. In particular, exotic astrophysical systems and exotic types of new physics +models. +24 + +• In this work, we only consider fermion pair radiation; however, the results can be modified +to also include bosonic pair radiation. All that needs to be done is to calculate the relevant +matrix elements. It is expected to result in a different kinematic dependence. +We conclude with the main message of our paper: If nature includes new light states, fermion +pair radiation can be one more tool in our toolbox to probe them. +ACKNOWLEDGEMENTS +We are grateful to Kfir Blum, Jeff Dror, Toby Opferkuch, Nadav Outmezguine, Ira Rothstein, +Ryosuke Sato, and Kohsaku Tobioka for useful discussions. The work of YG is supported in +part by the NSF grant PHY1316222. The research of WT is supported by NSF Grant No. +PHY-2013052. +Appendix A: Derivation of the power loss formula +We present below an explicit derivation of the power loss formula for the fermion pair +radiation by a point-like classical object on an elliptical orbit. We perform the calculation +separately for the case of vector and scalar mediators. In our calculation, we follow closely the +analysis in Ref. [6]. +1. +The case of a vector boson mediator +The power loss is a sum over different harmonics, as given by Eqs. (2.7) and (2.8). The matrix +element, at leading order, for a vector boson mediator, is given by Eq. (2.10). It includes the +Fourier Transform of the classical current Jµ +cl(x) defined in Eq. (2.1). We rewrite it here for +convenience: +Mn(s1, s2) = g2Qψ ¯u(k1, s1)γµv(k2, s2) i(−ηµν + (k1 + k2)µ(k1 + k2)ν/m2 +A) +(k1 + k2)2 − m2 +A + imAΓA +Jν +cl(Ωn) , +(A1) +where ηµν is the Minkowski metric tensor. Note that the contribution from the (k1 + k2)µ(k1 + +k2)ν term vanishes by means of the Dirac equation since the fermions are on-shell, that is, +¯u(/k1 + /k2)v = ¯u(mψ − mψ)v = 0. +(A2) +Squaring the amplitudes corresponding to different harmonics and summing over spins, we +25 + +find +|Mn|2 = +� +s1,s2 +|Mn|2 = +g4Q2 +ψ +((k1 + k2)2 − m2 +A)2 + m2 +AΓ2 +A +Jµ +cl(Ωn)J∗ν +cl (Ωn) Tr [(/k1 + mν)γµ(/k2 − mν)γν] += +4g4Q2 +ψ +((k1 + k2)2 − m2 +A)2 + m2 +AΓ2 +A +Jµ +cl(Ωn)J∗ν +cl (Ωn) +� +k1µk2ν + k1νk2µ − 1 +2(k1 + k2)2ηµν +� +.(A3) +Finally, we are ready to write the expression for the rate of energy loss due to ψ ¯ψ emission +at harmonic n by the classical source as +Pn = +�dE +dt +� +n += +� +Ωn dΓn += Ωn +� +d3k1 +(2π)3(2ω1) +d3k2 +(2π)3(2ω2)(2π)δ(Ωn − ω1 − ω2)|Mn|2 += Ωn +� +dΦ1dΦ2 +|k1|dω1 +2(2π)3 +|k2|dω2 +2(2π)3 (2π)δ(Ωn − ω1 − ω2)|Mn|2 , +(A4) +where |k1,2| = +� +ω2 +1,2 − m2 +ψ, we used Ωn = ω1 + ω2 for the total energy carried away by the +fermion pair, dΦ1,2 are the differential elements of solid angles in the fermion’s direction of +flight, and +��Mn +��2 is given in Eq. (A3). The total power radiated is found by summing over all +kinematically allowed harmonics: +P = +� +n +Pn. +(A5) +To calculate the power radiated in fermion pairs by a point-like source in an elliptical orbit, +we need to evaluate the integrals in Eq. (A4), after substituting in the explicit form of Jµ +cl(Ωn) +in Eq. (A3). Using Eqs. (2.1) and (2.4), we find the Fourier Transform Jµ +cl(Ωn) as: +Ji +cl(Ωn) = aΩQji +n, +J0 +cl(Ωn) = aΩQ +�jn · p +nΩ +� +, +(A6) +where the 3-vector jn is defined as +jn = +� +−iJ′ +n(ne), +√ +1 − e2 +e +Jn(ne), 0 +� +, +(A7) +with Jn(z) denoting a Bessel function, and p = k1 + k2. +The terms in the numerator of |M|2 in Eq. (A3), are then given by +(Jµ +cl(Ωn)k1µ) (Jν∗ +cl (Ωn)k2ν) = a2Ω2Q2ji +njj∗ +n +� ω1ω2 +(nΩ)2pipj − ω1 +nΩpikj +2 − ω2 +nΩki +1pj + ki +1kj +2 +� +, +(A8) +and +|Jµ +cl(Ωn)|2 = |J0 +cl(Ωn)|2 − |Jcl(Ωn)|2 = a2Ω2Q2ji +njj∗ +n +� pipj +(Ωn)2 − δij +� +, +(A9) +where we used Ωn = nΩ. +Note that all quantities above are 3-vectors with Latin indices +i = 1, 2, 3, and a sum over i and j is implicit. The expression for (Jµ +cl(Ωn)k2µ) (Jν∗ +cl (Ωn)k1ν) is +obtained from Eq. (A8) via complex conjugation. +26 + +Next we note that the denominator of |Mn|2, see Eq. (A3), depends only on mA, ΓA, ω1,2, the +magnitudes |k1,2| and the relative angle between the two momenta k1, and k2 that we denote +as γ. Because of this, it is convenient to perform the change of coordinates in the integral in +Eq. (A4) from the integration over the solid angles dΦ1dΦ2 to the integration over dΦ1dΦr +2 +where the solid angle of the second neutrino is measured relative to the direction of k1, hence +the super index r. (Equivalently, one can also choose to integrate over dΦr +1dΦ2.) The Jacobian +of this coordinate change is unity since the transformation is simply a coordinate rotation, and +thus +dΦ1dΦ2 = dΦ1dΦr +2. +(A10) +Defining +dΦb = sin θbdθbdφb, +dΦr +2 = sin γdγdδ, +b = 1, 2 , +(A11) +we find the following relations between the two sets of integration variables +cos γ = cos θ1 cos θ2 + sin θ1 sin θ2 cos (φ2 − φ1) , +sin δ = sin θ2 sin (φ2 − φ1) +sin γ +. +(A12) +Since, out of all the angular variables, the denominator only depends on the relative angle γ, +the integrals over θ1, φ1 and δ can be taken easily using the following relations +� +dΦ1dΦ2ki +akj +a = +� +dΦ1dΦr +2ki +akj +a = δij 8π2 +3 k2 +a +� +sin γdγ, +� +dΦ1dΦ2ki +1kj +2 = +� +dΦ1dΦr +1ki +1kj +2 = δij 8π2 +3 (k1 · k2) +� +sin γdγ, +� +dΦ1dΦ2 = +� +dΦ1dΦr +2 = 8π2 +� +sin γdγ . +(A13) +Using this and the results of Eqs. (A8) and (A9), we perform the integration over θ1, φ1 and δ +in Eq. (A4), and find the following expression for the power radiated in harmonic n, +Pn = g4 (nΩ) +12π3 a2Ω2Q2 +ψQ2 |jn|2 +� +δ(nΩ − ω1 − ω2) +((k1 + k2)2 − m2 +A)2 + m2 +AΓ2 +A +× +� +−1 +2 (k1 + k2)2 � +(k1 + k2)2 / +� +(nΩ)2 − 3 +�� ++ 2 ω1ω2 +(nΩ)2 (k1 + k2)2 +−2 ω1 +nΩ +� +k2 +2 + k1 · k2 +� +− 2 ω2 +nΩ +� +k2 +1 + k1 · k2 +� ++ 2k1 · k2 +� +× +ω1ω2 +� +1 − m2 +ψ +ω2 +1 +�1/2 � +1 − m2 +ψ +ω2 +2 +�1/2 +sin γ dγdω1dω2 , +(A14) +where the only integrals left are the integrals over γ, ω1 and ω2. +Next, we introduce the following dimensionless variables and parameters +x1 = ω1 +Ω , +x2 = ω2 +Ω , +nψ = mψ +Ω , +nA = mA +Ω , +nΓ = ΓA +Ω . +(A15) +27 + +Performing the change of variables in Eq. (A14) from (ω1, ω2) to (x1, x2), we rewrite the ex- +pression for the power radiated in harmonic n as follows: +Pn = +g4 +12π3a2Ω4Q2 +ψ|jn|2 +� +sin γ dγ dx1 dx2 δ(n − x1 − x2) F(cos γ, x1, x2) . +(A16) +Upon taking the integral over x2 and performing the replacement x1 → x, we obtain +Pn = +g4 +12π3a2Ω4Q2 +ψQ2|jn|2 +� n−nψ +nψ +dx +� 1 +−1 +d(cos γ) F(cos γ, x) , +(A17) +where function F(cos γ, x) is given by +F(cos γ, x) = b(x) +2n +1 +2b2(x) cos2 γ + b(x)c(x) cos γ + d(x) +(a(x) − b(x) cos γ)2 + g2 +, +(A18) +with +a(x) = 2n2 +ψ + 2x(n − x) − n2 +A , +b(x) = 2 +� +x2 − n2 +ψ +� +(n − x)2 − n2 +ψ , +c(x) = − +� +n2 + 2n2 +ψ +� +, +d(x) = 2(x(n3 − 2n2x + 2nx2 − x3) + 2n2n2 +ψ + n4 +ψ), +g2 = n2 +An2 +Γ . +(A19) +The variable x here is the ratio of the energy of one of the fermions to the fundamental oscillation +frequency. It can be at least nψ or at most n − nψ, hence the limits on the integral. Also note +that F also depends on the parameters of the problem namely nA, nψ, nΓ defined in Eq. (A15), +but we do not write them explicitly for brevity. Lastly, note that the γ-dependence of the +numerator of function F is through a term quadratic in cos γ and a term linear in cos γ. This +behavior is attributed to the theory that we pick – renormalizable theories such as in the case +considered here would only contribute at most two powers of momentum in the matrix element, +leading to a cos γ dependence that is at most quadratic. However non-renormalizable theories +have more momenta in the matrix element, and will give us a different cos γ dependence in the +F. +Now, we define +F A(x) ≡ F A(n, x, nψ, nA, nΓ) = +� 1 +−1 +d (cos γ) F (cos γ, x, n) , +(A20) +where the superscript A denotes the vector boson mediator. +The integral over cos γ can be taken analytically. Then, we find that the function F A(x), +has the form: +F A(x) = F A +0 (x) + F A +1 (x) +nMnΓ +� +tan−1 +�a(x) + b(x) +nMnΓ +� +− tan−1 +�a(x) − b(x) +nMnΓ +�� ++ F A +2 (x) tanh−1 +� +2a(x)b(x) +a(x)2 + b(x)2 + n2 +Mn2 +Γ +� +, +(A21) +28 + +with: +F A +0 (x) = b(x)/2n , +F A +1 (x) = 1 +4n +� +n4 +A + 4n2n2 +ψ − n2 +An2 +Γ + 2n2 +An2 − 4nxn2 +A + 4x2n2 +A +� +, +F A +2 (x) = 1 +2n +� +n2 +A + n2 − 2nx + 2x2� +. +(A22) +Consequently, the power loss formula of each mode with n > 2nψ becomes +Pn = 2g4Q2 +ψQ2 +3(2π)3 a2Ω4 +� +J′ +n(ne)2 + 1 − e2 +e2 +Jn(ne)2 +� � n−nψ +nψ +dxF A(x), +(A23) +which gives us Eq. (2.18) for the case M = A, where we define for mediator M +BM +n (nM, nν, nΓ) ≡ +� +J′ +n(ne)2 + 1 − e2 +e2 +Jn(ne)2 +� � n−nψ +nψ +dx F M(x, n, nM, nν, nΓ), +(A24) +where Jn(z) is a Bessel function of order n in the variable z. +2. +The case of the scalar mediator +The derivation for the power loss in the scalar mediator is similar to the vector case, but the +matrix element is different, as shown in Eq. (2.14). This matrix element contains the number +density ρcl(x) of source particles, instead of a current. As such, the difference in the calculation +in this case comes from the calculation of the squared matrix element, which in this case, is +given by: +� +s1,s2 +|Mn(s1, s2)|2 = +g2g′2 +((k1 + k2)2 − m2 +φ)2 + m2 +φΓ2 +φ +Tr(( /k1 + mψ)( /k2 − mψ))|ρcl(Ωn)|2 += +4g2g′2 +((k1 + k2)2 − m2 +φ)2 + m2 +φΓ2 +φ +(k1 · k2 − m2 +ψ)|ρcl(Ωn)|2 ]. +(A25) +The power loss is again given by Eq. (A4). +Using Eqs. (2.2) and (2.4), we find the Fourier Transform ρµ +cl(Ωn) as: +ρ0 +cl(Ωn) = aΩN +�jn · p +nΩ +� +, +(A26) +where, like in the vector case, we define the 3-vector ji +n as follows: +jn = +� +−iJ′ +n(ne), +√ +1 − e2 +e +Jn(ne), 0 +� +, +(A27) +with Jn(z) denoting a Bessel function, amd p = k1 + k2. +After performing all the steps analogous to Eqns. (A4)–(A20) in the previous section, i.e, +after performing the angular integration, we get: +Pn = g2g′2 +12π3 a2Ω4N 2|jn|2 +� n−nψ +nψ +dx +� 1 +−1 +d cos γ F(cos γ, x) , +(A28) +29 + +where function F(cos γ, x) is given by +F(cos γ, x) = −b(x) +2n +1 +2b2(x) cos2 γ + b(x)c(x) cos γ + d(x) +(a(x) − b(x) cos γ)2 + g2 +, +(A29) +with +a(x) = 2n2 +ψ + 2x(n − x) − n2 +φ , +b(x) = 2 +� +x2 − n2 +ψ +� +(n − x)2 − n2 +ψ , +c(x) = (n − 2x)2 +2 +, +d(x) = (n2 +ψ − nx + x2)(n2 − 2n2 +ψ − 2nx + 2x2), +g2 = n2 +φn2 +Γ . +(A30) +Like before, we define +F φ(x) ≡ F φ(n, x, nψ, nφ, nΓ) = +� 1 +−1 +d (cos γ) F (cos γ, x, n) , +(A31) +where the superscript φ denotes the scalar mediator. +The integral over cos γ can be taken analytically to find a form for F φ: +F φ(x) = F φ +0 (x) + F φ +1 (x) +nMnΓ +� +tan−1 +�a(x) + b(x) +nMnΓ +� +− tan−1 +�a(x) − b(x) +nMnΓ +�� ++ F φ +2 (x) tanh−1 +� +2a(x)b(x) +a(x)2 + b(x)2 + n2 +Mn2 +Γ +� +, +(A32) +with: +F φ +0 (x) = −b(x)/2n , +F φ +1 (x) = 1 +4n +� +n2 +φn2 +Γ + (n2 − n2 +φ)(n2 +φ − 4n2 +ν) +� +, +F φ +2 (x) = 1 +4n +� +n2 + 4n2 +ν − 2n2 +φ +� +. +(A33) +Consequently, the power loss formula of each mode with n > 2nψ becomes +Pn = 2g2g′2 +3(2π)3a2Ω4N 2 +� +J′ +n(ne)2 + 1 − e2 +e2 +Jn(ne)2 +� � n−nψ +nψ +dxF φ(x), +(A34) +which gives us Eq. (2.19) for the case M = φ +P φ +n = g2g′2 +12π3 a2Ω4 +�N1 +m1 +− N2 +m2 +�2 +Bφ +n(nA, nν, nΓ). +(A35) +We find that the form of the function F M is general for the two types of mediators, the +difference lying in the explicit forms of the functions F M +0 , F M +1 +and F M +2 . This is due to the fact +that the cos γ dependence of the function F is the same in both cases, as in both cases, the +theory considered is a renormalizable one. As we explained in the previous sub-section, this +30 + +general form of F M is not what we will have when we consider non-renormalizable theories that +give us higher powers of momenta in the numerator of F. +[1] J. S. b. Larmor, Philosophical Magazine Series 1 44, 503 (1897). +[2] J. D. Jackson, Classical Electrodynamics (Wiley, 1998). +[3] D. Krause, H. T. Kloor, and E. Fischbach, Phys. Rev. D 49, 6892 (1994). +[4] S. Mohanty and P. Kumar Panda, Phys. Rev. D 53, 5723 (1996), arXiv:hep-ph/9403205. +[5] J. A. Dror, R. Laha, +and T. Opferkuch, Phys. Rev. D 102, 023005 (2020), arXiv:1909.12845 +[hep-ph]. +[6] T. Kumar Poddar, S. Mohanty, and S. Jana, Phys. Rev. D 100, 123023 (2019), arXiv:1908.09732 +[hep-ph]. +[7] J. Huang, M. C. Johnson, L. Sagunski, M. Sakellariadou, and J. Zhang, Phys. Rev. D 99, 063013 +(2019), arXiv:1807.02133 [hep-ph]. +[8] T. Kumar Poddar, S. Mohanty, and S. Jana, Phys. Rev. D 101, 083007 (2020), arXiv:1906.00666 +[hep-ph]. +[9] A. Hook and J. Huang, JHEP 06, 036 (2018), arXiv:1708.08464 [hep-ph]. +[10] R. Foot, Mod. Phys. Lett. A 6, 527 (1991). +[11] X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D 44, 2118 (1991). +[12] R. Foot, X. G. He, H. Lew, +and R. R. Volkas, Phys. Rev. D 50, 4571 (1994), arXiv:hep- +ph/9401250. +[13] J. Heeck and W. Rodejohann, Phys. Rev. D 84, 075007 (2011), arXiv:1107.5238 [hep-ph]. +[14] J. Kopp, R. Laha, T. Opferkuch, +and W. Shepherd, JHEP 11, 096 (2018), arXiv:1807.02527 +[hep-ph]. +[15] S. Alexander, E. McDonough, R. Sims, and N. Yunes, Class. Quant. Grav. 35, 235012 (2018), +arXiv:1808.05286 [gr-qc]. +[16] H. G. Choi and S. Jung, Phys. Rev. D 99, 015013 (2019), arXiv:1810.01421 [hep-ph]. +[17] M. Fabbrichesi and A. Urbano, JCAP 06, 007 (2020), arXiv:1902.07914 [hep-ph]. +[18] B. C. Seymour and K. Yagi, Class. Quant. Grav. 37, 145008 (2020), arXiv:1908.03353 [gr-qc]. +[19] G. Feinberg and J. Sucher, Phys. Rev. 166, 1638 (1968). +[20] S. D. H. Hsu and P. Sikivie, Phys. Rev. D 49, 4951 (1994), arXiv:hep-ph/9211301. +[21] M. Ghosh, Y. Grossman, and W. Tangarife, Phys. Rev. D 101, 116006 (2020), arXiv:1912.09444 +[hep-ph]. +[22] M. Ghosh, Y. Grossman, W. Tangarife, X.-J. Xu, and B. Yu, (2022), arXiv:2209.07082 [hep-ph]. +31 + +[23] W. van Straten, M. Bailes, M. C. Britton, S. R. Kulkarni, S. B. Anderson, R. N. Manchester, +and J. Sarkissian, Nature 412, 158 (2001), arXiv:astro-ph/0108254. +[24] M. Kramer et al., Science 314, 97 (2006), arXiv:astro-ph/0609417. +[25] I. H. Stairs, S. E. Thorsett, J. H. Taylor, +and A. Wolszczan, Astrophys. J. 581, 501 (2002), +arXiv:astro-ph/0208357. +[26] R. M. Shannon, S. Johnston, +and R. N. Manchester, Mon. Not. Roy. Astron. Soc. 437, 3255 +(2014), arXiv:1311.0588 [astro-ph.SR]. +[27] J. Antoniadis et al., Science 340, 6131 (2013), arXiv:1304.6875 [astro-ph.HE]. +[28] N. D. R. Bhat, M. Bailes, and J. P. W. Verbiest, Phys. Rev. D 77, 124017 (2008), arXiv:0804.0956 +[astro-ph]. +[29] P. C. C. Freire, N. Wex, G. Esposito-Farese, J. P. W. Verbiest, M. Bailes, B. A. Jacoby, +M. Kramer, I. H. Stairs, J. Antoniadis, and G. H. Janssen, Mon. Not. Roy. Astron. Soc. 423, +3328 (2012), arXiv:1205.1450 [astro-ph.GA]. +[30] R. D. Ferdman et al., Mon. Not. Roy. Astron. Soc. 443, 2183 (2014), arXiv:1406.5507 [astro- +ph.SR]. +[31] J. van Leeuwen et al., Astrophys. J. 798, 118 (2015), arXiv:1411.1518 [astro-ph.SR]. +[32] B. A. Jacoby, P. B. Cameron, F. A. Jenet, S. B. Anderson, R. N. Murty, and S. R. Kulkarni, +Astrophys. J. Lett. 644, L113 (2006), arXiv:astro-ph/0605375. +[33] H. Davoudiasl and P. B. Denton, Phys. Rev. Lett. 123, 021102 (2019), arXiv:1904.09242 [astro- +ph.CO]. +[34] R. A. Hulse and J. H. Taylor, Astrophys. J. Lett. 195, L51 (1975). +[35] J. H. Taylor and J. M. Weisberg, Astrophys. J. 253, 908 (1982). +[36] J. M. Weisberg and Y. Huang, The Astrophysical Journal 829, 55 (2016). +[37] M. Kilic, J. Hermes, A. Gianninas, and W. R. Brown, Monthly Notices of the Royal Astronomical +Society: Letters 446, L26 (2015). +[38] P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963). +[39] D. G. Yakovlev, A. D. Kaminker, O. Y. Gnedin, +and P. Haensel, Phys. Rept. 354, 1 (2001), +arXiv:astro-ph/0012122. +[40] R. Garani and J. Heeck, Phys. Rev. D 100, 035039 (2019), arXiv:1906.10145 [hep-ph]. +[41] J. M. Pearson, N. Chamel, A. Y. Potekhin, A. F. Fantina, C. Ducoin, A. K. Dutta, and S. Goriely, +Mon. Not. Roy. Astron. Soc. 481, 2994 (2018), [Erratum: Mon.Not.Roy.Astron.Soc. 486, 768 +(2019)], arXiv:1903.04981 [astro-ph.HE]. +[42] I. Harry and T. Hinderer, Class. Quant. Grav. 35, 145010 (2018), arXiv:1801.09972 [gr-qc]. +[43] F. Zhang and L. Chen, Chin. Phys. Lett. 18, 142 (2001), arXiv:nucl-th/0011017. +32 + +[44] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28, 870 (1962). +33 + diff --git a/OtAzT4oBgHgl3EQfWfy0/content/tmp_files/load_file.txt b/OtAzT4oBgHgl3EQfWfy0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e65aca9c005625ab0687d449e7fc91d7ab3c380 --- /dev/null +++ b/OtAzT4oBgHgl3EQfWfy0/content/tmp_files/load_file.txt @@ -0,0 +1,1138 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf,len=1137 +page_content='Fermion pair radiation from accelerating classical systems Margarita Gavrilova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ∗ Mitrajyoti Ghosh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' † Yuval Grossman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ‡ Walter Tangarife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' § and Tien-Hsueh Tsai3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ¶ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' LEPP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' NY 14853,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' USA 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Loyola University Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' IL 60660,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' National Tsing Hua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Hsinchu 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Taiwan Abstract Accelerating classical systems that couple to a fermion-antifermion pair at the microscopic level can radiate pairs of fermions and lose energy in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this work, we derive the generalization of the Larmor formula for fermion pair radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We focus on the case of a point-like classical source in an elliptical orbit that emits fermions through vector and scalar mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Ultra-light fermion emission from such systems becomes relevant when the mass of the mediator is larger than the frequency of the periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This enables us to probe regions of the parameter space that are inaccessible in on- shell bosonic radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We apply our results to pulsar binaries with mediators that couple to muons and neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using current data on binary period decays, we extract bounds on the parameters of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ∗ mg2333@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='edu † mg2338@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='edu ‡ yg73@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='edu § wtangarife@luc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='edu ¶ s106022901@m106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='nthu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='tw 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='01303v1 [hep-ph] 3 Jan 2023 CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fermion pair radiation by a point-like object 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' General formalism 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Power loss formulae 7 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Discussion of the power-loss formula 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' General features of the power-loss formula 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Asymptotic behavior for the case of circular orbits 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fermion-pair radiation in the SM 14 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fermion pair radiation by pulsar binaries 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Pulsar binaries as a classical source 16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Neutrino pair radiation by pulsar binaries in the SM 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' New physics constraints from the neutrino pair radiation by pulsar binaries 19 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Conclusion 24 Acknowledgements 25 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Derivation of the power loss formula 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The case of a vector boson mediator 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The case of the scalar mediator 29 References 31 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' INTRODUCTION Radiation by a classical system is a well-known phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Probably the most familiar example is the radiation of electromagnetic waves by an accelerating point-like particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss, in this case, is calculated using the famous Larmor formula [1, 2], which, in natural units, is given by Ploss = 1 6πq2a2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) where q is the electric charge of the particle and a is its acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The Larmor formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) has been also generalized to other types of radiation by accelerating classical sources, such as radiation of massive vector and scalar bosons [3–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2 Generalizations of the Larmor formula to exotic types of radiation are motivated, among other things, by their applications to new physics searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The basic idea is that if a new physics radiation accompanies an accelerating astrophysical object, the power loss effect can be enhanced thanks to the large number density of an object, even if the coupling between the new physics and the Standard Model (SM) is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This expected enhancement can be used to obtain constraints on various new physics scenarios using astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' One example is the radiation of an ultra-light gauged Lµ − Lτ vector boson [10–13] by pulsar binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The measurement of the orbital period decay, when compared to the prediction due to the gravitational wave (GW) radiation, was used to constrain the mass of the Lµ −Lτ gauge boson and its couplings to the SM [5, 6, 14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this paper, we extend the previous work and derive the generalization of the Larmor formula to the case of fermion-antifermion pair radiation by classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The interest in this scenario is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' First, it is interesting theoretically since it is one more example of a case where a fermion pair behaves like a boson (other cases are Cooper pairs in superconductors and the mediation of forces between objects via 2-fermion forces [19–22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus we can study the coherent radiation of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The key point is that single-fermion emission changes the source and thus can not be treated classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fermion-pair emission, however, can take place without changing any quantum degrees of freedom of the emitting system (such as spin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, fermion-pair emission (or emission of any even number of fermions) can be treated classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The second aspect is phenomenological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, we consider radiation by astrophysical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the SM, as we show below, the effect of the fermion pair radiation is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In beyond the SM (BSM) theories, however, such processes can be enhanced, enabling us to probe various new physics scenarios using astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, fermion- pair radiation can become significant in models with a new light mediator (a vector or scalar boson) that couples to some light fermionic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' These fermionic degrees of freedom can be the well-known neutrinos or some new BSM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The effects of this radiation can become relevant when the mediator is too heavy to be produced on-shell, but the fermions are much lighter and can be radiated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Since fermion pairs can be produced via off-shell mediators, the fermion pair radiation can be used to probe broader regions of the parameter space of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As a particular application of our result for the fermion-pair radiation, we consider two models: (i) a model with a gauged Lµ − Lτ symmetry and (ii) a model with a muonophilic scalar that couples to the muon and the muon neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We study the implications of these scenarios for the power loss by pulsar binaries and compare our results to the cases of on-shell vector boson radiation [3, 5, 6] and on-shell scalar radiation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A stark difference is that 3 the emission of neutrino pairs in a particular harmonic mode of the periodic system is not kinematically forbidden when the mediator mass becomes larger than the frequency of that particular mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the case of on-shell bosonic radiation, radiation from a harmonic mode is cut off once the boson mass exceeds the frequency of that particular mode due to energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We use the available period decay data for pulsar binaries to demonstrate how neutrino pair radiation, mediated by BSM bosons, can be used to probe a broader parameter space than the on-shell boson emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We, however, do not perform a comprehensive study of other bounds on the models we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' II, we discuss the general machinery required for calculating fermion-pair radiation from a classical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' III, we discuss the main fea- tures of the power-loss formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' IV, we perform the computation for the particular case where the classical system is a binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We then use available data to place constraints on the parameters of a few models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The detailed calculations are shown in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' FERMION PAIR RADIATION BY A POINT-LIKE OBJECT In this section, we outline the calculation of the power of fermion-pair radiation that accom- panies a non-relativistic point-like object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We formulate a general approach to the derivation of the power loss formula with a focus on the case of elliptical orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The fermion pair radiation is realized in our analysis via the coupling of the classical object to a massive boson: a vector, or a scalar, which is unstable and decays into a fermion pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We consider the emission of Dirac fermions and generalize our result to the case of Weyl fermions when we discuss the application of our result to the SM in Section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' While a point-like object is a purely theoretical entity, it is worthwhile to perform this calculation since the approximation of a radiating extended object as a point is valid in the limit of long-wavelength radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' General formalism We describe a point-like object as a classical source using classical current, Jµ cl(x) and classical density, ρcl(x), which are given by Jµ cl(x) = Qδ3(x − x(t))uµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) ρcl(x) = Nδ3(x − x(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2) Here, Q is the total charge of the object under the symmetry of interest, N is the number of the relevant microscopic constituents, x(t) is its position as a function of time, t, and uµ is its 4 four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Assuming motion in the x − y plane, in the non-relativistic limit, the four-velocity of the object is given by uµ = (1, ˙x, ˙y, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='3) We focus on the case of the elliptical motion in the x − y plane, which can be parametrically described by x = a(cos ξ − e), y = a √ 1 − e2 sin ξ, Ω t = ξ − e sin ξ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='4) where e is the eccentricity, a is the semi-major axis of the ellipse, and Ω is the fundamental frequency of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' One full revolution around the ellipse corresponds to changing the parameter ξ from 0 to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss due to the fermion-pair radiation is calculated using Ploss = � (ω1 + ω2) dΓ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='5) where ω1 and ω2 are the energies of the emitted fermion and anti-fermion, respectively, and dΓ is the differential rate of the fermion-pair emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The rate depends on the type of mediator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=', a scalar or a vector, and the specific form of the classical current or density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In general, the acceleration is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the case of periodic orbits, the motion can be decomposed into harmonic modes with frequencies Ωn = nΩ, where Ω is the fundamental frequency of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The total emission rate can then be written as a sum of emission rates at different harmonics n, dΓ = � n dΓn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6) The sum goes over all kinematically allowed harmonics n > 2mψ/Ω, where mψ is the mass of the emitted fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The emission rate at harmonic n is found using dΓn = � s1,s2 |Mn(s1, s2)|2(2π)δ(Ωn − ω1 − ω2) d3k1 (2π)3ω1 d3k2 (2π)3ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='7) Here, k1 = (ω1, k1) and k2 = (ω2, k2) are the four-momenta of the fermion and anti-fermion respectively, and s1(s2) is the spin of the fermion (anti-fermion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The microscopic physics enters via Mn (s1, s2), which is the matrix element of the fermion-pair emission at harmonic n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' At leading order, this matrix element is obtained from the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the diagram, ⊗ denotes the classical source, which is given by the classical current, Jµ cl(x), in the case of vector mediator and by the density, ρcl(x), in the case of the scalar mediated radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The total power loss via fermion-pair radiation is simply a sum of power losses over all harmonics Ploss = � n Pn, Pn = � (ω1 + ω2) dΓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) 5 ψ, k1 ψ, k2 mediator FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Feynman diagram for a fermion pair emission by a classical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Here, Pn is the power loss of the nth harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In what follows, we consider two types of mediators: a massive gauge boson and a massive scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We only consider s-channel exchange and remark on t-channel exchange at the end of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' First, we consider a vector mediator, Aµ, that corresponds to a broken U(1)′ and has mass mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This gauge boson couples to a classical current Jµ cl(x), which has charge Q under U(1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The gauge boson Aµ is unstable and decays into a fermion pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The terms in the effective Lagrangian, relevant for the fermion-pair radiation via Aµ, are Leff ⊃ gAµJµ cl + gqψ ¯ψγµAµψ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9) where qψ is the U(1)′ charge of the fermion ψ, g is a dimensionless coupling constant, and Jµ cl(x) is the classical current defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Both the vector boson and the fermions are assumed to be massive with masses mA and mψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The leading order matrix element for the emission, at the n−th harmonic, is given by Mn(s1, s2) = g2qψ ¯u(k1, s1)γµv(k2, s2) i(−ηµν + (k1 + k2)µ(k1 + k2)ν/m2 A) (k1 + k2)2 − m2 A + imAΓA Jν cl(Ωn) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='10) where Jν cl(Ωn) is the Fourier transform of Jν cl(x), given by Jν cl(Ωn) = Ω 2π � 2π/Ω 0 dt � d3x ei(nΩt−p·x)Jν cl(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='11) with p = k1 + k2, ΓA is the decay width of the gauge boson, and 2π/Ω is the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We assume that the decay into a ¯ψψ pair is the only decay channel for the gauge boson Aµ, and that the fermion mass mψ is negligible compared to the gauge boson mass mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Under these assumptions, the decay width of Aµ is given by ΓA = g2q2 ψmA 12π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='12) The other case we consider is that of a scalar mediator, φ, for which the relevant terms in the Lagrangian are L ⊃ gφρcl + g′φ ¯ψψ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='13) 6 where g is the dimensionless coupling between the scalar φ and the classical source, g′ is the Yukawa coupling of the fermion ψ to the scalar φ, and ρcl(x) is the number density of relevant particles in the classical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Both the scalar and the fermions are assumed to be massive with masses mφ and mψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The matrix element in this case is given by Mn(s1, s2) = gg′¯u(k1, s1)v(k2, s2) iρcl(Ωn) (k1 + k2)2 − m2 φ + imφΓφ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14) where ρcl(Ωn) is the Fourier transform of ρcl(x), ρcl(Ωn) = Ω 2π � 2π/Ω 0 dt � d3x ei(nΩt−p·x)ρcl(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='15) and the decay width of the scalar is Γφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As in the case of the vector mediator, we assume that the fermionic decay mode is the only available mode, and the fermion mass mψ can be neglected compared to the mass of a scalar mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus we have Γφ = g′2mφ 8π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='16) So far, we have only considered the s-channel contribution to the fermion pair radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fermion pair radiation via t−channel process mediated by a vector or scalar is also a possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Such contributions, however, are highly suppressed for mS ≫ Ω, mM, where mS is the mass of the particles in the source that couple to the fermion pairs ¯ψψ at the microscopic level, and mM is the mediator mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Since the emitted fermions have energy of the order of Ω, the fundamental frequency of the system, the t-channel contribution to the momentum entering the propagator is of the order of mS − Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus the t-channel propagator is schematically given by Π ∼ 1 (mS − Ω)2 − m2 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='17) In the case where mS is much larger than both Ω and mM, the propagator is dominated by the mass of the source particles, and the process is heavily suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this paper, we assume that the mass hierarchy mS ≫ Ω, mM and neglect the t−channel contributions to the fermion pair radiation everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Power loss formulae Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14), we can calculate the power loss via fermion-pair radiation from a point-like object moving in an elliptical orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The detailed derivations are shown in Ap- pendix A, and here we only quote the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss due to fermion-pair emission 7 in harmonic n > 2mψ/Ω, for the cases of the vector and scalar mediator, can be written as P A n = g4q2 ψQ2 12π3 a2Ω4 BA n (nA, nψ, nΓ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18) P φ n = g2g′2N 2 12π3 a2Ω4 Bφ n(nφ, nψ, nΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='19) The functions BM n (nA, nψ, nΓ), where M = A, φ, are given by BM n (nM, nψ, nΓ) ≡ � J′ n(ne)2 + 1 − e2 e2 Jn(ne)2 � � n−nψ nψ dx F M(x, n, nM, nψ, nΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20) Here nM ≡ mM/Ω, nψ ≡ mψ/Ω, nΓ ≡ ΓM/Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='21) and Jn(ne) is a Bessel function of order n with argument ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The integration variable in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20) is defined by x ≡ ω1/Ω, where ω1 is the energy of one of the final-state fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In what follows, for brevity, we use the notation F M(x) ≡ F M(x, n, nM, nψ, nΓ), BM n ≡ BM n (nM, nψ, nΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='22) The functions F M(x) have the general form F M(x) = F M 0 (x) + F M 1 (x) nMnΓ � tan−1 �a(x) + b(x) nMnΓ � − tan−1 �a(x) − b(x) nMnΓ �� + F M 2 (x) tanh−1 � 2a(x)b(x) a(x)2 + b(x)2 + n2 Mn2 Γ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='23) with a(x) and b(x) being universal for both gauge boson and scalar mediators, a(x) = 2n2 ψ − n2 M + 2nx − 2x2 , b(x) = 2 � x2 − n2 ψ � (n − x)2 − n2 ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='24) The functions F M 0 (x), F M 1 (x), and F M 2 (x) are different for the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For a gauge boson mediator, we obtain F A 0 (x) = b(x)/2n , F A 1 (x) = 1 4n � n4 A + 4n2n2 ψ − n2 An2 Γ + 2n2 An2 − 4nxn2 A + 4x2n2 A � , F A 2 (x) = 1 2n � n2 A + n2 − 2nx + 2x2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='25) while for a scalar mediator, F φ 0 (x) = −b(x)/2n , F φ 1 (x) = 1 4n � n2 φn2 Γ + (n2 − n2 φ)(n2 φ − 4n2 ψ) � , F φ 2 (x) = 1 4n � n2 + 4n2 ψ − 2n2 φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='26) 8 Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='26) are the main results of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Analytical integration of F A(x) and F φ(x) is challenging, but it still can be performed in certain limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' III B, we consider two limiting cases: the case of nM ≪ 1, which reproduces the Larmor formula, and nM ≫ 1, which is relevant for the fermion pair radiation in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In general, however, calculating the power loss requires numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We perform such an analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' IV when we discuss a particular phenomenological application of our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' DISCUSSION OF THE POWER-LOSS FORMULA The power loss due to fermion-pair emission by a classical source on an elliptical orbit is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Below we discuss the main features and the asymptotic behavior of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' General features of the power-loss formula We start with the general features that hold for both the vector and scalar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The radiation rate is proportional to the charge-squared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' that is, the functions P A n and P φ n depend on Q2 and N 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is a manifestation of the fact that the fermion-pair radiation that we are considering is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The form of F M(x), with M = A, φ, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='23) is somewhat general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We show in Appendix A that the overall form of F M(x), at the tree level, is the same for any renormalizable theory that couples fermions to a classical source moving in an elliptical orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Note that the functions a(x) and b(x) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='24) are purely kinematic and thus have the same form for any theory of fermion pair emission, while the form of F M 0 (x), F M 1 (x), and F M 2 (x) vary with the theory considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For instance, considering non-renormalizable interactions would lead to a different momentum dependence of the matrix element that could, in principle, change the form of F M(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss for both vector and scalar mediators behaves qualitatively the same way despite the different functional forms of F A i (x) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' F φ i (x), with i = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is not surprising since there is nothing fundamentally different between the matrix elements for the vector and scalar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Energy conservation implies that the functions F M(x) are invariant under x → (n − x) exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The reason is that the total energy radiated in fermion pairs in the n-th harmonic is nΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The transformation x → (n − x) exchanges the energies of the emitted fermion and anti-fermion, and the emission rate is the same regardless of the order in which the integrals 9 are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This invariance results from the fact that the fermion-antifermion emission from a classical system is essentially a 2-body decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Note that this has nothing to do with the details of the considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For nA < n, the power loss has a very weak dependence on nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is true for the particular models that we chose here but is not expected to be true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For an example when this is not the case, see the discussion of Proca fields in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [6], where dependence on nA appears due to the absence of gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' There is an interplay of three energy scales: The mass of the mediator, mM, the mass of the fermion, mψ, and the frequency of the harmonics, nΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The fermions cannot be produced when 2mψ > nΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the opposite limit, when 2mψ < nΩ, the production rate depends strongly on the mediator mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For mM < 2mψ < nΩ, fermion production is strongly suppressed since the on-shell boson is kinematically forbidden from decaying into fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (Note that strictly speaking, our result cannot be straightforwardly applied in this case as everywhere we assume ΓM > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=') For 2mψ < mM < nΩ, the fermions are produced via decay of the on-shell mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus the power loss in the fermion-pair radiation is equal to that of the on-shell boson radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The region of the parameter space where mM > nΩ > 2mψ is of the most interest to us, as in this region the fermions are kinematically allowed, the mediator is off-shell, and therefore the fermion pair emission is most significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As an example that illustrates the qualitative features of the power loss, consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' It shows BA n , defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20), as a function of nA for massless fermions for the first four harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The most striking feature of the plots is a sharp drop at nA ∼ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This behavior follows from the fact that at nA ∼ n, the radiation regime switches from the radiation dominated by on-shell boson production (nA < n), which is proportional to g2 to the off-shell production (nA > n) proportional to g4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss in the regime dominated by fermion-pair radiation is thus suppressed by g2 compared to the power loss in the regime dominated by the on-shell boson radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss in the case of the scalar mediator exhibits the same behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Comparing our results to the cases of vector [3, 5, 6] and scalar radiation [3], we note that from kinematic considerations alone, boson radiation drops to zero as soon as nM = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is not what we observe for the fermion-pair emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the case of fermion-pair radiation, off-shell boson production is possible, even though there is an extra suppression by g2 for a vector and g′2 for a scalar compared to on-shell boson radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As a result, the regime nM > n opens up new regions of the parameter space for each harmonic n and is of particular phenomenological interest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Next, we remark on the dependence of the power loss on the eccentricity in the case of orbits close to circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For that, we note that the eccentricity only enters the power loss 10 ����� ����� �� ���� ��-�� ��-�� ��-�� �� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' BA n vs nA for fixed eccentricity, e = 10−3, coupling constant g = 10−15, and massless final state fermions, mψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' See Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='25) for the definition of BA n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' through the Bessel function prefactor of BM n in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20), which we denote as K(n, e), K(n, e) = J′ n(ne)2 + 1 − e2 e2 Jn(ne)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) We recall that Jn(z) and J′ n(z) behave asymptotically, in the limit z ≪ 1, as Jn(z) ≈ 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' �z 2 �n , J′ n(z) ≈ n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 1 2 �z 2 �n−1 ≈ n z Jn(z), z ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2), we find for the eccentricity dependent prefactor K(n, e), in the limit ne ≪ 1, that K(n, e) = J′ n(ne)2 + n2 − (ne)2 (ne)2 Jn(z)2 ≈ J′ n(ne)2 + n2 (ne)2Jn(z)2 = 2n2 z2 Jn(ne)2 = (ne)2n−2 22n−1 n2 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' )2 = (ne)2n−2 22n−1 ((n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='3) Thus we learn that in the limit ne ≪ 1, prefactor K(n, e) scales with the eccentricity as K(n, e) ∝ (ne)2n−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='4) This shows that for small eccentricities (and thus orbits close to circular ones), the contributions from higher harmonics die away very fast as n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For n = 1 and e ≪ 1, we have K(1, e) ≈ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For each subsequent harmonic power drops by a factor of order e2, until the factorial in the denominator of K(n, e) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='3)) starts to dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Then the contributions from the higher harmonics start to decay away even faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2 illustrates the behavior of the power loss for the first four harmonics in the case of small eccentricity e = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The case of highly eccentric orbits e ∼ 1 is significantly more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' First, the contri- butions from different modes do not follow the simple hierarchy of the low eccentricity case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 11 � � �� �� ��-�� ��-�� ��-� ��-� ��� � � � �� �� ��-�� ��-�� ��-�� ��-�� ��-�� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Left: BA n as a function of n in the regime where the radiation is dominated by on-shell boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Different colors correspond to different values of eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The values of nψ, nA and g are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Right: BA n as a function of n for a highly eccentric orbit with e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6 in the regime where the radiation is dominated by off-shell boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The contributions from higher modes can be of the same order or even larger than the first mode depending on the values of other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' See the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 3 to compare the n-dependence of BA n for different eccentricity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Second, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 3 demonstrates, the hierarchy of modes in the on-shell dominated part of the parameter space does not carry into the off-shell dominated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Consider the green line corresponding to a highly eccentric orbit with e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For nA = 10−1 (left panel), the maximum contribution to the power loss comes from the mode with n = 2 and the first 5 modes contribute at about the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The situation is drastically different for nA = 50 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The maximum contribution to the power loss comes from the n = 8 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We learn that for e ∼ 1, generally speaking, the power loss per mode first increases as we increase n and then starts decreasing after reaching a certain value of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Where this maximum occurs depends on other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Asymptotic behavior for the case of circular orbits We now move to the discussion of the asymptotic behaviour of the power loss in two limiting cases mM ≪ Ω and mM ≫ Ω, where mM is the mass of the mediator, M = A, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this subsection, for simplicity we consider the straightforward case of circular orbits (e = 0) and massless fermions (mψ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For the eccentricity dependent part of the power loss, K(n, e), we have lim e → 0 K(n, e) = lim e → 0 � J′ n(ne)2 + 1 − e2 e2 Jn(ne)2 � = 1 2δn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='5) Thus the only mode that contributes to the power loss in the circular orbit limit is the mode with n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 12 First, let us consider the regime of light mediators, mM ≪ Ω, or equivalently nM ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this limit, F M(x) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='23) is dominated by the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We thus neglect the first and the third terms of F M(x) and take the second term’s limit nM → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' After that, the integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20) can be performed analytically, yielding the following asymptotic expressions for the power radiated via vector and scalar, respectively: P A(mA ≪ Ω) ≈ g2 6πQ2a2Ω4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6) P φ(mφ ≪ Ω) ≈ g2 12πN 2a2Ω4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='7) The asymptotic behavior that we find for P A and P φ reproduces the known results for the on-shell vector [3, 5, 6] and scalar [3] radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is expected as, in the regime mM ≪ Ω, the fermion pair radiation is dominated by on-shell boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Additionally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6) also reproduces the Larmor formula for the power of the electromagnetic wave radiation given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' To see this, recall that the acceleration on a circular orbit is equal to aΩ2, where a is the radius of the orbit and Ω is the frequency of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Next, we study the regime when on-shell boson production is kinematically forbidden, and the fermion pair radiation takes place through the off-shell mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is the limit of heavy mediators, mM ≫ Ω, or equivalently nM ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As in the case of the light mediators, we take the nM → ∞ limit of F M(x) and find that the resulting expression can be integrated analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Upon performing the integration, we find that the vector and scalar-mediated radiation behave as P A(mA ≫ Ω) ≈ g4q2 ψQ2 210π3 a2Ω8 m4 A = 1 35π2 g2q2 ψΩ4 m4 A × P A(mA ≪ Ω), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) P φ(mφ ≫ Ω) ≈ g2g′2N 2 840π3 a2Ω8 m4 φ = 1 70π2 g′2Ω4 m4 φ × P φ(mφ ≪ Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9) We learn that in the limit of heavy mediators, the fermion pair radiation is suppressed compared to on-shell boson radiation by the following factors: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A factor of g2q2 ψ or g′2, which, at the amplitude level, comes from the coupling of the mediator to the fermion pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A factor of Ω4/m4 φ, which comes from the propagator of the mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A phase space factor of 1/35π2 or 1/70π2, which arises from the fact that there are more particles in the final state in the case of the off-shell pair production than in the case of the on-shell boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Note that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9) can be interpreted as integrating out the heavy mediator, resulting in an effective 4-Fermi interaction with a coefficient proportional to g2/m2 A or gg′/m2 φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, it is also valid for t-channel and u-channel interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 13 Last, we compare the results of the vector to that of the scalar mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Consider mA = mφ, Q2 = N 2 and g′ = gqψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this case, the power radiated via the vector mediator is greater than the power radiated via the scalar mediator in both radiation regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, we have P A(mA ≪ Ω) P φ(mφ ≪ Ω) ≈ 2, P A(mA ≫ Ω) P φ(mφ ≫ Ω) ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='10) These factors are related to the different number of degrees of freedom between the vector and scalar cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' There are two polarization states for an on-shell massless vector, while the scalar has only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For the deeply off-shell mediator, the correspondence is not so clear, but it seems to us that it is related to the fact that off shell gauge boson, Aµ, has four degrees of freedom C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fermion-pair radiation in the SM The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) can be used to estimate the power loss due to fermion pair radiation by classical sources within the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this subsection, we consider neutrino pair radiation mediated by Z-boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The contribution due to W-boson mediated pair emission is qualitatively the same as the Z-boson contribution and is expected to be of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The main difference between the two contributions is due to the fact that W-boson mediated radiation is only relevant for leptons in the source while Z-boson contribution is present for all types of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Consider a source made of NΨ fermions of type Ψ with the total weak charge Q = NΨqΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' To apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) to the neutrino pair radiation in the SM, we need to recall that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) was derived under the assumption of vectorial couplings, while the SM is a chiral theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The relevant parts of the SM Lagrangian are different from the Lagrangian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' in particular, in the SM we have LSM ⊃ −i g 2 cos θW � ¯Ψγµ(cΨ V − cψ A)Ψ + ¯νγµ(cν V − cν A)ν � Zµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='11) Thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) yields the following expression for the Z-boson mediated power loss due to the neutrino pair radiation in the SM P Z(mZ ≫ Ω) ≈ 1 210π3 g4q2 νq2 ΨN 2 Ψ 16 cos4 θW a2Ω8 m4 Z , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='12) where we perform the replacement g → g/(2 cos θW) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) and define q2 ψ = q2 ν = (cν V )2 + (cν A)2, qΨ = cΨ V , mA = mZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='13) Note that, for the source, only vectorial coupling cΨ V enters the power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is because we consider coherent radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 14 The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='12) can be rewritten as P Z(mZ ≫ Ω) ≈ G2 effq2 Ψq2 νN 2 Ψ a2Ω8 210π3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14) where Geff = √ 2GF and GF is the Fermi constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' When the power loss is written in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14), it becomes clear that it is the same as what one would obtain by performing the calculation for the effective Fermi theory with the effective Lagrangian given by LZ eff ⊃ Geff[¯Ψγµ(cΨ V − cΨ Aγ5)Ψ][¯νγµ(cν V − cν Aγ5)ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='15) This, of course, is not surprising as we consider radiation at the energy Ω, which is much less than the electroweak scale, Ω ≪ mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In fact, the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14) applies to any effective 4-Fermi interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' While we derive our results for s-channel exchange, in the limit where the mediator is much heavier than the orbit frequency, we do not need to distinguish between s- channel and t-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14) can also be used for t-channel W-exchange in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Finally, we discuss the situation when there are several different types of fermions in the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this case, we need to first add all the amplitudes that correspond to the radiation by different fermions Ψ (for leptons, we add both Z-boson and W-boson contributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Then, we square the sum of the relevant amplitudes to obtain the total emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We end this subsection with the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The power loss due to neutrino pair radiation in the SM was estimated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [4] to be P Z SM ∼ G2 FΩ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using the explicit calculation, however, we find that P Z SM ∼ G2 Fa2Ω8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' That is, there is an extra factor of a2Ω2 compared to the estimation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In fact, our result includes the semi-major axis a as an additional energy scale of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' FERMION PAIR RADIATION BY PULSAR BINARIES We now move to discuss the phenomenological applications of our results to astrophysical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We focus on the neutrino-pair emission from pulsar binaries [23–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A pulsar binary is a binary system of a pulsar and companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This choice is motivated by the availability of extensive period decay data for such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, we apply our results to two binaries: Hulse-Taylor binary PSR B1913+16 [34–36] (a system of a pulsar and a neutron star) and PSR J1738+0333 [29, 37] (a system of a pulsar and a white dwarf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The parameters characterizing the two systems are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In what follows, we first discuss the applicability of our results of Section II B to pulsar binaries in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Then we estimate the contribution to the power loss due to neutrino pair 15 Binary system PSR B1913+16 [36] PSR J1738+0333 [29] Eccentricity e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6171340(4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='4(11) × 10−7 Pulsar mass m1 (M⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='438(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='46(6) Companion mass m2 (M⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='390(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='181(8) Binary period Tb (GeV−1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='240 × 1028 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='657 × 1028 Intrinsic period decay ˙Tb −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='398(4) × 10−12 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='59(32) × 10−14 Predicted period decay due to GW ˙TGW −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='40263(5) × 10−12 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='77(19) × 10−14 Ratio of period decays R = ˙Tb/ ˙TGW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9983(16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='94(13) Orbital frequency Ω = 2π/Tb (GeV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='482 × 10−28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='349 × 10−28 Semi-major axis a (GeV−1) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='878 × 1024 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='77 × 1024 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The relevant parameters for the PSR B1913+16 and PSR J1738+0333 binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Figures in parenthesis are the 1σ uncertainties in the last quoted digit, where all the uncertainties are symmetrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' M⊙ is the mass of the sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The relative experimental error of the binary period Tb is ∼ 10−12 for PSR B1913+16, and ∼ 10−11 for PSR J1738+0333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The double line separates binary parameters quoted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [29, 36] and the ones we derive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Values of the semi-major axis a are calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' emission in the SM and show that it is negligible compared to the gravitational wave radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We then consider neutrino pair radiation in two BSM scenarios via ultralight vector and scalar mediators and apply our results to the pulsar binaries with the parameters in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Pulsar binaries as a classical source The results for the fermion pair radiation, summarized in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='26), were derived for the case of classical current describing non-relativistic point-like object following an elliptical orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' To justify the application of our results to pulsar binaries, we note the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A pulsar binary can be treated as a classical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The typical size of a pulsar binary can be estimated as the size of the semi-major axis which varies between 106 and 108 km, that is, a ∼ 1024 − 1026 GeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The wavelength of the radiation is determined by the fundamental frequency of the orbit, and for a typical pulsar binary with periods in the range of 10−1 − 103 days, the wavelength is λ ∼ 1028 − 1032 GeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, λ ≫ a and we conclude that pulsar binaries can be treated as classical radiation sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Stars of the pulsar binary can be treated as point-like objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Typical sizes of stars in a binary vary from r ∼ 10 km ∼ 1019 GeV−1, for neutron stars, and r ∼ 103 km ∼ 16 1021 GeV−1, for white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus r ≪ a, λ and both pulsar and its companion can be treated as point-like objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Moreover, r ≪ λ implies the coherence of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The motion of the pulsar and its companion in the binary system is non-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We can roughly estimate the orbital velocity of the stars in a binary as v ∼ aΩ, which for characteristic values quoted above implies v ≲ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For a wide range of pulsar binary systems, the observed power loss is such that it has no significant effect on the eccentricity of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus we can treat the orbit as elliptical over the time of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For example, the Hulse-Taylor binary has e ∼ 1, with Tb(de/dt) ≲ 10−11, where Tb is the binary period and de/dt is the time derivative of the eccentricity [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Now that we have established that the results of Section II B can be applied to pulsar binaries, we proceed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' First, we modify our expressions for the classical current and number density in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2) to the case of two point-like objects on an elliptical orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Second, we perform the standard reduction of the two-body problem to a one-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We write the classical current and number density as Jµ cl(x) = � b=1,2 Qb δ3(x − xb(t))uµ b , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) and ρcl(x) = � b=1,2 Nb δ3(x − xb(t) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Here, b = 1, 2 is the index that labels the stars of the binary system, xb(t) is the position of the b-th star at time t, and uµ b is its four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Next, we move to the binary system’s Center-of-Mass (CoM) frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For that, we define R, the coordinate of center of mass, and r, the distance between the two stars, R = m1 m1 + m2 x1 + m2 m1 + m2 x2, r = x1 − x2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='3) where m1 and m2 are the masses of the two stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As we are not concerned with the translational motion of the system as a whole, which is described by R, we can solely focus on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is the standard two-body to one-body problem reduction for central force motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The non-relativistic classical trajectory of the stars in the CoM frame can thus be described by the vector r = (x, y, 0) and is given by elliptical orbits as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='3): x = a(cos ξ − e), y = a √ 1 − e2 sin ξ, Ωt = ξ − e sin ξ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='4) 17 where e is the eccentricity, a is the semi-major axis of the elliptical orbit, and the fundamental frequency of revolution is given by Ω = � GN(m1 + m2) a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='5) The results of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='26) generalize to the case of binary systems via the following replacements that follow from the 2-body to 1-body reduction procedure: Q2 → M 2 �Q1 m1 − Q2 m2 �2 , N 2 → M 2 �N1 m1 − N2 m2 �2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6) where M = m1m2 m1 + m2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='7) is the reduced mass of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As a result we obtain the following expressions for the power loss in n-th harmonic for a vector and scalar mediators respectively: P A n = g4q2 ψ 12π3M 2 �Q1 m1 − Q2 m2 �2 a2Ω4 BA n (nA, nψ, nΓ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) P φ n = g2g′2 12π3 M 2 �N1 m1 − N2 m2 �2 a2Ω4 Bφ n(nφ, nψ, nΓ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9) where the functions BA n and Bφ n are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Neutrino pair radiation by pulsar binaries in the SM In the SM, for the pulsar binary, the power loss via electroweak mediators is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Here, we simply generalize it to the case of 2-body motion using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We obtain the following expression for the power loss in neutrino pair radiation via Z-exchange in the SM PSM ≈ G2 F � cν V 2 + cν A 2� 105π3 cos2 θW M 2a2Ω8 � 1 m1 � i=n,p,e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ci V N1iQ1i − 1 m2 � i=n,p,e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ci V N2iQ2i �2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='10) where the sum goes over all microscopic constituents of binary stars, such as neutrons (n), protons (p), electrons (e), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' To perform a numerical estimate, we consider a pulsar binary with a neutron star companion and assume that all of the neutron star mass is in the form of neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We consider a typical pulsar-neutron star binary with m1,2 ∼ M⊙ ∼ 1057GeV, a ∼ 1025 GeV−1, Ω ∼ 10−28 GeV, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='11) and non-zero dipole moment M 2 �Q1 m1 − Q2 m2 �2 ∼ Q2 1,2 ∼ 10114, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='12) 18 where Qb = Nb(n)−Nb(¯n) ≈ Nb(n) ≈ M⊙/mn ≈ 1057, with b = 1, 2, are the neutron charges of the neutron stars, Nb(n) and Nb(¯n) are the numbers of neutrons and anti-neutrons respectively, mn is the neutron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using cν V = cν A = 1/2, cn V = −1/2, and the measured values of mn, GF, and θW, we find the following numerical estimate for the radiated power PSM ∼ 10−56eV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='13) To see if the above result is significant, we compare it to the power loss in the form of gravitational wave (GW) radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using the quadrupole formula for the GW radiation [38] for the case of circular orbit (e = 0) we have PGW = 32 5 GNM 2a4Ω6 ∼ 108 GeV2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14) where GN is Newton’s gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The rough estimates in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14) show that, in the SM, the fermion-pair radiation by astrophysical objects is completely negligible compared to the gravitational wave radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We close the subsection with one remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Within the SM, neutron stars also emit syn- chrotron radiation of fermion-antifermion pairs in their self-produced magnetic fields, as shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This phenomenon is different from the one we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Synchrotron radiation is an incoherent effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, the power loss, in this case, scales as N, the number of neutrons in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the case we are considering, the radiation is coherent and comes from the star’s acceleration as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Then, the net power that is radiated is proportional to N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' New physics constraints from the neutrino pair radiation by pulsar binaries Since extra radiation in the SM is negligible, any observed deviation from the gravitational wave radiation would be strong evidence for the physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, fermion- pair radiation can be enhanced in BSM models with light vector or scalar mediators, with mA,φ ≪ mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' To explain why such light bosonic states have evaded detection so far, we must require that they have small couplings, thus evading all the available constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The smallness of couplings, however, still can be compensated in cases where the object has a large charge under the new symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This can be the case for astrophysical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, such objects are our prime focus in the rest of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, in this subsection, we demonstrate how our results can be used to derive new physics bounds from the neutrino pair radiation by pulsar binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As we mentioned above, we use two distinct pulsar binary systems, the Hulse-Taylor binary PSR B1913+16 and PSR J1738+0333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The relevant properties of the two systems are summarized in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The 19 Hulse-Taylor binary is a pulsar binary with a neutron star companion, it is highly eccentric, and the mass ratio of the two stars is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The PSR J1738+0333, on the other hand, is a pulsar-white dwarf binary with an almost circular orbit and a high pulsar-to-companion mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For both systems, the data on the orbital period decay is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Both binaries lie within 1σ of the general relativity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In our analysis, we exploit the fact that typical neutron stars contain a very large number of muons, N(µ) ∼ 1055 [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus, the effects of muonophilic new physics can be significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The presence of the large muon number in neutron stars is attributed to the fact that when the electron chemical potential, µe, is larger than the muon mass µe > mµ, it becomes energetically favorable for relativistic electrons at the Fermi surface to decay into muons via e− → µ− + ¯νµ + νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Moreover, the muonic beta-decay n → p + µ− + ¯νµ and inverse beta-decay p + µ− → n + νµ reactions become energetically favorable, while the muon decay µ− → e− + ¯νe + νµ is forbidden by Fermi statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Being motivated by the neutron star muonic content, we consider neutrino pair emission by pulsar binaries via the following two types of BSM mediators: U(1)Lµ−Lτ massive gauge boson with L ⊃ gAα (¯µγαµ − ¯τγατ + ¯νµγανµ − ¯ντγαντ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='15) Massive muonophilic scalar with L ⊃ gφ¯µµ + g′φ¯νµνµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='16) It is known that at least two of the SM neutrinos are massive, while the third neutrino can be very light or massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This means that only one neutrino mass eigenstate can be radiated in the two scenarios we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A realistic treatment of neutrino emission would include insertions of the corresponding PMNS matrix elements [44], resulting in an additional factor of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Since we already neglecting an O(1) factor coming from the estimate of the muon number density in the neutron stars, we also ignore any PMNS factors in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Note also that in a theory with general couplings to the left and right-handed neutrinos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=', gAα¯νγα(cV − cAγ5)ν, the results for the power loss are qualitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Moreover, in the case of massless neutrinos, the power loss for the case of the general coupling is the same as the power loss for the case of purely vectorial coupling up to g2 → g2(c2 A + c2 V ) replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is why in what follows, for simplicity, we consider the case of the vectorial coupling only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' These two BSM models imply the possibility for the neutrino pair radiation at rates enhanced compared to the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Our results from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9) thus can be used to set bounds on the coupling constants and masses of the new bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 20 The presence of the muonophilic new physics, however, not only alters the radiation patterns of pulsar binaries, but it also has important implications for the neutron star’s equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, the presence of a repulsive (vector) or attractive (scalar) interaction between muons could affect the muon number, which depends on the coupling g to the new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the following, we write the muon number as N(µ, g) to keep the dependence on g explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The number of muons becomes g-dependent as the interactions change the muon chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The muon interaction due to the Lµ − Lτ vector boson is repulsive, and thus the chemical potential is increased compared to its SM value by ε ∼ g2N(µ, g)/R, where R is the radius of the neutron star the boson mass is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' When the coupling g is small, such that ε ≪ mµ, the effect of the new interaction is insignificant, and the number of muons is approximately given by its value in the limit of no interaction N(µ, g = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' When the interaction is strong, such that ε ≫ mµ, it becomes energetically less favorable to have muons inside the neutron star and thus N(µ, g) < N(µ, g = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Similar reasoning applies to the case of the scalar mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The only difference is the sign of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the scalar case, the interaction between muons is attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thus the muon chemical potential is decreased by ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This leads to the increase of the muon number for larger couplings N(µ, g) > N(µ, g = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In both cases, the change from the regime when N(µ, g) ≈ N(µ, g = 0) to the situation when the interaction starts to affect the muon number happens for couplings such that ε ∼ mµ, or numerically g ∼ 10−18 for a typical neutron star [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' However, in what follows, we ignore the effect of the new physics on the muon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Everywhere in our analysis, we use the muon number in the limit of no new physics interaction, that is we set N(µ) = N(µ, g = 0) ∼ 1055 [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In principle, g-independence of muon number can be achieved in models with both vector and scalar mediators with fine-tuned coupling constants such that the repulsive and attractive interactions cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' To apply Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9), we define Nb(µ) and Nb(¯µ) as the number of muons and antimuons respectively in neutron star labeled by b = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Then, as there are almost no tau leptons in neutron stars, Qb = Nb(µ) − Nb(¯µ) is the total charge of the neutron star under the Lµ−Lτ gauge symmetry, and Nb = Nb(µ)+Nb(¯µ) is the total number of muons and anti-muons in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Additionally, since Nb(¯µ) ≈ 0, we have Qb ≈ Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The energy lost through radiation in a binary star system can be directly probed by measur- ing the decay of the orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Assuming that the attractive gravitational force between the two stars is such that their orbits stay Keplerian, the decay rate of the period of revolution Tb is related directly to the energy lost via radiation [6]: ˙Tb = −6πa5/2G−3/2 N (m1m2)−1(m1 + m2)−1/2 × Ploss, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='17) 21 where ˙Tb is the time derivative of the binary period, GN is the gravitational constant, m1 and m2 are the masses of the stars in the binary system, a is the semi-major axis of the elliptical orbit, and Ploss is the total power radiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The decay of the period per unit of time is dimensionless and is measured experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' GW emission is the dominant source of power loss in a binary star system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Assuming that the GW emission and neutrino pair emission are the only sources of energy loss, we have Ploss = PGW + P¯νν, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18) where P¯νν is the power loss due to the neutrino pair radiation and PGW is the power loss due to GW emission, which, to the leading order, is given by the GW quadrupole radiation formula [38], P GW loss = 32 5 GΩ6M 2a4(1 − e2)−7/2 � 1 + 73 24e2 + 37 96e4 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='19) where M is the reduced mass of the system, as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The binary period decay ˙Tb thus can be written as a sum of two contributions, ˙Tb = ˙TGW + ˙T¯νν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='20) We next introduce the period decay ratio R as the ratio of the measured period decay to the theoretical prediction of the period decay due to GW radiation, R = ˙Tb ˙TGW = 1 + ˙T¯νν ˙TGW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='21) We use the measured value of R to set 2σ limits on the masses and couplings of the BSM mediators of neutrino pair radiation as ˙T¯νν ˙TGW ≤ (R − 1) + 2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='22) The resulting constraints on the parameter space (g, mA) and (g, mφ) that we derive from the period decay data for the Hulse-Taylor binary and PSR J1738+033 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' When deriving the constraints, we use Qb = Nb = 1055 with b = 1, 2 and qν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For the gauge boson mediator (left panel), we calculate the period decay due to neutrino pair emission, ˙T¯νν, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As we take all three neutrinos to be massless, and as Lµ −Lτ boson couples to two neutrino types, there is an extra factor of 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Similarly, for the case of the scalar mediator (right panel), we use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As there is no symmetry that requires equality of g and g′ in the case of the scalar mediator, we present our results for the scalar case in the (g, mφ) plane for four different values of g′ that vary from 10−7 to 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' First, let us discuss the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 4, which shows constraints on the mass and coupling of the gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For the PSR J1738+0333 (red line), whose orbit is very close to circular, the 22 ��� �����+�� ��� �����+���� ��-�� ��-�� ��-�� ��-�� ��-�� ��-�� ��-�� ��-�� ��-�� ��-�� ��-� ��-� PSR J1738+0333 10-22 10-20 10-18 10-16 10-20 10-18 10-16 10-14 10-12 10-10 10-8 10-6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Left: Constraints on g vs mA from the highly eccentric PSR B1913+16 (Hulse-Taylor) Bounds from the neutrino pair radiation (solid) and vector boson radiation (dashed) are shown such that the region above the curves is excluded by the measurements of the period decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The system parameters are taken from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Right: Constraints on g vs mφ from PSR J1738+033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The dashed gray line corresponds to the bound set by the emission of the scalar boson only, while the solid lines show the bounds from including a coupling g′ to the neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' effect of neutrino pair radiation becomes significant for the mediator masses greater than the second harmonic frequency, mA > 2Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For the highly eccentric Hulse-Taylor binary, off-shell radiation dominates for mA > 85Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In the region mA > 2Ω (mA > 85Ω) for PSR J1738+0333 (Hulse-Taylor binary), the boundary of the excluded region is approximately quadratic in the mediator mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is in stark contrast with the case of the on-shell boson emission discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [3, 5, 6], where the boundary of the excluded region jumps in steps at mA = nΩ, with n being an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' For comparison, the dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 4 show the bounds due to the on-shell boson radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Finally, we comment on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 4, which shows the constraints on the mass mφ and coupling g for different values of g′ in the case of the scalar mediated radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We only demonstrate the constraints for PSR J1738+0333;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' the results for the Hulse-Taylor binary are qualitatively the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Depending on the value of g′ the off-shell scalar radiation starts to dominate for mφ > Ω (g′ ≳ 10−4) or mφ > 2Ω (g′ ≲ 10−8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As one can see from the plot, g′ = 10−1 provides the strongest bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We conclude this section by noting that we do not perform a detailed analysis of the bounds on muonophilic light states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We only remark that very strong bounds on light states are derived 23 from fifth force searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Most of these bounds do not apply in our case as these experiments are done using materials made out of protons, neutrons, and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' CONCLUSION It is well known that fermion pairs can behave as bosons in several circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this work, we show that fermion pairs can also constitute classical radiation just like bosonic states do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We use this understanding to derive the generalization of the Larmor formula for the case of the fermion pair emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Being motivated by the potential of applying fermion pair radiation to astrophysical objects, we consider the case of classical sources following elliptical orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The most interesting regime of fermion pair radiation is when the mediator is off-shell, which takes place when the mass of the mediator is much smaller than the frequency of the periodic motion of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In this regime, the fermion pair emission takes over from on-shell boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This opens up a window into a broader region of parameter space for various models that allow for the fermion pair radiation by classical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Subsequently, we apply our results to neutrino-antineutrino emission by two pulsar binary systems PSR B1913+16 and PSR J1738+0333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Neutrino pair emission by binary systems is highly suppressed in the SM compared to GW radiation, but can be significantly enhanced in various BSM scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, we consider two possibilities: light muonophilic vector and scalar mediators that couple to the SM neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using period decay data for the two binary systems, we derive bounds on the parameters of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' While we did not perform a comprehensive study of the relevance of these bounds, the key point is that they provide a demonstration of the fact that fermion pair radiation can be used to enhance BSM probes using astrophysical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' There are several future directions to go from here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Here are a few that we find particularly interesting: A thorough and detailed study of the bounds that we find on specific models is called for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This, however, is complicated by the large uncertainties that come from the estimates on the neutron star constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, new physics interactions alter the equation of state of a neutron star and, currently, there is no precise quantitative understanding of how this affects its content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' It also would be interesting to see if we can find more systems to which our results can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In particular, exotic astrophysical systems and exotic types of new physics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 24 In this work, we only consider fermion pair radiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' however, the results can be modified to also include bosonic pair radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' All that needs to be done is to calculate the relevant matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' It is expected to result in a different kinematic dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We conclude with the main message of our paper: If nature includes new light states, fermion pair radiation can be one more tool in our toolbox to probe them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We are grateful to Kfir Blum, Jeff Dror, Toby Opferkuch, Nadav Outmezguine, Ira Rothstein, Ryosuke Sato, and Kohsaku Tobioka for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The work of YG is supported in part by the NSF grant PHY1316222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The research of WT is supported by NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' PHY-2013052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Appendix A: Derivation of the power loss formula We present below an explicit derivation of the power loss formula for the fermion pair radiation by a point-like classical object on an elliptical orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We perform the calculation separately for the case of vector and scalar mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' In our calculation, we follow closely the analysis in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The case of a vector boson mediator The power loss is a sum over different harmonics, as given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The matrix element, at leading order, for a vector boson mediator, is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' It includes the Fourier Transform of the classical current Jµ cl(x) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' We rewrite it here for convenience: Mn(s1, s2) = g2Qψ ¯u(k1, s1)γµv(k2, s2) i(−ηµν + (k1 + k2)µ(k1 + k2)ν/m2 A) (k1 + k2)2 − m2 A + imAΓA Jν cl(Ωn) , (A1) where ηµν is the Minkowski metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Note that the contribution from the (k1 + k2)µ(k1 + k2)ν term vanishes by means of the Dirac equation since the fermions are on-shell, that is, ¯u(/k1 + /k2)v = ¯u(mψ − mψ)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A2) Squaring the amplitudes corresponding to different harmonics and summing over spins, we 25 find |Mn|2 = � s1,s2 |Mn|2 = g4Q2 ψ ((k1 + k2)2 − m2 A)2 + m2 AΓ2 A Jµ cl(Ωn)J∗ν cl (Ωn) Tr [(/k1 + mν)γµ(/k2 − mν)γν] = 4g4Q2 ψ ((k1 + k2)2 − m2 A)2 + m2 AΓ2 A Jµ cl(Ωn)J∗ν cl (Ωn) � k1µk2ν + k1νk2µ − 1 2(k1 + k2)2ηµν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A3) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' we are ready to write the expression for the rate of energy loss due to ψ ¯ψ emission at harmonic n by the classical source as Pn = �dE dt � n = � Ωn dΓn = Ωn � d3k1 (2π)3(2ω1) d3k2 (2π)3(2ω2)(2π)δ(Ωn − ω1 − ω2)|Mn|2 = Ωn � dΦ1dΦ2 |k1|dω1 2(2π)3 |k2|dω2 2(2π)3 (2π)δ(Ωn − ω1 − ω2)|Mn|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A4) where |k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2| = � ω2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2 − m2 ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' we used Ωn = ω1 + ω2 for the total energy carried away by the fermion pair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' dΦ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2 are the differential elements of solid angles in the fermion’s direction of flight,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' and ��Mn ��2 is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The total power radiated is found by summing over all kinematically allowed harmonics: P = � n Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A5) To calculate the power radiated in fermion pairs by a point-like source in an elliptical orbit, we need to evaluate the integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A4), after substituting in the explicit form of Jµ cl(Ωn) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='4), we find the Fourier Transform Jµ cl(Ωn) as: Ji cl(Ωn) = aΩQji n, J0 cl(Ωn) = aΩQ �jn · p nΩ � , (A6) where the 3-vector jn is defined as jn = � −iJ′ n(ne), √ 1 − e2 e Jn(ne), 0 � , (A7) with Jn(z) denoting a Bessel function, and p = k1 + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The terms in the numerator of |M|2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A3), are then given by (Jµ cl(Ωn)k1µ) (Jν∗ cl (Ωn)k2ν) = a2Ω2Q2ji njj∗ n � ω1ω2 (nΩ)2pipj − ω1 nΩpikj 2 − ω2 nΩki 1pj + ki 1kj 2 � , (A8) and |Jµ cl(Ωn)|2 = |J0 cl(Ωn)|2 − |Jcl(Ωn)|2 = a2Ω2Q2ji njj∗ n � pipj (Ωn)2 − δij � , (A9) where we used Ωn = nΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Note that all quantities above are 3-vectors with Latin indices i = 1, 2, 3, and a sum over i and j is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The expression for (Jµ cl(Ωn)k2µ) (Jν∗ cl (Ωn)k1ν) is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A8) via complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 26 Next we note that the denominator of |Mn|2, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A3), depends only on mA, ΓA, ω1,2, the magnitudes |k1,2| and the relative angle between the two momenta k1, and k2 that we denote as γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Because of this, it is convenient to perform the change of coordinates in the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A4) from the integration over the solid angles dΦ1dΦ2 to the integration over dΦ1dΦr 2 where the solid angle of the second neutrino is measured relative to the direction of k1, hence the super index r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (Equivalently, one can also choose to integrate over dΦr 1dΦ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=') The Jacobian of this coordinate change is unity since the transformation is simply a coordinate rotation, and thus dΦ1dΦ2 = dΦ1dΦr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A10) Defining dΦb = sin θbdθbdφb, dΦr 2 = sin γdγdδ, b = 1, 2 , (A11) we find the following relations between the two sets of integration variables cos γ = cos θ1 cos θ2 + sin θ1 sin θ2 cos (φ2 − φ1) , sin δ = sin θ2 sin (φ2 − φ1) sin γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A12) Since, out of all the angular variables, the denominator only depends on the relative angle γ, the integrals over θ1, φ1 and δ can be taken easily using the following relations � dΦ1dΦ2ki akj a = � dΦ1dΦr 2ki akj a = δij 8π2 3 k2 a � sin γdγ, � dΦ1dΦ2ki 1kj 2 = � dΦ1dΦr 1ki 1kj 2 = δij 8π2 3 (k1 · k2) � sin γdγ, � dΦ1dΦ2 = � dΦ1dΦr 2 = 8π2 � sin γdγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A13) Using this and the results of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A8) and (A9), we perform the integration over θ1, φ1 and δ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A4), and find the following expression for the power radiated in harmonic n, Pn = g4 (nΩ) 12π3 a2Ω2Q2 ψQ2 |jn|2 � δ(nΩ − ω1 − ω2) ((k1 + k2)2 − m2 A)2 + m2 AΓ2 A × � −1 2 (k1 + k2)2 � (k1 + k2)2 / � (nΩ)2 − 3 �� + 2 ω1ω2 (nΩ)2 (k1 + k2)2 −2 ω1 nΩ � k2 2 + k1 · k2 � − 2 ω2 nΩ � k2 1 + k1 · k2 � + 2k1 · k2 � × ω1ω2 � 1 − m2 ψ ω2 1 �1/2 � 1 − m2 ψ ω2 2 �1/2 sin γ dγdω1dω2 , (A14) where the only integrals left are the integrals over γ, ω1 and ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Next, we introduce the following dimensionless variables and parameters x1 = ω1 Ω , x2 = ω2 Ω , nψ = mψ Ω , nA = mA Ω , nΓ = ΓA Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A15) 27 Performing the change of variables in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A14) from (ω1, ω2) to (x1, x2), we rewrite the ex- pression for the power radiated in harmonic n as follows: Pn = g4 12π3a2Ω4Q2 ψ|jn|2 � sin γ dγ dx1 dx2 δ(n − x1 − x2) F(cos γ, x1, x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A16) Upon taking the integral over x2 and performing the replacement x1 → x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' we obtain Pn = g4 12π3a2Ω4Q2 ψQ2|jn|2 � n−nψ nψ dx � 1 −1 d(cos γ) F(cos γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' x) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A17) where function F(cos γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' x) is given by F(cos γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' x) = b(x) 2n 1 2b2(x) cos2 γ + b(x)c(x) cos γ + d(x) (a(x) − b(x) cos γ)2 + g2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A18) with a(x) = 2n2 ψ + 2x(n − x) − n2 A ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' b(x) = 2 � x2 − n2 ψ � (n − x)2 − n2 ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' c(x) = − � n2 + 2n2 ψ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' d(x) = 2(x(n3 − 2n2x + 2nx2 − x3) + 2n2n2 ψ + n4 ψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' g2 = n2 An2 Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A19) The variable x here is the ratio of the energy of one of the fermions to the fundamental oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' It can be at least nψ or at most n − nψ, hence the limits on the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Also note that F also depends on the parameters of the problem namely nA, nψ, nΓ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A15), but we do not write them explicitly for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lastly, note that the γ-dependence of the numerator of function F is through a term quadratic in cos γ and a term linear in cos γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This behavior is attributed to the theory that we pick – renormalizable theories such as in the case considered here would only contribute at most two powers of momentum in the matrix element, leading to a cos γ dependence that is at most quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' However non-renormalizable theories have more momenta in the matrix element, and will give us a different cos γ dependence in the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Now, we define F A(x) ≡ F A(n, x, nψ, nA, nΓ) = � 1 −1 d (cos γ) F (cos γ, x, n) , (A20) where the superscript A denotes the vector boson mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The integral over cos γ can be taken analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Then, we find that the function F A(x), has the form: F A(x) = F A 0 (x) + F A 1 (x) nMnΓ � tan−1 �a(x) + b(x) nMnΓ � − tan−1 �a(x) − b(x) nMnΓ �� + F A 2 (x) tanh−1 � 2a(x)b(x) a(x)2 + b(x)2 + n2 Mn2 Γ � , (A21) 28 with: F A 0 (x) = b(x)/2n , F A 1 (x) = 1 4n � n4 A + 4n2n2 ψ − n2 An2 Γ + 2n2 An2 − 4nxn2 A + 4x2n2 A � , F A 2 (x) = 1 2n � n2 A + n2 − 2nx + 2x2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A22) Consequently, the power loss formula of each mode with n > 2nψ becomes Pn = 2g4Q2 ψQ2 3(2π)3 a2Ω4 � J′ n(ne)2 + 1 − e2 e2 Jn(ne)2 � � n−nψ nψ dxF A(x), (A23) which gives us Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='18) for the case M = A, where we define for mediator M BM n (nM, nν, nΓ) ≡ � J′ n(ne)2 + 1 − e2 e2 Jn(ne)2 � � n−nψ nψ dx F M(x, n, nM, nν, nΓ), (A24) where Jn(z) is a Bessel function of order n in the variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The case of the scalar mediator The derivation for the power loss in the scalar mediator is similar to the vector case, but the matrix element is different, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This matrix element contains the number density ρcl(x) of source particles, instead of a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As such, the difference in the calculation in this case comes from the calculation of the squared matrix element, which in this case, is given by: � s1,s2 |Mn(s1, s2)|2 = g2g′2 ((k1 + k2)2 − m2 φ)2 + m2 φΓ2 φ Tr(( /k1 + mψ)( /k2 − mψ))|ρcl(Ωn)|2 = 4g2g′2 ((k1 + k2)2 − m2 φ)2 + m2 φΓ2 φ (k1 · k2 − m2 ψ)|ρcl(Ωn)|2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A25) The power loss is again given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='4), we find the Fourier Transform ρµ cl(Ωn) as: ρ0 cl(Ωn) = aΩN �jn · p nΩ � , (A26) where, like in the vector case, we define the 3-vector ji n as follows: jn = � −iJ′ n(ne), √ 1 − e2 e Jn(ne), 0 � , (A27) with Jn(z) denoting a Bessel function, amd p = k1 + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' After performing all the steps analogous to Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A4)–(A20) in the previous section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='e, after performing the angular integration, we get: Pn = g2g′2 12π3 a2Ω4N 2|jn|2 � n−nψ nψ dx � 1 −1 d cos γ F(cos γ, x) , (A28) 29 where function F(cos γ, x) is given by F(cos γ, x) = −b(x) 2n 1 2b2(x) cos2 γ + b(x)c(x) cos γ + d(x) (a(x) − b(x) cos γ)2 + g2 , (A29) with a(x) = 2n2 ψ + 2x(n − x) − n2 φ , b(x) = 2 � x2 − n2 ψ � (n − x)2 − n2 ψ , c(x) = (n − 2x)2 2 , d(x) = (n2 ψ − nx + x2)(n2 − 2n2 ψ − 2nx + 2x2), g2 = n2 φn2 Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A30) Like before, we define F φ(x) ≡ F φ(n, x, nψ, nφ, nΓ) = � 1 −1 d (cos γ) F (cos γ, x, n) , (A31) where the superscript φ denotes the scalar mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' The integral over cos γ can be taken analytically to find a form for F φ: F φ(x) = F φ 0 (x) + F φ 1 (x) nMnΓ � tan−1 �a(x) + b(x) nMnΓ � − tan−1 �a(x) − b(x) nMnΓ �� + F φ 2 (x) tanh−1 � 2a(x)b(x) a(x)2 + b(x)2 + n2 Mn2 Γ � , (A32) with: F φ 0 (x) = −b(x)/2n , F φ 1 (x) = 1 4n � n2 φn2 Γ + (n2 − n2 φ)(n2 φ − 4n2 ν) � , F φ 2 (x) = 1 4n � n2 + 4n2 ν − 2n2 φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A33) Consequently, the power loss formula of each mode with n > 2nψ becomes Pn = 2g2g′2 3(2π)3a2Ω4N 2 � J′ n(ne)2 + 1 − e2 e2 Jn(ne)2 � � n−nψ nψ dxF φ(x), (A34) which gives us Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='19) for the case M = φ P φ n = g2g′2 12π3 a2Ω4 �N1 m1 − N2 m2 �2 Bφ n(nA, nν, nΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' (A35) We find that the form of the function F M is general for the two types of mediators, the difference lying in the explicit forms of the functions F M 0 , F M 1 and F M 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' This is due to the fact that the cos γ dependence of the function F is the same in both cases, as in both cases, the theory considered is a renormalizable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' As we explained in the previous sub-section, this 30 general form of F M is not what we will have when we consider non-renormalizable theories that give us higher powers of momenta in the numerator of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Larmor, Philosophical Magazine Series 1 44, 503 (1897).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jackson, Classical Electrodynamics (Wiley, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Krause, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kloor, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fischbach, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 49, 6892 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Mohanty and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kumar Panda, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 53, 5723 (1996), arXiv:hep-ph/9403205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Dror, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Laha, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Opferkuch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 102, 023005 (2020), arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='12845 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kumar Poddar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Mohanty, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jana, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 100, 123023 (2019), arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='09732 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Johnson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sagunski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sakellariadou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Zhang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 99, 063013 (2019), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='02133 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kumar Poddar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Mohanty, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jana, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 101, 083007 (2020), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='00666 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Hook and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Huang, JHEP 06, 036 (2018), arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='08464 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Foot, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A 6, 527 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [11] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Joshi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lew, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Volkas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 44, 2118 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Foot, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lew, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Volkas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 50, 4571 (1994), arXiv:hep- ph/9401250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Heeck and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rodejohann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 84, 075007 (2011), arXiv:1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='5238 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kopp, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Laha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Opferkuch, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Shepherd, JHEP 11, 096 (2018), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='02527 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Alexander, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' McDonough, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sims, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Yunes, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 35, 235012 (2018), arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='05286 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Choi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jung, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 99, 015013 (2019), arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='01421 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fabbrichesi and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Urbano, JCAP 06, 007 (2020), arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='07914 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Seymour and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Yagi, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 37, 145008 (2020), arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='03353 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Feinberg and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sucher, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 166, 1638 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Hsu and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sikivie, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 49, 4951 (1994), arXiv:hep-ph/9211301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Ghosh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Grossman, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Tangarife, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 101, 116006 (2020), arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='09444 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Ghosh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Grossman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Tangarife, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Xu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Yu, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='07082 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 31 [23] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' van Straten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Bailes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Britton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kulkarni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Anderson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Manchester, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sarkissian, Nature 412, 158 (2001), arXiv:astro-ph/0108254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kramer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=', Science 314, 97 (2006), arXiv:astro-ph/0609417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [25] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Stairs, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Thorsett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Taylor, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Wolszczan, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 581, 501 (2002), arXiv:astro-ph/0208357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Shannon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Johnston, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Manchester, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 437, 3255 (2014), arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='0588 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Antoniadis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=', Science 340, 6131 (2013), arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='6875 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [28] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Bhat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Bailes, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Verbiest, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 77, 124017 (2008), arXiv:0804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='0956 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Freire, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Wex, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Esposito-Farese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Verbiest, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Bailes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jacoby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kramer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Stairs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Antoniadis, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Janssen, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 423, 3328 (2012), arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1450 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Ferdman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=', Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 443, 2183 (2014), arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='5507 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=', Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 798, 118 (2015), arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='1518 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [32] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jacoby, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Cameron, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Jenet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Anderson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Murty, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kulkarni, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 644, L113 (2006), arXiv:astro-ph/0605375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [33] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Davoudiasl and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Denton, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 123, 021102 (2019), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='09242 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Hulse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Taylor, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 195, L51 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Taylor and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Weisberg, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 253, 908 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Weisberg and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Huang, The Astrophysical Journal 829, 55 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kilic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Hermes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Gianninas, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Brown, Monthly Notices of the Royal Astronomical Society: Letters 446, L26 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [38] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Peters and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Mathews, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 131, 435 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Yakovlev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Kaminker, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Gnedin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Haensel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 354, 1 (2001), arXiv:astro-ph/0012122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Garani and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Heeck, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' D 100, 035039 (2019), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='10145 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Pearson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Chamel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Potekhin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Fantina, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Ducoin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Dutta, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Goriely, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 481, 2994 (2018), [Erratum: Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 486, 768 (2019)], arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='04981 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [42] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Harry and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Hinderer, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 35, 145010 (2018), arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content='09972 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' [43] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Zhang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Chen, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 18, 142 (2001), arXiv:nucl-th/0011017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 32 [44] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Maki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Nakagawa, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Sakata, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 28, 870 (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfWfy0/content/2301.01303v1.pdf'} diff --git a/UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/2301.02018v1.pdf.txt b/UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/2301.02018v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d92add0155afdeebc5995267f2c40171322d41b --- /dev/null +++ b/UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/2301.02018v1.pdf.txt @@ -0,0 +1,1037 @@ +Trajectory Optimization on Matrix Lie Groups +with Differential Dynamic Programming and +Nonlinear Constraints +Gokhan Alcana, Fares J. Abu-Dakkab and Ville Kyrkia +Abstract— Matrix Lie groups are an important class of +manifolds commonly used in control and robotics, and the +optimization of control policies on these manifolds is a +fundamental problem. In this work, we propose a novel +approach for trajectory optimization on matrix Lie groups +using an augmented Lagrangian based constrained dis- +crete Differential Dynamic Programming (DDP) algorithm. +Our method involves lifting the optimization problem to the +Lie algebra in the backward pass and retracting back to +the manifold in the forward pass. In contrast to previous +approaches which only addressed constraint handling for +specific classes of matrix Lie groups, our method provides +a general approach for nonlinear constraint handling for a +generic matrix Lie groups. We also demonstrate the effec- +tiveness of our method in handling external disturbances +through its application as a Lie-algebraic feedback control +policy on SE(3). The results show that our approach is able +to effectively handle configuration, velocity and input con- +straints and maintain stability in the presence of external +disturbances. +I. INTRODUCTION AND RELATED WORK +T +HE configuration space of a physical system represents +the set of all possible configurations. Modeling this space +using local coordinates may lead to several potential problems. +One issue is that these coordinate systems may suffer from +singularities or degeneracies, where the coordinates become +ill-defined or degenerate. For example, Euler angles can +experience gimbal lock, where two of the angles become +degenerate, leading to a loss of degree of freedom and making +it difficult to represent certain configurations of the system +[1], [2]. Similarly, quaternions can experience a similar loss +of degree of freedom when their magnitude becomes close to +zero and multiple different representations can describe the +same configuration [3], although it has global representation. +These types of singularities can make it difficult to accurately +represent the transition between certain configurations of the +system, and may require the use of additional techniques to +avoid or mitigate these issues. +This work was supported by the Academy of Finland B-REAL Project +under Grant 328399. +Corresponding author: Gokhan Alcan (gokhan.alcan@aalto.fi) +a The authors are with the Intelligent Robotics Group, Department of +Electrical Engineering and Automation (EEA), Aalto University, 02150 +Espoo, Finland +b Munich Institute of Robotics and Machine Intelligence, Technische +Universit¨at M¨unchen, 80992 M¨unchen, Germany. Part of the research +presented in this work has been conducted when Abu-Dakka, F. was at +Intelligent Robotics Group, EEA, Aalto University, 02150 Espoo, Finland +On the other hand, a natural way to model the geometry +of the configuration space is through the use of matrix Lie +groups, which offer a continuous and structured framework +for understanding the structure and motion of the underlying +system [4], [5]. However, it brings the difficulties to deal with +non-flatness of manifolds in a coordinate-free manner [6]. +The use of geometric control techniques, which seamlessly +combine differential geometry and control theory, has become +increasingly prevalent in the field of robotics and control +systems [7]–[9]. Optimality conditions for geometric control +techniques are often simplified and numerical ill-conditioning +is avoided through the use of specific details about the control +problem. +Differential dynamic programming (DDP) is a numerical +method for solving optimal control problems that has gained +widespread use in various engineering fields. Originally pro- +posed by Mayne and Jacobson [10], DDP has been applied to +a wide range of complex, high-dimensional systems [11]. One +of the key advantages of DDP is its scalability, which allows +it to handle large and complex systems with many degrees +of freedom. In addition, DDP has a fast convergence rate, +which allows it to quickly find near-optimal solutions to the +control problem. Another important attribute of DDP is its +ability to generate feedback control policies, which can be +used to implement the optimal control solution in real-time. +Several early works have investigated the use of DDP for +geometric control [9], [12]. These studies focused on specific +matrix Lie groups, including SE(3), in order to derive the +final form of the DDP algorithm for applications in geo- +metric control. Boutselis and Theodorou [13] extended the +original DDP method by using quadratic expansion schemes +for cost functions and dynamics defined on Lie groups. They +demonstrated that DDP has significantly better convergence +rates compared to sequential quadratic programming (SQP) +methods. Teng et al. [18] further improved the convergence +performance of DDP for matrix groups by designing the +control objective in its Lie algebra. Both of these approaches +[13], [18] formulate the trajectory optimization on matrix Lie +groups in an unconstrained framework. In order to address +this limitation, Liu et al. [14] extended the work [13] by +imposing SO(3) pose constraints. However, this method is not +generalizable to nonlinear constraints for generic matrix Lie +groups. Table I provides a comparison of those methods in +terms of their cost definitions and constraints. +arXiv:2301.02018v1 [eess.SY] 5 Jan 2023 + +TABLE I +COMPARISON OF DDP METHODS FOR MATRIX LIE GROUPS +Cost +SO(3) +Any Group +Described in +Constraints +Constraints +Boutselis et al. [13] +manifold + + +Liu et al. [14] +manifold + + +Teng et al. [18] +tangent space + + +Our method +tangent space + + +The present paper aims to solve the problem of generic +constraints by extending the idea of Lie algebric cost definition +[18] and developing a DDP method for matrix Lie groups +under nonlinear constraints. The main contributions of our +work are: +1) Development of an augmented Lagrangian based con- +strained DDP algorithm for trajectory optimization on +matrix Lie groups. +2) A principled approach for nonlinear constraint handling +for generic matrix Lie groups unlike [14], which only ad- +dressed constraint handling for SO(3) pose constraints. +3) Evaluating the effectiveness of the proposed DDP +method in handling external disturbances through its +application in a numerical simulation as a Lie-algebraic +feedback control policy on SE(3). +The rest of this paper is organized as follows. In Section II, +we provide preliminaries regarding matrix Lie groups. Section +III defines the trajectory optimization problem for matrix +Lie groups. We detail our proposed method in Section IV. +In Section V, we provide numerical simulation experiments +for SE(3) to demonstrate the effectiveness of our approach. +Finally, the paper is concluded with potential directions for +future work in Section VI. +II. PRELIMINARIES +Consider G is an n-dimensional matrix Lie group, and its +associated Lie algebra, i.e., tangent space at the identity, is +denoted as g, where dim g = n. Isomorphism between the +vector space Rn and g can be defined through the following +operators: +(.)∧ : Rn �→ g +(.)∨ : g �→ Rn +(1) +Mapping between Rn and G can be defined using the +functions Exp(.) : Rn �→ G and Log(.) : G �→ Rn for any +φ ∈ Rn and X ∈ G as follows: +Exp(φ) = expm(φ∧) = X +Log(X) = logm(X)∨ = φ +(2) +where expm and logm are the exponential and logarithm of +square matrices, respectively. +The adjoint action, denoted as AdX : g �→ g for any X ∈ G, +is a Lie algebra isomorphism that allows change of frames. +Given φ, η ∈ Rn and φ∧, η∧ ∈ g, the adjoint action can be +expressed in the function form as +AdX (φ) = Xφ∧X −1 +(3) +or in the matrix form as +(AdX φ)∧ = Xφ∧X −1 +(4) +The adjoint map is the derivative of the adjoint action with +respect to X at the identity element and is defined as +adφ η = [φ∧, η∧] +(5) +where [φ∧, η∧] is the Lie bracket, calculated as +[φ∧, η∧] = φ∧η∧ − η∧φ∧ +(6) +III. PROBLEM DEFINITION +We consider the systems whose states reside in the tangent +bundle of a matrix Lie group. This encompasses a diverse +array of systems [15] whose states can be represented as pairs +{X, ξ∧} ∈ G × g, where X represents the configuration and +ξ∧ represents the velocity. The continuous equations of motion +for such systems can be written as: +˙Xt = Xtξ∧ +t +˙ξt = f +� +ξt, ut +� +(7) +where ut ∈ Rm is the generalized control input and f(.) is +the function of velocity dynamics. For a given initial state +{X0, ξ0}, a goal state {Xg, ξg} and a time horizon N, we +define the discrete-time constrained optimal control problem +as +min +u0,...,uN−1 +ℓf(XN, ξN) + +N−1 +� +k=0 +ℓ(Xk, ξk, uk) +subject to +Xk+1 = FX (Xk, ξk), +k = 0, ..., N − 1 +ξk+1 = Fξ(ξk, uk), +k = 0, ..., N − 1 +umin ≤ uk ≤ umax, +∀k, +g(Xk, ξk, uk) ≤ 0, +∀k, +given +X0, ξ0, +(8) +where ℓf : G × Rn �→ R and ℓ : G × Rn × Rm �→ R +are the final cost and the running cost, respectively. FX and +Fξ are the discretized form of the configuration and velocity +dynamics, which can be obtained by using either a zero- +order hold or Euler first-order integration method with a fixed +time step of ∆t. Lastly, g is a vector of p constraints in +the form of differentiable nonlinear functions representing the +state constraints. +IV. PROPOSED METHOD +It is often difficult to solve the general problem outlined +in Section III analytically. Additionally, finding the global +minimum numerically can be time-consuming, particularly for +systems with high dimensions. Therefore, we aim to propose a +method that provides feasible solutions, even if they may not +be globally optimal. As such, we aim to propose a method +that yields feasible solutions, even if they may not be globally +optimal. To accomplish this, we utilize the Differential Dy- +namic Programming (DDP) framework [20], which iteratively +solves sub-optimization problems in the backward pass and +generates a new trajectory in the forward pass based on the +found optimal policy, in order to approach a local optimum. + +In this work, we propose augmenting the cost function with +multiplier and penalty terms from the augmented Lagrangian +in order to account for constraints imposed on the system, +whose states lie in the tangent bundle of a matrix Lie group. +Our approach involves lifting the problem to the Lie algebra +in the backward pass by computing the gradient of the cost +function within the corresponding Lie algebra, and retracting +back to the manifold in the forward pass by integrating the +dynamics using the optimal policy obtained in the backward +pass. +A. Dynamics on Tangent Space +The central concept of DDP is that, at each iteration, all +nonlinear constraints and objectives are approximated using +first or second order Taylor series expansions. This allows the +approximate functions, which now operate on deviations from +the nominal trajectory, to be solved using discrete Linear- +Quadratic Regulator (LQR) techniques. In order to define +the cost and constraint functions in Lie algebra, we need +to determine the error dynamics for the configuration. To +obtain the perturbed state dynamics, we followed the approach +proposed by Teng et al. [17]. For completeness, we outline the +necessary steps here. Interested readers may refer to [17], [18] +for more information. +Consider a perturbed state {Xp, ξ∧ +p } that is in the vicinity +of a nominal state {X, ξ∧}. Then, the configuration error can +be defined as +Ψ = X −1Xp ∈ G +(9) +Differentiating both sides of (9) yields the configuration error +dynamics as +˙Ψ = X −1 d +dt +� +Xp +� ++ d +dt +� +X −1� +Xp += X −1 ˙Xp − X −1 ˙XX −1Xp += X −1Xpξ∧ +p − X −1Xξ∧X −1Xp += Ψξ∧ +p − ξ∧Ψ +(10) +Here, we can define a vector ψ in Rn such that the matrix ex- +ponential of ψ∧ corresponds to Ψ, denoted as Ψ = expm(ψ∧). +Using the first-order approximation of the matrix exponential, +which states that expm(ψ∧) ≈ In + ψ∧, the dynamics of the +configuration error in (10) can be linearized as follows: +˙Ψ = Ψξ∧ +p − ξ∧Ψ += (In + ψ∧)ξ∧ +p − ξ∧(In + ψ∧) += ξ∧ +p + ψ∧ξ∧ +p − ξ∧ − ξ∧ψ∧ += ξ∧ +p − ξ∧ + ψ∧ξ∧ +p − ξ∧ψ∧ += ξ∧ +p − ξ∧ + ψ∧(ξ∧ − ξ∧ + ξ∧ +p ) − ξ∧ψ∧ += ξ∧ +p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧ + ψ∧(ξ∧ +p − ξ∧) += ξ∧ +p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧ += ξ∧ +p − ξ∧ + adψ ξ +˙ψ = ξ∧ +p − ξ∧ − adξ ψ +(11) +Note that the second order term of ψ∧(ξ∧ +p − ξ∧) is also +discarded to obtain the linear dynamics of the configuration +error. ψ in (11) is the perturbed configuration represented in +Lie algebra. The perturbed velocity and control input are also +defined as +δξ = ξp − ξ, +and +δu = up − u +(12) +The perturbed velocity dynamics then become: +δ ˙ξt = Γtδξt + Λtδut +(13) +where Γt and Λt are the Jacobians of f(ξt, ut) defined in (7) +around the nominal trajectory about ξt and ut. +Defining the perturbed states as concatenation +x = +�ψ +δξ +� +, +¯u = δu +(14) +the perturbed state dynamics are expressed as +˙x = h(x, u) +˙x = +�− adξ +In +0n×n +Γt +� +� +�� +� +≜At +x + +�0n×m +Λt +� +� +�� +� +≜Bt +¯u +(15) +The discretized versions of the matrices At and Bt can be +simply obtained by applying a zero-order hold or a first-order +Euler integration method with a fixed time step of ∆t. +B. Constraint Handling +By decomposing the state into configuration and velocity, +we can also decompose the constraints in vector g(Xk, ξk, uk) +in equation (8) into two types: those that constrain the velocity +and those that specify configurations to be avoided. This allows +us to separately handle the constraints on velocity (cξ) and on +configuration (cX ) as +cξ(ξk, uk) ≤ 0 +cX (Xk, ξk) ≤ 0 +(16) +The velocity component of the state (ξ) resides in Rn, and +as a result, constraints involving any metric in Euler space +produce a distance vector in Rn. Therefore, any boundary +velocity constraint can be written as +δξb = ξb − ξ, +¯cξ(βδξb) ≤ 0 +(17) +where +β = +� +−1 +if ξb is upper bound, ++1 +if ξb is lower bound +(18) +On the other hand, the difference between two group +elements in the configuration state produces a geodesic in +the group. To handle this, we propose mapping the distance +geodesic to the tangent space of the configuration at the current +time step and addressing the constraint in that vector space. +Configuration avoidance constraints can typically be formu- +lated as inequality constraints using an n-spherical function, +with the center of the n-sphere located at the configuration +to be avoided (Xc) and the radius (rc) defining the restricted +region. This allows us to specify a region of configurations +that should be avoided. The distance between the nominal and +restricted configurations in the tangent space of the nominal +trajectory, ψc, is given by: +ψc = logm(X −1Xc) +(19) + +Then, the configuration avoidance constraint can be written as +¯cX (ψc) = (r2 +c − ∥ψc∥2) ≤ 0 +(20) +In this approach, we consider the same restricted region +for each axis in the n-dimensional sphere. However, it is +also possible to specify different radius values for each axis, +resulting in an n-dimensional ellipsoid as the restricted region. +Our method can accommodate these types of configuration +constraints as well. +In order to handle the constraints in DDP framework, +we need the first-order approximations of them around the +perturbed state dynamics introduced in (15) as follows: +¯c(x + δx, u + δu) ≈ ¯c(x, u) ++ ¯cx(x, u)δx + ¯cu(x, u)δu +(21) +where +¯c(x) = +�¯cX (ψc) +¯cξ(δξb) +� +(22) +¯cx and ¯cu are the derivative of ¯c with respect to x and +u, respectively. If the constraints are designed according to +equations (18) and (20), the derivatives can be calculated as +follows: +¯cx(x, u) = +�−2(adξ ψc)⊤ +2ψc⊤ +01×n +β(Γδξb)⊤ +� +¯cu(x, u) = +� +01×m +β(Λ⊤δξb)⊤ +� +(23) +C. Constrained Differential Dynamic Programming +Using the perturbed state dynamics defined in (15), the +backward pass of differential dynamic programming (DDP) is +lifted to the tangent space. The backward pass of DDP involves +computing the cost-to-go function at each time step in a given +trajectory. Unlike [18], our algorithm not only considers the +objective function when calculating the cost-to-go function, +but also takes into account any constraints on the state and +control variables. +An effective method for solving constrained optimization +problems is to transform the constraints into the objective +function and iteratively increase the penalty for violating or +approaching them. This technique, known as penalty method, +guarantees convergence to the optimal solution as the penalty +terms increase indefinitely. However, this may not be prac- +tical to implement in numerical optimization routines due +to the limitations of finite precision arithmetic. Augmented +Lagrangian methods [19] offer an alternative solution by +maintaining estimates of the Lagrange multipliers associated +with the constraints, allowing for convergence to the optimal +solution without requiring the penalty terms to increase indef- +initely. +Here we obtain the augmented Lagrangian as +LA = LN(xN) + +N−1 +� +k=0 +Lk(xk, uk) +LN(xN) = ¯ℓf(xN) + (λ + 1 +2 ¯g(xN)Iµ)⊤¯g(xN) +Lk(xk, uk) = ¯ℓ(xk, uk) + (λ + 1 +2 ¯g(xk, uk)Iµ)⊤¯g(xk, uk) +(24) +where ¯ℓf : R2n �→ R and ¯ℓ : R2n ×Rm �→ R are the final cost +and the running cost functions for perturbed system dynamics, +respectively. A typical design of such functions are +¯ℓf(x) = 1 +2∥δx∥SV +¯ℓ(x, u) = 1 +2∥δx∥SQ + 1 +2∥δu∥SU +(25) +where SV ∈ R2n×2n, SQ ∈ R2n×2n and SU ∈ Rm×m are +the cost matrices that are specified by the user and remain +constant throughout all iterations. +In (24), ¯g is a vector of p constraints for perturbed state +dynamics as introduced in (21). λ ∈ Rp is a Lagrange +multiplier, µ ∈ Rp is a penalty weight and Iµ ∈ Rp×p is +the penalty matrix defined as +Iµ = +� +0 +if gi(.) < 0 and λi = 0, +µi +otherwise +(26) +In general, one can define time varying Lagrange multiplier +and penalty weight, but we kept them constant for each time +step during the same iteration for simplicity. +We define the cost-to-go and action-value functions as +Vk(xk) = min +uk {Lk(xk, uk)} + Vk+1(Akxk + Bkuk) += min +uk Q(xk, uk)) +(27) +The matrices Ak and Bk represent the discretized versions +of At and Bt in equation (15). Second order Taylor series +expansion of cost-to-go function can be written as +δVk(x) ≈ 1 +2δx⊤ +k Vxx,kδxk + V ⊤ +x,kδxk +(28) +where Vxx,k and Vx,k are the Hessian and gradient of the +cost-to-go at time step k, respectively. Action-value function +defined in (27) can be also approximated as a quadratic +function as +Q(x + δx, u + δu) ≈ Q(x, u) + Q⊤ +x δx + Q⊤ +uδu ++ 1 +2(δx⊤Qxxδx + δx⊤Qxxδx) ++ δx⊤Qxuδu +(29) +To compute the derivative matrices in (29): +Qx = ¯ℓx + A⊤V ′ +x + ¯g⊤ +x (λ + Iµ¯g) +Qu = ¯ℓu + B⊤V ′ +x + ¯g⊤ +u (λ + Iµ¯g) +Qxx = ¯ℓxx + A⊤V ′ +xxA + ¯g⊤ +x Iµ¯gx + (V ′ +xhxx) +Quu = ¯ℓuu + B⊤V ′ +xxB + ¯g⊤ +u Iµ¯gu + (V ′ +xhuu) +Qux = ¯ℓux + B⊤V ′ +xxA + ¯g⊤ +u Iµ¯gx + (V ′ +xhux) +(30) +To simplify the notation, we have omitted the time indices +on all variables. All variables in this expression are evaluated +at time step k, except for those marked with ′, which are +evaluated at time step k + 1. +Calculating the full second-order expansion of the state +dynamics (hxx, huu, hux), can be computationally expensive, +particularly for systems with complex dynamics and high- +dimensional states. DDP refers to iterative LQR by discarding +the second-order dynamics and computing only the first-order + +expansion. This results in a Gauss-Newton approximation of +the true Hessian, which reduces the local fidelity and requires +more iterations. However, these iterations are less expensive +to compute and often lead to a faster overall convergence +rate. Therefore, in this work, we eliminated the second order +dynamics as approximating the perturbed state dynamics as +described in (11). +Minimizing (29) with respect to δu results in an affine +controller +δu∗ = −Q−1 +uu(Quxδx + Qu) ≜ Kδx + d +(31) +Substituting δu∗ into (29) yields the derivatives of the cost- +to-go at time step k in terms of the derivatives of the action +value function as: +Vx = Qx + KQu + K⊤Quud + Q⊤ +uxd, +Vxx = Qxx + K⊤QuuK + K⊤Qux + Q⊤ +uxK +(32) +At the final step, Vx and Vxx can be easily computed as the +first and second derivatives of the final cost function (¯ℓf). This +way, the derivatives of the action-value function (30) and in +turn the local optimal control policy (31) at each step can be +calculated backwards starting from the final step. +After determining the optimal control policy for each time +step, we update the nominal trajectories by simulating the +dynamics forward on the manifold itself starting from the +initial state as: +δxk = +�logm(Xk−1 ¯ +Xk) +¯ξk − ξk +� +δuk = Kkδxk + αdk +¯uk = uk + δuk +¯ξk = ξ + f(ξ, ¯uk)∆t +¯ +Xk = X expm(¯ξk∆t) +(33) +where {Xk, ξk, uk} and { ¯ +Xk, ¯ξk, ¯uk} represent the nominal +state-actions and the updated state-actions at time step k, +respectively. In the above expression, 0 ≤ α ≤ 1 is a +scaling term for simple linear search on the feedforward term. +Practically, the parameter α is initially set to 1, but if the +cost of the updated trajectory does not decrease, it will be +decreased. Once the cost of the updated trajectory is decreased, +the forward pass is succesfully completed, the new trajectory +is accepted as nominal trajectory and the backward pass is +triggered. +To optimize the performance of DDP-based algorithms, +there are a few implementation practices to consider. In the +backward pass, Quu may need to be regularized as Quu +ρI +if it is invertible or the former forward pass is unsuccesful, i.e., +the cost is not decreased. After the DDP iterations converge, +the parameters λ and µ can be updated as: +λ+ = max(0, λi + µi¯gi(x∗, u∗)) +µ+ = γµ, +γ > 0 +(34) +The DDP iterations can then be restarted until convergence is +achieved. For more information, see reference [19]. +V. EXPERIMENTS +The goal of this section is to devise a trajectory on SE(3) +that satisfies both position and orientation constraints while +avoiding unsafe configurations and obstacles. +A. Dynamics on SE(3) +We consider a 3D rigid body in SE(3) where the states of +the system can be represented by a rotation matrix +R ∈ SO(3) = {R ∈ R3×3|R⊤R = I3, det(R) = 1} +(35) +and position p ∈ R3. The homogeneous representation of a +typical group element in SE(3) is +X = +�R +p +0 +1 +� +∈ SE(3) +(36) +The velocity vector ξ in SE(3) is known as a “twist” and is +composed of both angular (ω) and linear (v) velocities in body +frame as +ξ = +� +ω +v +� +∈ R6, +ξ∧ = +� +ω∧ +v +0 +0 +� +∈ se(3) +(37) +The forced Euler-Poincar´e equations [16] define the twist +dynamics as +Jb ˙ξ = ad∗ +ξ Jbξ + u +(38) +In this expression, Jb represents the generalized inertia matrix +in the body fixed principal axes, while u ∈ g∗ represents +the generalized control input forces applied to these axes. +g∗ denotes the cotangent space and the coadjoint map is +represented by ad∗ +ξ. The matrix representations of the adjoint +action in the Lie algebra (adξ) and the coadjoint map (ad∗ +ξ) +are as follows: +adξ = +�ω∧ +0 +v∧ +ω∧ +� +, +ad∗ +ξ = ad⊤ +ξ = − +�ω∧ +v∧ +0 +ω∧ +� +. +(39) +The continuous equations of motion described in (7) can be +written for SE(3) group as +˙X = Xξ∧, +˙ξ = Jb +−1� +ad∗ +ξ Jbξ + u +� +(40) +The linearized twist dynamics as given in [17]: +˙ξ = Γtξ + Λtu + bt +(41) +where Γt, Λt and bt are: +Γt := J−1 +b +ad∗ +ξ Jb + J−1 +b +�(Ibω)∧ +mv∧ +mv∧ +0 +� +Λt := Jb +−1 +bt := −Jb +−1 +�(Ibω)∧ +mv∧ +mv∧ +0 +� +ξ +(42) +assuming the inertia matrix Jb is defined as +Jb := +�Ib +0 +0 +mI3 +� +(43) +where Ib and m are the moment of inertia in the body frame +and the body mass, respectively. + +Fig. 1. Constrained (red) and unconstrained (blue) trajectories generated by proposed DDP method. +Fig. 2. +Control input signals of the planned trajectories. Torques refers +to inputs acting on angular velocity and Forces refers to inputs acting on +linear velocity. Left column: unconstrained, Right column: constrained +B. Simulations +We now test the proposed algorithm for planning a safe path +for a SE(3) rigid body. The task we have defined involves +rotating the body to the configuration Rz(180) from the +identity and translating it to the position {2, 2, 2} from the +initial position {1, 1, 1} in 3 seconds with a fixed time step +of ∆t = 0.01. To denote the rotation around the x, y, and z +axes of the body-fixed frame, we use Rx(.), Ry(.), and Rz(.), +respectively. During the motion, the configuration Rz(90) is +considered to be unsafe and must be avoided. Additionally, it is +assumed that there is a spherical obstacle at {1.25, 1.25, 1.25} +with a radius of r = 0.5 that must be avoided as well. +We only penalized the final state and control inputs, so we +set the controller parameters as SV = 1000I12, SU = 0.001I6, +and SQ = 0. In order to assess the performance of the +proposed DDP in terms of constraint handling, we conducted +an experiment with the same trajectory optimization task in +both constrained and unconstrained cases, and the resulting +trajectories are depicted in Fig. 1. +The +configuration +trajectories +were +converted +to +Eu- +ler angles (φ, θ, ψ in degrees) using the rotation order +Rx(.)Ry(.)Rz(.). In the unconstrained case (shown in blue), it +is shown that the ψ angle goes directly from 0 to 180 degrees +while the φ and θ angles remain constant (Fig. 1), as the +task only involves rotating around the z-axis. This can also be +observed in the upper left part of Fig. 2, where only a single +input is non-zero to achieve the desired motion, while the +others remain zero. However, in the constrained case (shown +in red), rotations around x and y axes (changes in φ and θ +angles) were also observed to avoid the unsafe configuration +of Rx(90) (Fig. 1). +As shown in the last row of Fig. 1, the shortest paths for +the positional states were obtained in the unconstrained case. +However, the presence of a sphere at {1.25, 1.25, 1.25} with +a radius of r = 0.5 forced the positional trajectories to take +a circuitous route around obstacle by deviating along y and x +axes. +To evaluate the effectiveness of the proposed DDP method +in handling external disturbances, we extend the analysis done +in [13] to SE(3) and employ the proposed DDP method as a +Lie-algebraic feedback control: +ufb +k := u∗ +k + Kk logm((X ∗ +k )−1X ϵ +k) +(44) +where u∗ +k and Kk represent the (sub)optimal control sequence +and time-varying feedback gains, respectively, which are ob- +tained through DDP. The feed-forward term dk in (31) is +not explicitly shown in the feedback policy (44) because it +is already included in u∗ +k. The variables X ∗ +k and X ϵ +k represent +the (sub)optimal states obtained through DDP convergence and +the perturbed states due to disturbance, respectively. +We test the policy (44) on the following stochastic version +of the SE(3) dynamics: +X ϵ +k+1 = X ϵ +k expmSE(3)(ξϵ +k∆t) +ξϵ +k+1 = ξϵ +k + f(ξϵ +k, uk)∆t + σωω +(45) +where ω is assumed to be spatially uncorrelated independent +and identically distributed noise, drawn from a zero mean +Gaussian distribution, ω ∼ N(06×1, I6), σω = 0.001. The +performance of the proposed DDP method under stochas- +tic conditions was evaluated by allowing the optimizer to +converge on the deterministic system and then testing its +performance on the stochastic system. Fig. 3 shows the results +of this comparison, using 1000 sampled trajectories under +noisy dynamics to compare the open-loop policy u∗ +k with the +feedback policy ufb +k +(44). The results demonstrate that the +use of the obtained feedback gains significantly reduces state +variance, particularly in the vicinity of the the goal points, as +expected. + +50 +0 +[6ap] +[deg] +[deg] +100 +0 +-50 +0 +-50 +0 +2.0 +2.0 +2.0 +1.5 +× 1.5 +y +N 1.5 +1.0 +1.0 +1.0 +0 +1 +2 +3 +0 +1 +2 +3 +0 +1 +2 +3 +Time [sec] +Time [sec] +Time [sec]20 +Torques +25 +0 +0 +-25 +-20 +0.5 +Forces +0 +0.0 +-10 +-0.5 +0 +2 +0 +2 +Time [sec] +Time [sec]Fig. 3. Evaluation of DDP’s control sequence under random disturbances. Left Column: Open-loop control, Right column: Closed-loop control +VI. CONCLUSION +In conclusion, the optimization of control policies on matrix +Lie groups is an important problem in robotics and control +with numerous applications. In this work, we have proposed a +novel approach for tackling this problem using an augmented +Lagrangian based constrained discrete DDP algorithm. Our +method involves lifting the optimization problem to the Lie +algebra in the backward pass, allowing us to compute the +gradient of the objective and constraint functions within the +corresponding tangent space. In the forward pass, we retract +back to the manifold by integrating the dynamics using the +optimal policy obtained in the backward pass. +One of the key contributions of our work is the development +of a general approach for nonlinear constraint handling for +a wide range of matrix Lie groups. Previous methods were +only able to handle constraints for specific classes of matrix +Lie groups, such as SO(3) pose constraints. Our method, on +the other hand, is able to handle a wide range of nonlinear +constraints, making it a more widely applicable and flexible +solution. +In addition, we have demonstrated the effectiveness of our +method in handling external disturbances through its applica- +tion as a Lie-algebraic feedback control policy on SE(3). The +results show that our approach is able to effectively handle +constraints and maintain its stability in the presence of external +disturbances. +Overall, our proposed DDP algorithm represents a signif- +icant step forward in the field of optimization on matrix +Lie groups. It provides a flexible and general approach for +nonlinear constraint handling and has the potential to be +applied to a wide range of problems in robotics and control. +As a future direction, it would be interesting to investigate +the closed loop uncertainty propagation through the prediction +horizon, as has been done for Euclidean models [11], in order +to design a robust Model Predictive Control (MPC) framework +for matrix Lie groups. This would allow the proposed method +to be used in more challenging and dynamic environments. +Additionally, further experimentation with different types of +matrix groups and constraints would provide valuable insights +into the capabilities and limitations of the proposed DDP +algorithm. +REFERENCES +[1] E. G. Hemingway and O. M. O‘Reilly, “Perspectives on Euler angle +singularities, gimbal lock, and the orthogonality of applied forces and +applied moments,” Multibody Syst. Dyn., 44, 31–56, 2018. +[2] M. D. Shuster, “A survey of attitude representations.” Navigation, 8(9), +439-517, 1993 +[3] J. Diebel, “Representing attitude: Euler angles, unit quaternions, and +rotation vectors.” Matrix, 58(15-16), 1-35, 2006. +[4] A. M. Bloch, Nonholonomic Mechanics and Control, P. S. Krishnaprasad +and R. M. Murray, Eds. Springer, New York, NY, 2015. +[5] K. M. Lynch and F. C. Park, Modern robotics. Cambridge University +Press, 2017 +[6] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems: +Modeling, Analysis, and Design for Simple Mechanical Control Systems, +49, Springer, 2019. +[7] M. Kobilarov, M. Desbrun, J. E. Marsden, and G. S. Sukhatme, “A dis- +crete geometric optimal control framework for systems with symmetries,” +in Proc. Robot., Sci. Syst.,1–8, 2008. +[8] A. Saccon, J. Hauser, and A. P. Aguiar, “Optimal control on Lie groups: +The projection operator approach,” IEEE Trans. Autom. Control, 58, 9, +2230-–2245, 2013. +[9] M. Kobilarov, “Discrete optimal control on lie groups and applications to +robotic vehicles,” in Proc. IEEE Int. Conf. Robot. Autom., 5523–5529, +2014. +[10] D. H. Jacobson and D. Q. Mayne, Differential Dynamic Programming. +New York, NY, USA: Elsevier, 1970. +[11] G. Alcan and V. Kyrki, “Differential Dynamic Programming With +Nonlinear Safety Constraints Under System Uncertainties,” in IEEE +Robotics and Automation Letters, vol. 7, no. 2, pp. 1760-1767, 2022 + +150 +150 +Φ,e, [deg] +100 +,e,w [deg] +100 +50 +50 +0 +0 +-50 +-50 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time [sec] +Time [sec] +1.0 +1.0 +0.8 +0.8 +0.6 +0.6 +X,y,z +z'Kx +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0.2 +0.2 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time [sec] +Time [sec][12] M. Kobilarov, D.N. Ta, and F. Dellaert, “Differential dynamic program- +ming for optimal estimation,” in Proc. IEEE Int. Conf. Robot. Autom., +863-–869,2015 +[13] G.I. Boutselis, E. Theodorou, “Discrete-time differential dynamic pro- +gramming on lie groups: Derivation, convergence analysis, and numerical +results.” IEEE Transactions on Automatic Control, 66(10), 4636–4651, +2020. +[14] S. Liu, and D. Liu, “Discrete-Time Differential Dynamic Programming +on SO(3) With Pose Constraints.” IEEE Access, 10, 112921-112933, 2022 +[15] T. Lee, “Computational geometric mechanics and control of rigid +bodies,” Ph.D. dissertation, Univ. Michigan, Ann Arbor, MI, USA, 2008. +[16] A. Bloch, P. S. Krishnaprasad, J. E. Marsden, and T. S. Ratiu, “The +Euler-Poincar´e equations and double bracket dissipation,” Communica- +tions in Mathematical Physics, 175(1), 1–42, 1996. +[17] S. Teng, D. Chen, W. Clark and M. Ghaffari, “An Error-State Model +Predictive Control on Connected Matrix Lie Groups for Legged Robot +Control,” IEEE/RSJ International Conference on Intelligent Robots and +Systems, 8850-8857, 2022. +[18] S. Teng, W. Clark, A. Bloch, R. Vasudevan and M. Ghaffari, +“Lie Algebraic Cost Function Design for Control on Lie Groups.” +arXiv:2204.09177, 2022 +[19] T. A. Howell, B. E. Jackson and Z. Manchester, “ALTRO: A Fast +Solver for Constrained Trajectory Optimization,” IEEE/RSJ International +Conference on Intelligent Robots and Systems, pp. 7674-7679,2019. + diff --git a/UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/load_file.txt b/UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de4a99be806e7609c5ea28d6bb6454ed2fe833bb --- /dev/null +++ b/UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/load_file.txt @@ -0,0 +1,398 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf,len=397 +page_content='Trajectory Optimization on Matrix Lie Groups with Differential Dynamic Programming and Nonlinear Constraints Gokhan Alcana, Fares J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Abu-Dakkab and Ville Kyrkia Abstract— Matrix Lie groups are an important class of manifolds commonly used in control and robotics, and the optimization of control policies on these manifolds is a fundamental problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In this work, we propose a novel approach for trajectory optimization on matrix Lie groups using an augmented Lagrangian based constrained dis- crete Differential Dynamic Programming (DDP) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Our method involves lifting the optimization problem to the Lie algebra in the backward pass and retracting back to the manifold in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In contrast to previous approaches which only addressed constraint handling for specific classes of matrix Lie groups, our method provides a general approach for nonlinear constraint handling for a generic matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' We also demonstrate the effec- tiveness of our method in handling external disturbances through its application as a Lie-algebraic feedback control policy on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The results show that our approach is able to effectively handle configuration, velocity and input con- straints and maintain stability in the presence of external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' INTRODUCTION AND RELATED WORK T HE configuration space of a physical system represents the set of all possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Modeling this space using local coordinates may lead to several potential problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' One issue is that these coordinate systems may suffer from singularities or degeneracies, where the coordinates become ill-defined or degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' For example, Euler angles can experience gimbal lock, where two of the angles become degenerate, leading to a loss of degree of freedom and making it difficult to represent certain configurations of the system [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Similarly, quaternions can experience a similar loss of degree of freedom when their magnitude becomes close to zero and multiple different representations can describe the same configuration [3], although it has global representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' These types of singularities can make it difficult to accurately represent the transition between certain configurations of the system, and may require the use of additional techniques to avoid or mitigate these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This work was supported by the Academy of Finland B-REAL Project under Grant 328399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Corresponding author: Gokhan Alcan (gokhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='alcan@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='fi) a The authors are with the Intelligent Robotics Group, Department of Electrical Engineering and Automation (EEA), Aalto University, 02150 Espoo, Finland b Munich Institute of Robotics and Machine Intelligence, Technische Universit¨at M¨unchen, 80992 M¨unchen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Part of the research presented in this work has been conducted when Abu-Dakka, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' was at Intelligent Robotics Group, EEA, Aalto University, 02150 Espoo, Finland On the other hand, a natural way to model the geometry of the configuration space is through the use of matrix Lie groups, which offer a continuous and structured framework for understanding the structure and motion of the underlying system [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, it brings the difficulties to deal with non-flatness of manifolds in a coordinate-free manner [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The use of geometric control techniques, which seamlessly combine differential geometry and control theory, has become increasingly prevalent in the field of robotics and control systems [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Optimality conditions for geometric control techniques are often simplified and numerical ill-conditioning is avoided through the use of specific details about the control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Differential dynamic programming (DDP) is a numerical method for solving optimal control problems that has gained widespread use in various engineering fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Originally pro- posed by Mayne and Jacobson [10], DDP has been applied to a wide range of complex, high-dimensional systems [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' One of the key advantages of DDP is its scalability, which allows it to handle large and complex systems with many degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In addition, DDP has a fast convergence rate, which allows it to quickly find near-optimal solutions to the control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Another important attribute of DDP is its ability to generate feedback control policies, which can be used to implement the optimal control solution in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Several early works have investigated the use of DDP for geometric control [9], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' These studies focused on specific matrix Lie groups, including SE(3), in order to derive the final form of the DDP algorithm for applications in geo- metric control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Boutselis and Theodorou [13] extended the original DDP method by using quadratic expansion schemes for cost functions and dynamics defined on Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' They demonstrated that DDP has significantly better convergence rates compared to sequential quadratic programming (SQP) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [18] further improved the convergence performance of DDP for matrix groups by designing the control objective in its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Both of these approaches [13], [18] formulate the trajectory optimization on matrix Lie groups in an unconstrained framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In order to address this limitation, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [14] extended the work [13] by imposing SO(3) pose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, this method is not generalizable to nonlinear constraints for generic matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Table I provides a comparison of those methods in terms of their cost definitions and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='02018v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='SY] 5 Jan 2023 TABLE I COMPARISON OF DDP METHODS FOR MATRIX LIE GROUPS Cost SO(3) Any Group Described in Constraints Constraints Boutselis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [13] manifold \x17 \x17 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [14] manifold \x13 \x17 Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [18] tangent space \x17 \x17 Our method tangent space \x13 \x13 The present paper aims to solve the problem of generic constraints by extending the idea of Lie algebric cost definition [18] and developing a DDP method for matrix Lie groups under nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The main contributions of our work are: 1) Development of an augmented Lagrangian based con- strained DDP algorithm for trajectory optimization on matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 2) A principled approach for nonlinear constraint handling for generic matrix Lie groups unlike [14], which only ad- dressed constraint handling for SO(3) pose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 3) Evaluating the effectiveness of the proposed DDP method in handling external disturbances through its application in a numerical simulation as a Lie-algebraic feedback control policy on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In Section II, we provide preliminaries regarding matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Section III defines the trajectory optimization problem for matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' We detail our proposed method in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In Section V, we provide numerical simulation experiments for SE(3) to demonstrate the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Finally, the paper is concluded with potential directions for future work in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' PRELIMINARIES Consider G is an n-dimensional matrix Lie group, and its associated Lie algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', tangent space at the identity, is denoted as g, where dim g = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Isomorphism between the vector space Rn and g can be defined through the following operators: (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' )∧ : Rn �→ g (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' )∨ : g �→ Rn (1) Mapping between Rn and G can be defined using the functions Exp(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=') : Rn �→ G and Log(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=') : G �→ Rn for any φ ∈ Rn and X ∈ G as follows: Exp(φ) = expm(φ∧) = X Log(X) = logm(X)∨ = φ (2) where expm and logm are the exponential and logarithm of square matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The adjoint action, denoted as AdX : g �→ g for any X ∈ G, is a Lie algebra isomorphism that allows change of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Given φ, η ∈ Rn and φ∧, η∧ ∈ g, the adjoint action can be expressed in the function form as AdX (φ) = Xφ∧X −1 (3) or in the matrix form as (AdX φ)∧ = Xφ∧X −1 (4) The adjoint map is the derivative of the adjoint action with respect to X at the identity element and is defined as adφ η = [φ∧, η∧] (5) where [φ∧, η∧] is the Lie bracket, calculated as [φ∧, η∧] = φ∧η∧ − η∧φ∧ (6) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' PROBLEM DEFINITION We consider the systems whose states reside in the tangent bundle of a matrix Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This encompasses a diverse array of systems [15] whose states can be represented as pairs {X, ξ∧} ∈ G × g, where X represents the configuration and ξ∧ represents the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The continuous equations of motion for such systems can be written as: ˙Xt = Xtξ∧ t ˙ξt = f � ξt, ut � (7) where ut ∈ Rm is the generalized control input and f(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=') is the function of velocity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' For a given initial state {X0, ξ0}, a goal state {Xg, ξg} and a time horizon N, we define the discrete-time constrained optimal control problem as min u0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=',uN−1 ℓf(XN, ξN) + N−1 � k=0 ℓ(Xk, ξk, uk) subject to Xk+1 = FX (Xk, ξk), k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', N − 1 ξk+1 = Fξ(ξk, uk), k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', N − 1 umin ≤ uk ≤ umax, ∀k, g(Xk, ξk, uk) ≤ 0, ∀k, given X0, ξ0, (8) where ℓf : G × Rn �→ R and ℓ : G × Rn × Rm �→ R are the final cost and the running cost, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' FX and Fξ are the discretized form of the configuration and velocity dynamics, which can be obtained by using either a zero- order hold or Euler first-order integration method with a fixed time step of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Lastly, g is a vector of p constraints in the form of differentiable nonlinear functions representing the state constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' PROPOSED METHOD It is often difficult to solve the general problem outlined in Section III analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Additionally, finding the global minimum numerically can be time-consuming, particularly for systems with high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Therefore, we aim to propose a method that provides feasible solutions, even if they may not be globally optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' As such, we aim to propose a method that yields feasible solutions, even if they may not be globally optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' To accomplish this, we utilize the Differential Dy- namic Programming (DDP) framework [20], which iteratively solves sub-optimization problems in the backward pass and generates a new trajectory in the forward pass based on the found optimal policy, in order to approach a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In this work, we propose augmenting the cost function with multiplier and penalty terms from the augmented Lagrangian in order to account for constraints imposed on the system, whose states lie in the tangent bundle of a matrix Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Our approach involves lifting the problem to the Lie algebra in the backward pass by computing the gradient of the cost function within the corresponding Lie algebra, and retracting back to the manifold in the forward pass by integrating the dynamics using the optimal policy obtained in the backward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Dynamics on Tangent Space The central concept of DDP is that, at each iteration, all nonlinear constraints and objectives are approximated using first or second order Taylor series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This allows the approximate functions, which now operate on deviations from the nominal trajectory, to be solved using discrete Linear- Quadratic Regulator (LQR) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In order to define the cost and constraint functions in Lie algebra, we need to determine the error dynamics for the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' To obtain the perturbed state dynamics, we followed the approach proposed by Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' For completeness, we outline the necessary steps here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Interested readers may refer to [17], [18] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Consider a perturbed state {Xp, ξ∧ p } that is in the vicinity of a nominal state {X, ξ∧}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Then, the configuration error can be defined as Ψ = X −1Xp ∈ G (9) Differentiating both sides of (9) yields the configuration error dynamics as ˙Ψ = X −1 d dt � Xp � + d dt � X −1� Xp = X −1 ˙Xp − X −1 ˙XX −1Xp = X −1Xpξ∧ p − X −1Xξ∧X −1Xp = Ψξ∧ p − ξ∧Ψ (10) Here, we can define a vector ψ in Rn such that the matrix ex- ponential of ψ∧ corresponds to Ψ, denoted as Ψ = expm(ψ∧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Using the first-order approximation of the matrix exponential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' which states that expm(ψ∧) ≈ In + ψ∧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' the dynamics of the configuration error in (10) can be linearized as follows: ˙Ψ = Ψξ∧ p − ξ∧Ψ = (In + ψ∧)ξ∧ p − ξ∧(In + ψ∧) = ξ∧ p + ψ∧ξ∧ p − ξ∧ − ξ∧ψ∧ = ξ∧ p − ξ∧ + ψ∧ξ∧ p − ξ∧ψ∧ = ξ∧ p − ξ∧ + ψ∧(ξ∧ − ξ∧ + ξ∧ p ) − ξ∧ψ∧ = ξ∧ p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧ + ψ∧(ξ∧ p − ξ∧) = ξ∧ p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧ = ξ∧ p − ξ∧ + adψ ξ ˙ψ = ξ∧ p − ξ∧ − adξ ψ (11) Note that the second order term of ψ∧(ξ∧ p − ξ∧) is also discarded to obtain the linear dynamics of the configuration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' ψ in (11) is the perturbed configuration represented in Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The perturbed velocity and control input are also defined as δξ = ξp − ξ, and δu = up − u (12) The perturbed velocity dynamics then become: δ ˙ξt = Γtδξt + Λtδut (13) where Γt and Λt are the Jacobians of f(ξt, ut) defined in (7) around the nominal trajectory about ξt and ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Defining the perturbed states as concatenation x = �ψ δξ � , ¯u = δu (14) the perturbed state dynamics are expressed as ˙x = h(x, u) ˙x = �− adξ In 0n×n Γt � � �� � ≜At x + �0n×m Λt � � �� � ≜Bt ¯u (15) The discretized versions of the matrices At and Bt can be simply obtained by applying a zero-order hold or a first-order Euler integration method with a fixed time step of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Constraint Handling By decomposing the state into configuration and velocity, we can also decompose the constraints in vector g(Xk, ξk, uk) in equation (8) into two types: those that constrain the velocity and those that specify configurations to be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This allows us to separately handle the constraints on velocity (cξ) and on configuration (cX ) as cξ(ξk, uk) ≤ 0 cX (Xk, ξk) ≤ 0 (16) The velocity component of the state (ξ) resides in Rn, and as a result, constraints involving any metric in Euler space produce a distance vector in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Therefore, any boundary velocity constraint can be written as δξb = ξb − ξ, ¯cξ(βδξb) ≤ 0 (17) where β = � −1 if ξb is upper bound, +1 if ξb is lower bound (18) On the other hand, the difference between two group elements in the configuration state produces a geodesic in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' To handle this, we propose mapping the distance geodesic to the tangent space of the configuration at the current time step and addressing the constraint in that vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Configuration avoidance constraints can typically be formu- lated as inequality constraints using an n-spherical function, with the center of the n-sphere located at the configuration to be avoided (Xc) and the radius (rc) defining the restricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This allows us to specify a region of configurations that should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The distance between the nominal and restricted configurations in the tangent space of the nominal trajectory, ψc, is given by: ψc = logm(X −1Xc) (19) Then, the configuration avoidance constraint can be written as ¯cX (ψc) = (r2 c − ∥ψc∥2) ≤ 0 (20) In this approach, we consider the same restricted region for each axis in the n-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, it is also possible to specify different radius values for each axis, resulting in an n-dimensional ellipsoid as the restricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Our method can accommodate these types of configuration constraints as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In order to handle the constraints in DDP framework, we need the first-order approximations of them around the perturbed state dynamics introduced in (15) as follows: ¯c(x + δx, u + δu) ≈ ¯c(x, u) + ¯cx(x, u)δx + ¯cu(x, u)δu (21) where ¯c(x) = �¯cX (ψc) ¯cξ(δξb) � (22) ¯cx and ¯cu are the derivative of ¯c with respect to x and u, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' If the constraints are designed according to equations (18) and (20), the derivatives can be calculated as follows: ¯cx(x, u) = �−2(adξ ψc)⊤ 2ψc⊤ 01×n β(Γδξb)⊤ � ¯cu(x, u) = � 01×m β(Λ⊤δξb)⊤ � (23) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Constrained Differential Dynamic Programming Using the perturbed state dynamics defined in (15), the backward pass of differential dynamic programming (DDP) is lifted to the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The backward pass of DDP involves computing the cost-to-go function at each time step in a given trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Unlike [18], our algorithm not only considers the objective function when calculating the cost-to-go function, but also takes into account any constraints on the state and control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' An effective method for solving constrained optimization problems is to transform the constraints into the objective function and iteratively increase the penalty for violating or approaching them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This technique, known as penalty method, guarantees convergence to the optimal solution as the penalty terms increase indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, this may not be prac- tical to implement in numerical optimization routines due to the limitations of finite precision arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Augmented Lagrangian methods [19] offer an alternative solution by maintaining estimates of the Lagrange multipliers associated with the constraints, allowing for convergence to the optimal solution without requiring the penalty terms to increase indef- initely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Here we obtain the augmented Lagrangian as LA = LN(xN) + N−1 � k=0 Lk(xk, uk) LN(xN) = ¯ℓf(xN) + (λ + 1 2 ¯g(xN)Iµ)⊤¯g(xN) Lk(xk, uk) = ¯ℓ(xk, uk) + (λ + 1 2 ¯g(xk, uk)Iµ)⊤¯g(xk, uk) (24) where ¯ℓf : R2n �→ R and ¯ℓ : R2n ×Rm �→ R are the final cost and the running cost functions for perturbed system dynamics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' A typical design of such functions are ¯ℓf(x) = 1 2∥δx∥SV ¯ℓ(x, u) = 1 2∥δx∥SQ + 1 2∥δu∥SU (25) where SV ∈ R2n×2n, SQ ∈ R2n×2n and SU ∈ Rm×m are the cost matrices that are specified by the user and remain constant throughout all iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In (24), ¯g is a vector of p constraints for perturbed state dynamics as introduced in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' λ ∈ Rp is a Lagrange multiplier, µ ∈ Rp is a penalty weight and Iµ ∈ Rp×p is the penalty matrix defined as Iµ = � 0 if gi(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=') < 0 and λi = 0, µi otherwise (26) In general, one can define time varying Lagrange multiplier and penalty weight, but we kept them constant for each time step during the same iteration for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' We define the cost-to-go and action-value functions as Vk(xk) = min uk {Lk(xk, uk)} + Vk+1(Akxk + Bkuk) = min uk Q(xk, uk)) (27) The matrices Ak and Bk represent the discretized versions of At and Bt in equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Second order Taylor series expansion of cost-to-go function can be written as δVk(x) ≈ 1 2δx⊤ k Vxx,kδxk + V ⊤ x,kδxk (28) where Vxx,k and Vx,k are the Hessian and gradient of the cost-to-go at time step k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Action-value function defined in (27) can be also approximated as a quadratic function as Q(x + δx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' u + δu) ≈ Q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' u) + Q⊤ x δx + Q⊤ uδu + 1 2(δx⊤Qxxδx + δx⊤Qxxδx) + δx⊤Qxuδu (29) To compute the derivative matrices in (29): Qx = ¯ℓx + A⊤V ′ x + ¯g⊤ x (λ + Iµ¯g) Qu = ¯ℓu + B⊤V ′ x + ¯g⊤ u (λ + Iµ¯g) Qxx = ¯ℓxx + A⊤V ′ xxA + ¯g⊤ x Iµ¯gx + (V ′ xhxx) Quu = ¯ℓuu + B⊤V ′ xxB + ¯g⊤ u Iµ¯gu + (V ′ xhuu) Qux = ¯ℓux + B⊤V ′ xxA + ¯g⊤ u Iµ¯gx + (V ′ xhux) (30) To simplify the notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' we have omitted the time indices on all variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' All variables in this expression are evaluated at time step k, except for those marked with ′, which are evaluated at time step k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Calculating the full second-order expansion of the state dynamics (hxx, huu, hux), can be computationally expensive, particularly for systems with complex dynamics and high- dimensional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' DDP refers to iterative LQR by discarding the second-order dynamics and computing only the first-order expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This results in a Gauss-Newton approximation of the true Hessian, which reduces the local fidelity and requires more iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, these iterations are less expensive to compute and often lead to a faster overall convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Therefore, in this work, we eliminated the second order dynamics as approximating the perturbed state dynamics as described in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Minimizing (29) with respect to δu results in an affine controller δu∗ = −Q−1 uu(Quxδx + Qu) ≜ Kδx + d (31) Substituting δu∗ into (29) yields the derivatives of the cost- to-go at time step k in terms of the derivatives of the action value function as: Vx = Qx + KQu + K⊤Quud + Q⊤ uxd, Vxx = Qxx + K⊤QuuK + K⊤Qux + Q⊤ uxK (32) At the final step, Vx and Vxx can be easily computed as the first and second derivatives of the final cost function (¯ℓf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This way, the derivatives of the action-value function (30) and in turn the local optimal control policy (31) at each step can be calculated backwards starting from the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' After determining the optimal control policy for each time step, we update the nominal trajectories by simulating the dynamics forward on the manifold itself starting from the initial state as: δxk = �logm(Xk−1 ¯ Xk) ¯ξk − ξk � δuk = Kkδxk + αdk ¯uk = uk + δuk ¯ξk = ξ + f(ξ, ¯uk)∆t ¯ Xk = X expm(¯ξk∆t) (33) where {Xk, ξk, uk} and { ¯ Xk, ¯ξk, ¯uk} represent the nominal state-actions and the updated state-actions at time step k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In the above expression, 0 ≤ α ≤ 1 is a scaling term for simple linear search on the feedforward term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Practically, the parameter α is initially set to 1, but if the cost of the updated trajectory does not decrease, it will be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Once the cost of the updated trajectory is decreased, the forward pass is succesfully completed, the new trajectory is accepted as nominal trajectory and the backward pass is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' To optimize the performance of DDP-based algorithms, there are a few implementation practices to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In the backward pass, Quu may need to be regularized as Quu +ρI if it is invertible or the former forward pass is unsuccesful, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', the cost is not decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' After the DDP iterations converge, the parameters λ and µ can be updated as: λ+ = max(0, λi + µi¯gi(x∗, u∗)) µ+ = γµ, γ > 0 (34) The DDP iterations can then be restarted until convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' For more information, see reference [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' EXPERIMENTS The goal of this section is to devise a trajectory on SE(3) that satisfies both position and orientation constraints while avoiding unsafe configurations and obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Dynamics on SE(3) We consider a 3D rigid body in SE(3) where the states of the system can be represented by a rotation matrix R ∈ SO(3) = {R ∈ R3×3|R⊤R = I3, det(R) = 1} (35) and position p ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The homogeneous representation of a typical group element in SE(3) is X = �R p 0 1 � ∈ SE(3) (36) The velocity vector ξ in SE(3) is known as a “twist” and is composed of both angular (ω) and linear (v) velocities in body frame as ξ = � ω v � ∈ R6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' ξ∧ = � ω∧ v 0 0 � ∈ se(3) (37) The forced Euler-Poincar´e equations [16] define the twist dynamics as Jb ˙ξ = ad∗ ξ Jbξ + u (38) In this expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Jb represents the generalized inertia matrix in the body fixed principal axes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' while u ∈ g∗ represents the generalized control input forces applied to these axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' g∗ denotes the cotangent space and the coadjoint map is represented by ad∗ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The matrix representations of the adjoint action in the Lie algebra (adξ) and the coadjoint map (ad∗ ξ) are as follows: adξ = �ω∧ 0 v∧ ω∧ � , ad∗ ξ = ad⊤ ξ = − �ω∧ v∧ 0 ω∧ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' (39) The continuous equations of motion described in (7) can be written for SE(3) group as ˙X = Xξ∧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' ˙ξ = Jb −1� ad∗ ξ Jbξ + u � (40) The linearized twist dynamics as given in [17]: ˙ξ = Γtξ + Λtu + bt (41) where Γt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Λt and bt are: Γt := J−1 b ad∗ ξ Jb + J−1 b �(Ibω)∧ mv∧ mv∧ 0 � Λt := Jb −1 bt := −Jb −1 �(Ibω)∧ mv∧ mv∧ 0 � ξ (42) assuming the inertia matrix Jb is defined as Jb := �Ib 0 0 mI3 � (43) where Ib and m are the moment of inertia in the body frame and the body mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Constrained (red) and unconstrained (blue) trajectories generated by proposed DDP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Control input signals of the planned trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Torques refers to inputs acting on angular velocity and Forces refers to inputs acting on linear velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Left column: unconstrained, Right column: constrained B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Simulations We now test the proposed algorithm for planning a safe path for a SE(3) rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The task we have defined involves rotating the body to the configuration Rz(180) from the identity and translating it to the position {2, 2, 2} from the initial position {1, 1, 1} in 3 seconds with a fixed time step of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' To denote the rotation around the x, y, and z axes of the body-fixed frame, we use Rx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' ), Ry(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' ), and Rz(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' During the motion, the configuration Rz(90) is considered to be unsafe and must be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Additionally, it is assumed that there is a spherical obstacle at {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='25} with a radius of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 that must be avoided as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' We only penalized the final state and control inputs, so we set the controller parameters as SV = 1000I12, SU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='001I6, and SQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In order to assess the performance of the proposed DDP in terms of constraint handling, we conducted an experiment with the same trajectory optimization task in both constrained and unconstrained cases, and the resulting trajectories are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The configuration trajectories were converted to Eu- ler angles (φ, θ, ψ in degrees) using the rotation order Rx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=')Ry(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=')Rz(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In the unconstrained case (shown in blue), it is shown that the ψ angle goes directly from 0 to 180 degrees while the φ and θ angles remain constant (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 1), as the task only involves rotating around the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This can also be observed in the upper left part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 2, where only a single input is non-zero to achieve the desired motion, while the others remain zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, in the constrained case (shown in red), rotations around x and y axes (changes in φ and θ angles) were also observed to avoid the unsafe configuration of Rx(90) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' As shown in the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 1, the shortest paths for the positional states were obtained in the unconstrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' However, the presence of a sphere at {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='25} with a radius of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 forced the positional trajectories to take a circuitous route around obstacle by deviating along y and x axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' To evaluate the effectiveness of the proposed DDP method in handling external disturbances, we extend the analysis done in [13] to SE(3) and employ the proposed DDP method as a Lie-algebraic feedback control: ufb k := u∗ k + Kk logm((X ∗ k )−1X ϵ k) (44) where u∗ k and Kk represent the (sub)optimal control sequence and time-varying feedback gains, respectively, which are ob- tained through DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The feed-forward term dk in (31) is not explicitly shown in the feedback policy (44) because it is already included in u∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The variables X ∗ k and X ϵ k represent the (sub)optimal states obtained through DDP convergence and the perturbed states due to disturbance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' We test the policy (44) on the following stochastic version of the SE(3) dynamics: X ϵ k+1 = X ϵ k expmSE(3)(ξϵ k∆t) ξϵ k+1 = ξϵ k + f(ξϵ k, uk)∆t + σωω (45) where ω is assumed to be spatially uncorrelated independent and identically distributed noise, drawn from a zero mean Gaussian distribution, ω ∼ N(06×1, I6), σω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The performance of the proposed DDP method under stochas- tic conditions was evaluated by allowing the optimizer to converge on the deterministic system and then testing its performance on the stochastic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 3 shows the results of this comparison, using 1000 sampled trajectories under noisy dynamics to compare the open-loop policy u∗ k with the feedback policy ufb k (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The results demonstrate that the use of the obtained feedback gains significantly reduces state variance, particularly in the vicinity of the the goal points, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 50 0 [6ap] [deg] [deg] 100 0 50 0 50 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 y N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 0 1 2 3 0 1 2 3 0 1 2 3 Time [sec] Time [sec] Time [sec]20 Torques 25 0 0 25 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 Forces 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='5 0 2 0 2 Time [sec] Time [sec]Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Evaluation of DDP’s control sequence under random disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Left Column: Open-loop control, Right column: Closed-loop control VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' CONCLUSION In conclusion, the optimization of control policies on matrix Lie groups is an important problem in robotics and control with numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In this work, we have proposed a novel approach for tackling this problem using an augmented Lagrangian based constrained discrete DDP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Our method involves lifting the optimization problem to the Lie algebra in the backward pass, allowing us to compute the gradient of the objective and constraint functions within the corresponding tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In the forward pass, we retract back to the manifold by integrating the dynamics using the optimal policy obtained in the backward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' One of the key contributions of our work is the development of a general approach for nonlinear constraint handling for a wide range of matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Previous methods were only able to handle constraints for specific classes of matrix Lie groups, such as SO(3) pose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Our method, on the other hand, is able to handle a wide range of nonlinear constraints, making it a more widely applicable and flexible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' In addition, we have demonstrated the effectiveness of our method in handling external disturbances through its applica- tion as a Lie-algebraic feedback control policy on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' The results show that our approach is able to effectively handle constraints and maintain its stability in the presence of external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Overall, our proposed DDP algorithm represents a signif- icant step forward in the field of optimization on matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' It provides a flexible and general approach for nonlinear constraint handling and has the potential to be applied to a wide range of problems in robotics and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' As a future direction, it would be interesting to investigate the closed loop uncertainty propagation through the prediction horizon, as has been done for Euclidean models [11], in order to design a robust Model Predictive Control (MPC) framework for matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' This would allow the proposed method to be used in more challenging and dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Additionally, further experimentation with different types of matrix groups and constraints would provide valuable insights into the capabilities and limitations of the proposed DDP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' REFERENCES [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Hemingway and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' O‘Reilly, “Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments,” Multibody Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', 44, 31–56, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Shuster, “A survey of attitude representations.” Navigation, 8(9), 439-517, 1993 [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Diebel, “Representing attitude: Euler angles, unit quaternions, and rotation vectors.” Matrix, 58(15-16), 1-35, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Bloch, Nonholonomic Mechanics and Control, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Krishnaprasad and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Murray, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Springer, New York, NY, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Lynch and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Park, Modern robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Cambridge University Press, 2017 [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Bullo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Lewis, Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems, 49, Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Kobilarov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Desbrun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Marsden, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Sukhatme, “A dis- crete geometric optimal control framework for systems with symmetries,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=',1–8, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Saccon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Hauser, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Aguiar, “Optimal control on Lie groups: The projection operator approach,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Control, 58, 9, 2230-–2245, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Kobilarov, “Discrete optimal control on lie groups and applications to robotic vehicles,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', 5523–5529, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Jacobson and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Mayne, Differential Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' New York, NY, USA: Elsevier, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Alcan and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Kyrki, “Differential Dynamic Programming With Nonlinear Safety Constraints Under System Uncertainties,” in IEEE Robotics and Automation Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 1760-1767, 2022 150 150 Φ,e, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Kobilarov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Ta, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Dellaert, “Differential dynamic program- ming for optimal estimation,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=', 863-–869,2015 [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Boutselis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Theodorou, “Discrete-time differential dynamic pro- gramming on lie groups: Derivation, convergence analysis, and numerical results.” IEEE Transactions on Automatic Control, 66(10), 4636–4651, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Liu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Liu, “Discrete-Time Differential Dynamic Programming on SO(3) With Pose Constraints.” IEEE Access, 10, 112921-112933, 2022 [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Lee, “Computational geometric mechanics and control of rigid bodies,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' dissertation, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Michigan, Ann Arbor, MI, USA, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Bloch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Krishnaprasad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Marsden, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Ratiu, “The Euler-Poincar´e equations and double bracket dissipation,” Communica- tions in Mathematical Physics, 175(1), 1–42, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Teng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Clark and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Ghaffari, “An Error-State Model Predictive Control on Connected Matrix Lie Groups for Legged Robot Control,” IEEE/RSJ International Conference on Intelligent Robots and Systems, 8850-8857, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Teng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Clark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Bloch, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Vasudevan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Ghaffari, “Lie Algebraic Cost Function Design for Control on Lie Groups.” arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content='09177, 2022 [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Howell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Jackson and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' Manchester, “ALTRO: A Fast Solver for Constrained Trajectory Optimization,” IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} +page_content=' 7674-7679,2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'} diff --git a/UdAyT4oBgHgl3EQfuvkh/content/tmp_files/2301.00617v1.pdf.txt b/UdAyT4oBgHgl3EQfuvkh/content/tmp_files/2301.00617v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c56355abe29bab5d68c674a953fb79619c25e256 --- /dev/null +++ b/UdAyT4oBgHgl3EQfuvkh/content/tmp_files/2301.00617v1.pdf.txt @@ -0,0 +1,1834 @@ +arXiv:2301.00617v1 [math.FA] 2 Jan 2023 +SOME REMARKS ON CONVEX BODY DOMINATION +TUOMAS P. HYTÖNEN +Dedicated, with admiration, to the Ukrainian people. +Abstract. Convex body domination is an important elaboration of the tech- +nique of sparse domination that has seen significant development and applica- +tions over the past ten years. In this paper, we present an abstract framework +for convex body domination, which also applies to Banach space -valued func- +tions, and yields matrix-weighted norm inequalities in this setting. We explore +applications to “generalised commutators”, obtaining new examples of bounded +operators among linear combinations of compositions of the form aiTbi, where +ai, bi are pointwise multipliers and T is a singular integral operator. +1. Introduction +The technique of sparse domination was developed to provide a simpler approach, +achieved by Lerner [23], to the “A2 conjecture” on sharp weighted norm inequalities +for Calderón–Zygmund operators, which was first proved with a different machinery +by the author [16]. However, beyond this original aim, sparse domination imme- +diately led to significant further consequences and has by now been applied to a +variety of new questions, of which [1, 2, 3, 7, 11] is only a sample. The method +consists of two main steps that are largely independent of each other and essentially +decouple the operator from the space or norm in which it should be estimated: +(1) Dominating an operator of interest by a suitable sparse operator/form. +(2) Estimating the sparse form with respect to relevant norms of interest. +While sparse domination very efficiently captures the local size of an object un- +der consideration, and this is precisely what is needed in many applications, it +loses information about directions, which is sometimes relevant when dealing with +vector-valued functions, and especially so, matrix-valued weights are involved. To +extend the method to such questions, Nazarov et al. [27] developed the so-called +convex body domination, where the numerical averages featuring in sparse dom- +ination are replaced by convex subsets of Rn, thus containing information about +different behaviour in different directions. Since its introduction in the context of +Calderón–Zygmund operators and matrix A2 weights by [27] (see also [10] for an- +other approach but based on the same key idea), convex body domination has been +applied to matrix Ap-weight and two-weight bounds by Cruz-Uribe et al. [8], and ex- +tended to commutators of Calderón–Zygmund operators by Isralowitz et al. [20, 21] +and rough singular integral operators by Di Plinio et al. [13] and Muller and Rivera- +Ríos [26]. In a recent breakthrough, Bownik and Cruz-Uribe [6] extended the Rubio +Date: January 3, 2023. +2010 Mathematics Subject Classification. 42B20, 46E40. +The author is supported the Academy of Finland via the Finnish Centre of Excellence in +Randomness and Structures “FiRST” (grant no. 346314). +1 + +2 +T. P. HYTÖNEN +de Francia algorithm, and its key application to weighted extrapolation, to matrix- +valued weights, by further development of the convex body philosophy. +The aim of this paper is to further explore this technique, providing extensions, +new applications and—hopefully—some additional insight into the abstract under- +lying mechanisms. We begin by developing a somewhat general framework, but +our claims for originality in this regard are relatively mild, as most of the ideas +are at least implicit in the previous works in the existing literature. +A certain +justification for this framework comes from the observation that it applies almost +verbatim to the case of Banach space -valued functions. To be precise, given a Ba- +nach space E, we consider functions taking values in En, and develop a version of +convex body domination applicable to weighted norm inequalities involving matrix +weights W : Rd → Rn×n, acting on En in the natural way. That is, we make no +attempt towards a fully operator-valued theory of weighted norm inequalities in +infinite dimensions, yet the results that we obtain are still new even in this more +modest generality. In particular, if E is a Banach space with the UMD property, +the classical Hilbert transform extends boundedly to the matrix-weighted space +L2(W; En) of En-valued functions; see Corollary 6.3 for the result, and Section 6 +for the relevant definitions and background. A key to this extension is the observa- +tion that the convex bodies arising from our framework are still Rn-valued in this +generality—and not, for instance, En-valued, as one might have (and this author +certainly had) initially expected. Thus the powerful Euclidean machinery, most +notably the John ellipsoid theorem, is still available in this setting. +As for new applications of the theory, we build on a recent observation from +Isralowitz et al. [20, 21] that convex body domination of an operator T bootstraps +to a domination of its commutators [b, T ] = bT − T b with pointwise multipliers. As +we will explore in Section 7, this phenomenon is far more general, and can be used +to estimate any operators of the form +f �→ +n +� +i=1 +aiT (bif), +where an operator T satisfying convex body domination is pre- and post-composed +with pointwise multipliers ai, bi. From this general principle, we can in particular +recover and sharpen a recent sufficient condition [17] for the boundedness of iterated +mixed commutators [b1, [b2, T ]] in terms of joint conditions on the pair of functions +(b1, b2), but also obtain new examples. +In contrast to the development of the abstract framework in the first part of the +paper, we have not strived for the greatest generality in terms of the applications +in the later sections. In many cases, it will be clear to an experienced reader that +several variants and extensions could be obtained, and some of them will most likely +be pursued in forthcoming works, by this author and others. Besides the concrete +results contained in this paper, our aim is to hint at the many rich directions for +the further development of the theory. +2. Norms and convex bodies +Let X be a real normed space. We denote by +¯BX := {x ∈ X : ∥x∥X ≤ 1} + +SOME REMARKS ON CONVEX BODY DOMINATION +3 +its closed unit ball, and by X∗ the normed dual, which is a Banach space. For +x∗ ∈ X∗, we define, as usual, +∥x∗∥X∗ := sup{|⟨x, x∗⟩| : x ∈ ¯BX}. +As a consequence of the Hahn–Banach theorem, we have +∥x∥X = sup{|⟨x, x∗⟩| : x∗ ∈ ¯BX∗} = max{⟨x, x∗⟩ : x∗ ∈ ¯BX∗}; +(2.1) +in particular, the supremum is reached as a maximum, and we have ∥x∥X = ⟨x, x∗⟩ +for some x∗ ∈ ¯BX∗. +For ⃗x = (xi)n +i=1 ∈ Xn and x∗ ∈ X∗, we define the Rn-valued pairing ⟨⃗x, x∗⟩ := +(⟨xi, x∗⟩)n +i=1 ∈ Rn and the set-valued “norm” +⟨⟨⃗x⟩⟩X := {⟨⃗x, x∗⟩ : x∗ ∈ ¯BX∗} ⊂ Rn. +2.2. Remark. The notation is adapted from Nazarov et al. [27], who introduced +the version with X = Ł1(Q), the space L1(Q) with the normalised norm +1 +|Q|∥ ∥1). +The extension to X = Łp(Q) (i.e., Lp(Q) with the normalised norm +1 +|Q|1/p ∥ ∥p) +is due to Di Plinio et al. [13]. Although our main applications will be concerned +with spaces of functions (living on a cube Q), we find it illuminating to develop the +basics of the theory on a completely abstract level. Among other things, this point +of view will make it clear that there will be essentially no difference in treating a +space X = Lp(Q; E) of E-valued functions for an arbitrary Banach space E; for +⃗f ∈ Xn, the corresponding ⟨⟨⃗f⟩⟩X will still be subsets of Rn and not, say, of En. +This will allow us to make effortless use of the powerful John ellipsoid theorem from +Euclidean geometry, even when working with functions taking values in an infinite- +dimensional Banach space! In other applications, a choice like X = L log L(Q) +might also be relevant. +For ⃗a ∈ Rn and ⃗x ∈ Xn, we define the X-valued dot product +⃗a · ⃗x := ⃗x · ⃗a := +n +� +i=1 +aixi. +We observe the easy identities +⃗a · ⟨⃗x, x∗⟩ = ⟨⃗a · ⃗x, x∗⟩, +∀⃗a ∈ Rn, ⃗x ∈ Xn, x∗ ∈ X∗, +and +spanX(⃗x) := span{xi}n +i=1 = {⃗a · ⃗x : a ∈ Rn} ⊂ X. +2.3. Lemma. For each ⃗x ∈ Xn, the set ⟨⟨⃗x⟩⟩X ⊂ Rn is convex, compact, and +symmetric about the origin. +Proof. Symmetry, convexity and boundedness are immediate from the fact that +¯BX∗ has these properties. For compactness in Rn, it remains to show closedness, +so suppose that ⟨⃗x, x∗ +k⟩ → ⃗e ∈ Rn as k → ∞, where each x∗ +k ∈ ¯BX∗; we need to +show that ⃗e ∈ ⟨⟨x⟩⟩X. For each ⃗a ∈ Rn, it follows that +|⃗a · ⃗e| = lim +k→∞ |⃗a · ⟨⃗x, x∗ +k⟩| = lim +k→∞ |⟨⃗a · ⃗x, x∗ +k⟩| ≤ ∥⃗a · ⃗x∥X. +This in turn implies that +Λ(⃗a · ⃗x) := ⃗a · ⃗e, +∀⃗a · ⃗x ∈ spanX(⃗x), + +4 +T. P. HYTÖNEN +gives a well-defined linear functional of norm 1 on the subspace spanX(⃗x) ⊂ X. By +the Hahn–Banach theorem, Λ is the restriction of some x∗ ∈ ¯BX∗. Hence +⃗a · ⃗e = Λ(⃗a · ⃗x) = ⟨⃗a · ⃗x, x∗⟩ = ⃗a · ⟨⃗x, x∗⟩ +∀⃗a ∈ Rn, +and thus limk→∞⟨⃗x, xk⟩ = ⃗e = ⟨⃗x, x∗⟩ ∈ ⟨⟨⃗x⟩⟩X, as we wanted to show. +□ +For A, B ⊂ Rn, we define the Minkowski dot product +A · B := {⃗a ·⃗b : ⃗a ∈ A,⃗b ∈ B} ⊂ R. +If A, B ⊂ Rn are convex, compact and symmetric, so is A · B ⊂ R. On R, such +sets are precisely intervals of the form [−c, c]. Hence we can, and sometimes will, +identify A · B = [−c, c] ⊂ R with its right end-point c ∈ [0, ∞). In particular, for +⃗x ∈ Xn and ⃗y ∈ Y n, we will use this identification when dealing with +⟨⟨⃗x⟩⟩X · ⟨⟨⃗y⟩⟩Y = {⟨⃗x, x∗⟩ · ⟨⃗y, y∗⟩ : x∗ ∈ ¯BX∗, y∗ ∈ ¯BY ∗} += +� +n +� +i=1 +⟨xi, x∗⟩⟨yi, y∗⟩ : x∗ ∈ ¯BX∗, y∗ ∈ ¯BY ∗ +� +. +3. Bi-linear forms +Let X, Y be real normed spaces, and suppose that we have a bilinear from +t : X × Y → R. We define its extension acting on pairs of vectors (⃗x, ⃗y) ∈ Xn × Y n +as follows. If ⃗e ∈ Rn and x ∈ Xn, we have ⃗x · ⃗e ∈ F by our previous convention +about the X-valued dot product. If (ei)n +i=1 is a fixed orthonormal basis of Rn, we +then define +t(⃗f,⃗g) := +n +� +i=1 +t(⃗f · ⃗ei,⃗g · ⃗ei), +For x ∈ X, y ∈ Y , and ⃗e, ⃗u ∈ Rn, it follows that +t(x⃗e, y⃗u) := +n +� +i=1 +t(x⃗e · ⃗ei, y⃗u · ⃗ei) = t(x, y) +n +� +i=1 +(⃗e · ⃗ei)(⃗u · ⃗ei) = t(x, y)⃗e · ⃗u. +If (⃗ui)n +i=1 is another orthonormal basis, then +n +� +i=1 +t(⃗x · ⃗ui, ⃗y · ⃗ui) = +n +� +i,j,k=1 +t(⃗x · ⃗ej, ⃗y · ⃗ek)(⃗ej · ⃗ui)(⃗ek · ⃗ui) += +n +� +j,k=1 +t(⃗x · ⃗ej, ⃗y · ⃗ek)(⃗ej · ⃗ek) = +n +� +j=1 +t(⃗x · ⃗ej, ⃗y · ⃗ej) =: t(⃗x, ⃗y), +so the definition of t(⃗f,⃗g) is independent of the chosen orthonormal basis. +If A ∈ Rn×n is a linear transformation of Rn, acting in a natural way on F n, +then +t(A⃗x, ⃗y) = +n +� +i=1 +t(A⃗f · ⃗ei,⃗g · ⃗ei) = +n +� +i=1 +t(⃗f · At⃗ei,⃗g · ⃗ei) += +n +� +i,j=1 +t(⃗f · ⃗ej,⃗g · ⃗ei)(⃗ej · At⃗ei) += +n +� +i,j=1 +t(⃗f · ⃗ej,⃗g · ⃗ei)(A⃗ej · ⃗ei) = +n +� +j=1 +t(⃗f · ⃗ej,⃗g · A⃗ej) = t(⃗f, At⃗g). + +SOME REMARKS ON CONVEX BODY DOMINATION +5 +4. From norm bounds to convex body bounds +The idea of the following lemma lies behind many of the existing convex body +domination results. To isolate the key point, we state it here in an operator-free +version, involving functions and therir norms only. +4.1. Lemma. Let X, Y be normed spaces, and ⃗f ∈ Xn,⃗g ∈ Y n. +Let EK be the John ellipsoid of K := ⟨⟨⃗f⟩⟩X such that +EK ⊂ K ⊂ √nEK, +and suppose that EK is non-degenerate (i.e., of full dimension). Let RK be a linear +transformation such that RKEK = ¯BRn, the closed unit ball of Rn, and let (⃗ei)n +i=1 +be an orthonormal basis of Rn. If +fi := RK ⃗f · ⃗ei, +gi := R−t +K ⃗g · ei, +i = 1, . . . , n, +then +n +� +i=1 +∥fi∥X∥gi∥Y ≤ n3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y . +Proof. If φ ∈ ¯BX∗, then +⟨φ, RK ⃗f · ⃗ei⟩ = RK⟨φ, ⃗f⟩ · ⃗ei +∈ RK⟨⟨⃗f⟩⟩X · ⃗ei ⊂ RK +√nEK · ⃗ei = √n ¯BRn · ⃗ei = √n[−1, 1], +and hence +∥fi∥X = ∥RK ⃗f · ⃗ei∥X ≤ √n. +If ψ ∈ ¯BY ∗, then +⟨ψ, R−t +K ⃗g · ⃗ei⟩ = R−t +K ⟨ψ,⃗g⟩ · ⃗ei +∈ R−t +K ⟨⟨⃗g⟩⟩Y · ⃗ei ⊂ [−M, M], +M := max{|⃗y| : ⃗y ∈ R−t +K ⟨⟨⃗g⟩⟩Y }. +It follows that |⟨ψ, R−t +K ⃗g · ⃗ei⟩| ≤ M, and hence +∥gi∥Y = ∥R−t +K ⃗g · ⃗ei∥Y ≤ M. +Combining the estimates, we have +n +� +i=1 +∥fi∥X∥gi∥Y ≤ +n +� +i=1 +√nM = n3/2M. +On the other hand, +⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y ⊃ EK · ⟨⟨⃗g⟩⟩Y = RKEK · R−t +K ⟨⟨⃗g⟩⟩Y = ¯BRn · R−t +K ⟨⟨⃗g⟩⟩Y = [−M, M], +and hence +M ≤ ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y , +using the identification of the symmetric interval ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y with its right end- +point in the last step. Substituting back, this completes the proof. +□ +The following proposition contains the basic idea of bootstrapping norm bounds +to convex body bounds. +Unfortunately, it is a bit too simple for most actual +applications, but we include it as an illustrative toy model for the more serious +result to be presented after it. +4.2. Proposition. Let X, Y be normed spaces with subspaces F ⊂ X and G ⊂ Y , +and let t : F × G → R be a bilinear form. Consider the following conditions: + +6 +T. P. HYTÖNEN +(1) For all (f, g) ∈ F × G, we have +|t(f, g)| ≤ C∥f∥X∥g∥Y . +(2) For all (⃗f,⃗g) ∈ F n × Gn, we have +|t(⃗f,⃗g)| ≤ Cn⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y . +For each n ∈ Z+, condition (1) implies condition (2) with Cn = Cn3/2. +Proof. Given ⃗f, consider the compact, convex, symmetric set +K := ⟨⟨⃗f⟩⟩X, +and denote by EK its John ellipsoid such that +EK ⊂ K ⊂ √nEK. +Case: EK is non-degenerate. Let RK be a linear transformation such that RKEK = +¯BRn, the closed unit ball of Rn. Let (⃗ei)n +i=1 be some orthonormal basis of Rn. We +then write +t(⃗f,⃗g) = t(R−1 +K RK ⃗f,⃗g) = t(RK ⃗f, R−t +K ⃗g) += +n +� +i=1 +t(RK ⃗f · ⃗ei, R−t +K ⃗g · ⃗ei) =: +n +� +i=1 +t(fi, gi), +(4.3) +where fi and gi are as in Lemma 4.1. +By assumption (1) and Lemma 4.1, it follows that +|t(⃗f,⃗g)| ≤ +n +� +i=1 +|t(fi, gi)| ≤ +n +� +i=1 +C∥fi∥X∥gi∥Y ≤ Cn3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y, +and this completes the proof in the case that EK is non-degenerate. +Case: EK is degenerate. Suppose then that EK is degenerate; hence H := span K +is a strict subspace of Rn. Let P denote the orthogonal projection of Rn onto H. +For each x∗ ∈ ¯BX∗, we have ⟨⃗f, x∗⟩ ∈ K ⊂ H, hence +⟨⃗f, x∗⟩ = P⟨⃗f, x∗⟩ = ⟨P ⃗f, x∗⟩, +and thus ⃗f = P ⃗f. It follows that +t(⃗f,⃗g) = t(P ⃗f,⃗g) = t(⃗f, P t⃗g) = t(⃗f, P⃗g), +(4.4) +and similarly +⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y = P⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y = ⟨⟨⃗f⟩⟩X · P t⟨⟨⃗g⟩⟩Y = ⟨⟨⃗f⟩⟩X · ⟨⟨P⃗g⟩⟩Y . +(4.5) +So it is enough to prove the claim with P⃗g in place of ⃗g, and hence we may assume +without loss of generality that also ⃗g = P⃗g. But then we can repeat the argument +in the non-degenerate case, but with Rn replaced by its subspace H throughout; +within this subspace, EK ⊂ H is non-degenerate, and the previous case applies to +give the desired result. +□ +In the following proposition, condition (1) is a typical intermediate step that +is established in the course of proving a sparse domination result for an operator, +while condition (2) is its convex body analogue. +The proposition says that (1) +in fact implies (2). It is essentially an abstraction (from L1 averages to general +dominating norms) of an idea already present in [27, Lemma 3.2]. + +SOME REMARKS ON CONVEX BODY DOMINATION +7 +4.6. Proposition. Let X, Y be normed spaces with subspaces F ⊂ X and G ⊂ Y . +Let Q0 ∈ D, and suppose that there are bilinear forms tQ : F × G → R indexed by +all Q ∈ D(Q0). Consider the following conditions: +(1) For all (f, g) ∈ F × G, there exist disjoint ˆQk ⊂ Q0 with � +k | ˆQk| ≤ ε|Q0| +and such that: whenever Qj ⊂ Q0 are disjoint, not strictly contained in +any ˆQk, and cover all ˆQk, then +|tQ0(f, g) − +� +j +tQj(f, g)| ≤ C∥f∥X∥g∥Y . +(2) For all (⃗f,⃗g) ∈ F n × Gn, there exist disjoint Qk ⊂ Q0 with � +k |Qk| ≤ +εn|Q0| and such that +|tQ0(⃗f,⃗g) − +� +k +tQk(⃗f,⃗g)| ≤ Cn⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y . +For each n ∈ Z+, condition (1) implies condition (2) with εn = nε and Cn = Cn3/2. +Of course, the condition � +k |Qk| ≤ εn|Q0| is only useful for εn < 1. For a fixed +ε and εn = nε, this would only allow us to conclude (2) for boundedly many values +of n; so in order to obtain (2) for all n ∈ N, we need (1) for arbitrarily small ε > 0. +This is seldom a problem in concrete situations. +Proof. As in the proof of Proposition 4.2, given ⃗f, we consider the compact, convex, +symmetric set +K := ⟨⟨⃗f⟩⟩X, +and denote by EK its John ellipsoid such that +EK ⊂ K ⊂ √nEK. +Case: EK is non-degenerate. Let RK be a linear transformation such that RKEK = +¯BRn, the closed unit ball of Rn. Let (⃗ei)n +i=1 be some orthonormal basis of Rn. As +in (4.3), we then write +tQ0(⃗f,⃗g) = tQ0(R−1 +K RK ⃗f,⃗g) = tQ0(RK ⃗f, R−t +K ⃗g) += +n +� +i=1 +tQ0(RK ⃗f · ⃗ei, R−t +K ⃗g · ⃗ei) =: +n +� +i=1 +tQ0(fi, gi), +(4.7) +where fi and gi are as in Lemma 4.1. +It is from this point on that the present proof requires some elaboration compared +to the proof of Proposition 4.2. According to assumption (1), for each of the pairs +of functions fi := RK ⃗f · ⃗ei and gi := R−t +K ⃗g · ⃗ei, we can find disjoint ˆQi,k ⊂ Q0 +with � +k | ˆQi,k| ≤ ε|Q0| and such that: whenever Qj ⊂ Q0 are disjoint, not strictly +contained in any ˆQi,k, and cover all ˆQi,k, then +|tQ0(fi, gi) − +� +j +tQj(fi, gi)| ≤ C∥fi∥X∥gi∥Y . +(4.8) +We make the following specific choice of the cubes Qj: Let {Qj}∞ +j=1 be the maximal +cubes among { ˆQi,k}1≤k<∞ +1≤i≤n . Then +� +j +|Qj| ≤ +n +� +i=1 +∞ +� +k=1 +| ˆQi,k| ≤ +n +� +k=1 +ε|Q0| = nε|Q0|, + +8 +T. P. HYTÖNEN +and (4.8) holds with these Qj for each i = 1, . . . , n. Using (4.7), and observing that +it also holds with Q0 replaced by Qj, it follows that +|tQ0(⃗f,⃗g) − +� +j +tQj(⃗f,⃗g)| ≤ +n +� +i=1 +|tQ0(fi, gi) − +� +j +tQj(fi, gi)| +≤ C +n +� +i=1 +∥fi∥X∥gi∥Y ≤ Cn3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y , +using Lemma 4.1 in the last step. This completes the proof under the assumption +that EK is non-degenerate. +Case: EK is degenerate. This follows the corresponding case in the proof of Propo- +sition 4.2 almost verbatim. Like there, let H := span K, and let P denote the +orthogonal projection of Rn onto H. We then have (4.4) for each t = tQ, as well as +(4.5). So it is again enough to prove the claim with P⃗g in place of ⃗g, and hence we +may assume without loss of generality that also ⃗g = P⃗g. But then we can repeat +the argument in the non-degenerate case, but with Rn replaced by its subspace H +throughout; within this subspace, EK ⊂ H is non-degenerate, and the previous case +applies to give the desired result. +□ +5. From single-scale bounds to global bounds +This passage is by now a relatively routine part of the theory, but we include some +details for completeness. The following lemma is again stated in an operator-free, +and even function-free way, simply as a criterion for dominating a real number by +sum over a sparse collection. A more concrete situation for applying this criterion +is presented afterwards. +5.1. Lemma. Consider numbers a ∈ R and aQ, cQ ∈ R indexed by dyadic cubes +Q ∈ D, with the following properties: +(1) There is a family Q of disjoint dyadic cubes such that +a = +� +Q∈Q +aQ. +(2) For some δ ∈ (0, 1) and each Q ∈ D that is contained in some P ∈ Q, +there is a family of disjoint Qk ∈ D(Q) such that +� +k +|Qk| ≤ δ|Q|, +���aQ − +� +k +aQk +��� ≤ cQ. +(3) For some α, C ∈ [1, ∞) and each Q ∈ D that is contained in some P ∈ Q, +we have |aQ| ≤ C|Q|α. +Then there is a (1 − δ)-sparse family of dyadic cubes S such that +S ⊂ +� +Q∈Q +D(Q), +|a| ≤ +� +S∈S +cS. +5.2. Remark. If Q = {Q0} consists of a single cube only, then condition (1) is +automatic with a = aQ0. + +SOME REMARKS ON CONVEX BODY DOMINATION +9 +Proof. Let Q ⊂ D be a disjoint collection provided by assumption (1). For each +P ∈ Q, denote S0(P) := {P}. +Assuming that a disjoint Sj(P) ⊂ D(P) has +already been constructed, for each Q ∈ Sj(P), let S ′(Q) := {Qk}∞ +k=1 be the +collection provided by assumption (2), and let Sj+1(P) := � +Q∈Sj(P ) S ′(Q). Let +also S (P) := �∞ +j=0 Sj(P), and S := � +P ∈Q S (P). +For Q ∈ S , let E(Q) := Q \ � +R∈S ′(Q) R. From the construction it is clear that +these sets E(Q) are pairwise disjoint, and by assumption (2) we have |E(Q)| ≥ +(1 − δ)|Q|. +By telescoping, for each P ∈ Q, we have +aP = +k−1 +� +j=0 +� +Q∈Sj(P ) +� +aQ − +� +R∈S ′(Q) +aR +� ++ +� +S∈Sk(P ) +aS. +and hence, using assumptions (2) and (3), +|aP | ≤ +k−1 +� +j=1 +� +Q∈Sj(P ) +cQ + +� +S∈Sk(P ) +C|S|α +By an elementary inequality and induction, we have +� +S∈Sk(P ) +|S|α ≤ +� +� +S∈Sk(P ) +|S| +�α +≤ (δk|P|)α, +and hence +|aP | ≤ lim +k→∞ +k−1 +� +j=1 +� +Q∈Sj(P ) +cQ = +� +Q∈S (P ) +cQ. +Substituting this into assumption (1), we obtain the claim. +□ +5.3. Lemma. Suppose that t is a bilinear form on L∞ +c (Rd; E) × L∞ +c (Rd; H), and +moreover bounded with respect to the norm of Lp(Rd; E) × Lq(Rd; H) for some +exponents with 1/p+1/q ≥ 1. For (⃗f,⃗g) ∈ L∞ +c (Rd; E)n ×L∞ +c (Rd; H)n, the numbers +a = t(⃗f,⃗g), +aQ = t(13Q ⃗f, 1Q⃗g) +satisfy assumptions (1) and (3) of Lemma 5.1, provided that D is a dyadic system +without quadrants. +Proof. Since D is without quadrants, each Q ∈ D is contained in some (large +enough) R ∈ D that contains supp ⃗f. Thus the collection Q of maximal cubes +that do not contain supp ⃗f form a cover of Rd. +By maximality, it follows that +supp ⃗f ⊂ 3Q, and hence ⃗f = 13Q ⃗f for every Q ∈ Q. On the other hand, any +Q with ℓ(Q) < diam(supp ⃗f) cannot contain supp ⃗f; hence any Q with ℓ(Q) < +1 +2 diam(supp ⃗f) cannot be among the maximal cubes Q, and thus every Q ∈ Q +will have to satisfy ℓ(Q) ≥ 1 +2 diam ⃗f. Since ⃗g ∈ L∞ +c (Rd; F)n, there are only finitely +many Q ∈ Q with 1Q⃗g ̸= 0. Hence, without any issues of convergence, we can write +t(⃗f,⃗g) = t +� +⃗f, +� +Q∈Q +1Q⃗g +� += +� +Q∈Q +t(⃗f, 1Q⃗g) = +� +Q∈Q +t(13Q ⃗f, 1Q⃗g), +which is condition (1). +If n = 1, the assumed boundedness directly implies that +|t(13Qf, 1Qg)| ≤ C∥13Qf∥Lp(Rd;E)∥1Qg∥Lq(Rd;F ) ≤ C3d/p∥f∥∞∥g∥∞|Q|1/p+1/q, + +10 +T. P. HYTÖNEN +where α := 1/p + 1/q ≥ 1, as required for condition (3). In general, if (⃗ei)n +i=1 is an +orthonormal basis of Rn and ⃗f = �n +i=1 fi⃗ei and similarly for ⃗g, we have +|t(13Q ⃗f, 1Q⃗g)| ≤ +n +� +i=1 +|t(13Qfi, 1Qgi)| ≤ Cn3d/p∥⃗f∥∞∥⃗g∥∞|Q|1/p+1/q, +using the previous bound in each component and trivial bounds like ∥fi∥∞ ≤ ∥⃗f∥∞. +□ +We are finally ready to state a semi-generic convex body domination principle. +Condition (1) below is a typical intermediate estimate in a number of sparse dom- +ination proofs for different operators. The conclusion is that it is already good +enough to conclude convex body domination as well. +5.4. Corollary. Let E and H be Banach spaces, and suppose that t is a bilinear +form defined on F × G := L∞ +c (Rd; E) × L∞ +c (Rd; H) and bounded with respect to +the norm of Lp(Rd; E) × Lq(Rd; H) for some exponents with 1/p + 1/q ≥ 1, and +suppose that +(1) for all (f, g) ∈ F × G and all Q ∈ D, there are disjoint ˆQk ⊂ Q with +� +k | ˆQk| ≤ ε|Q| and such that: whenever Qj ⊂ Q are disjoint, not strictly +contained in any ˆQk, and cover all ˆQk, then +|t(13Qf, 1Qg) − +� +j +t(13Qjf, 1Qjg)| ≤ c∥f∥X(Q)∥g∥Y (Q)|Q| +(5.5) +for some norms ∥ ∥X(Q) on L∞ +c (Rd; E) and ∥ ∥Y (Q) on L∞ +c (Rd; H). +Then for all (⃗f,⃗g) ∈ F n × Gn, there is a (1 − εn)-sparse collection S ⊂ D such +that +|t(⃗f,⃗g)| ≤ cn +� +S∈S +⟨⟨⃗f⟩⟩X(S) · ⟨⟨⃗g⟩⟩Y (S)|S|, +where εn = nε and cn = cn3/2. +Proof. Let us begin by considering a fixed cube Q = Q0 ∈ D. We observe that +assumption (1) of the present corollary coincides with condition (1) of Proposition +4.6 with +tQ(f, g) := t(13Qf, 1Qg), +C = c|Q|, +X = X(Q), +Y = Y (Q). +Hence the said proposition, applied to each fixed Q = Q0 ∈ D at a time, implies: +(2) For all (⃗f,⃗g) ∈ L∞ +c (Rd; E)n ×L∞ +c (Rd; H)n and all Q ∈ D, there are disjoint +Qk ⊂ Q with � +k |Qk| ≤ εn|Q| and such that +|t(13Q ⃗f, 1Q⃗g) − +� +j +t(13Qj ⃗f, 1Qj⃗g)| ≤ cn⟨⟨⃗f⟩⟩X(Q) · ⟨⟨g⟩⟩Y (Q)|Q|, +where εn = nε and cn = cn3/2. +Let us then consider a fixed pair (⃗f,⃗g) ∈ L∞ +c (Rd; E)n×L∞ +c (Rd; H)n. We observe +that condition (2) above coincides with condition (2) of Lemma 5.1 with the choices +aQ = t(13Q ⃗f, 1Q⃗g), +cQ = cn⟨⟨⃗f⟩⟩X(Q) · ⟨⟨g⟩⟩Y (Q)|Q|, +δ = εn. +On the other hand, Lemma 5.3 shows that these same aQ, together with a := +t(⃗f,⃗g), also satisfy conditions (1) and (3) of Lemma 5.1. Thus, all assumptions, +and hence the conclusions, of Lemma 5.1 are valid for the said quantities, and these + +SOME REMARKS ON CONVEX BODY DOMINATION +11 +conclusions agree with the claims of the result that we are proving. The proof is +thus complete. +□ +To facilitate the discussion of consequences of Corollary 5.4, we give +5.6. Definition. Suppose that a pair of normed spaces (X(Q), Y (Q)) is associated +to every dyadic cube Q ∈ D. We say that a bilinear form t : F ×G → R satisfies the +(X(Q), Y (Q)) convex body domination of order n ∈ N if F ⊆ X(Q) and G ⊆ Y (Q) +for every Q ∈ D, and if for every (f, g) ∈ F n×Gn, there exists a δn-sparse collection +S ⊂ D such that +|t(⃗f,⃗g)| ≤ Cn +� +Q∈S +|Q|⟨⟨⃗f⟩⟩X(Q) · ⟨⟨⃗g⟩⟩Y (Q). +We say that t : F × G → R satisfies the (X(Q), Y (Q)) convex body domination if +it satisfies this for every n ∈ N. We say that an operator T : F → G∗ satisfies these +properties if its associated bilinear form t(f, g) := ⟨T f, g⟩ does. +Let us now consider some examples: +5.7. Example (Calderón–Zygmund operators). Let T be a Dini–Calderón–Zygmund +operator, i.e., T is L2(Rd) bounded and has the representation +T f(x) = +ˆ +Rd K(x, y)f(y) dy, +x /∈ supp f, +where |K(x, y)| ≤ c|x − y|−d and, for |x − x′| ≤ 1 +2|x − y|, +|K(x, y) − K(x′, y)| + |K(y, x) − K(y, x′)| ≤ ω +�|x − x′| +|x − y| +� +1 +|x − y|d , +(5.8) +where ω : [0, 1 +2] → [0, ∞) is increasing, subadditive, and satisfies the Dini condition +ˆ 1/2 +0 +ω(t) dt +t < ∞. +Then (1) of Corollary 5.4 holds for t(f, g) = ⟨T f, g⟩ and E = H = R and X(Q) = +Ł1(3Q), Y (Q) = Ł1(Q), even in a stronger form. Namely, on the left oif (5.5), we +have +��� +� +T (13Qf), 1Qg +� +− +� +j +⟨T (13Qjf), 1Qjg⟩ +��� +≤ +���1QT (13Qf) − +� +j +1QjT (13Qjf) +��� +L∞(Q)∥g∥L1(Q), +(5.9) +and even the L∞ norm here is dominated by ∥f∥Ł1(3Q), as essentially shown in [24, +(3.4)]. (Strictly speaking, [24, (3.4)] is formally slightly weaker, but a straightfor- +ward modification of the argument gives the desired version, as observed in [27, +Proof of Theorem 3.4].) Thus Corollary 5.4 says that a Dini–Calderón–Zygmund +operator satisfies (Ł1(3Q), Ł1(Q)) convex body domination, but this was of course +already known from [27] by essentially the same argument. +5.10. Example (Banach space -valued Calderón–Zygmund operators). Let T be as +in Example 5.7 but now acting on the Bochner space L2(Rd; E) of Banach space E +-valued functions, and with an operator-valued kernel K(x, y) ∈ L (E) satisfying +the same estimates as above but for the operator norm in place of the absolute value, +e.g., ∥K(x, y)∥L (E) ≤ c|x − y|−d. It is in general a difficult problem to check the + +12 +T. P. HYTÖNEN +L2(Rd; E)-boundedness of such an operator, but we now take this as an assumption. +For g ∈ L2(Rd; E∗), we have (5.9) with L∞(Q; E) and L1(Q; E∗) in place of L∞(Q) +and L1(Q), and the same proof of [24, (3.4)] (with same modifications pointed +out in [27, Proof of Theorem 3.4]) shows that the L∞(Q; E) norm is dominated +by ∥f∥Ł1(3Q;E). Thus we find that (1) of Corollary 5.4 also holds with X(Q) = +Ł1(3Q; E) and Y (Q) = Ł1(Q; E∗). +The resulting sparse domination (i.e., case +n = 1 of the conclusion of Corollary 5.4) was known before, first in [15] for a +slightly smaller class of kernels, and since [22, discussion on page 193] in the present +generality. However, the convex body domination in this Banach space -valued +setting is completely new. +5.11. Example (Operators with grand maximal function control). Let 1 ≤ q ≤ r +and s ≥ 1. Suppose that T is a linear operator +T : L∞ +c (Rd) → L1 +loc(Rd), +(5.12) +that T has weak type (q, q), and that the bi-sublinear maximal operator +MT (f, g)(x) := sup +Q∋x + +Q +|T (1(3Q)cf)| · |g| +maps boundedly MT : Lr ×Ls → Lν,∞, where 1/ν = 1/r+1/s. Then condition (1) +of Corollary 5.4 holds for t(f, g) = ⟨T f, g⟩ and E = H = R and X(Q) = Łr(3Q), +Y (Q) = Łs(Q). This result is essentially contained in the proof of [25, Theorem +3.1], where it appears as an intermediate step towards the sparse domination (i.e., +case n = 1 of the conclusion of Corollary 5.4) for such operators. The extension +to convex body domination was recently achieved in [26], so Corollary 5.4 only +reproduces a known result here. A key example of concrete operators satisfying +these assumptions consists of rough homogeneous singular integrals +T f(x) = +ˆ +Rd +Ω(y) +|y|d f(x − y) dy, +where Ω(y) = Ω(y/|y|) is a bounded function with vanishing average over the unit +sphere. +As in Example 5.10, the abstract result above, involving a priori bounds of T +and MT , extends straightforwardly to the Banach space -valued setting; however, +verifying these bounds for concrete operators such as the rough homogeneous sin- +gular integrals may present a problem in this generality, since the scalar-valued +versions depend on deep results of Seeger [30], which so far lack a Banach space +-valued extension. +6. Matrix-weighted inequalities for Banach space -valued operators +A matrix weight is a locally integrable function W : Rd → Rn×n that is a.e. +positive definite -valued. The space Lp(W) consists of all measurable ⃗f : Rd → Rn +such that W 1/p ⃗f ∈ Lp(Rd; Rn), and ∥⃗f∥Lp(W) := ∥W 1/p ⃗f∥Lp(Rd;Rn). +For a Banach space E, we extend this definition in a natural way: The space +Lp(W; En) consists of all measurable ⃗f : Rd → En such that W 1/p ⃗f ∈ Lp(Rd; En), +and ∥⃗f∥Lp(W;En) := ∥W 1/p ⃗f∥Lp(Rd;En). Here, at each x ∈ Rd, we define (W 1/p ⃗f)(x) ∈ +En as the vector with components (W 1/p ⃗f)i(x) := �n +j=1(W 1/p(x))ijfj(x), i.e., the +matrix multiplication on Rn is extended to En in the natural way. + +SOME REMARKS ON CONVEX BODY DOMINATION +13 +We now concentrate on p = 2. For two matrix weights W, V : Rd → Rn×n, we +define +[W, V ]A2 := sup +Q +|⟨W⟩1/2 +Q ⟨V ⟩1/2 +Q |2, +[W]A2 := [W, W −1]A2, +where we denote the operator norm in Rn×n ≃ L (Rn) simply by | |. We denote by +A2(Rd; Rn) the class of matrix weights W : Rd → Rn×n for which [W]A2 < ∞. We +also define +[W]A∞ := sup +⃗e∈Rn[x �→ ⃗e · W(x)⃗e]A∞, +where on the right we have A∞ “norms” of some scalar weights, defined as usual by +[w]A∞ := sup +Q +1 +w(Q) +ˆ +Q +M(1Qw). +According to [27, Remark 4.4], we have +[W]A∞ ≤ 4[W]A2. +(6.1) +As a consequence of the Banach space -valued convex body domination from +Example 5.10, we obtain: +6.2. Theorem. Let E be a Banach space, and T ∈ L (L2(Rd; E)) be a Dini– +Calderón–Zygmund operator with L (E)-valued kernel. For any W ∈ A2(Rd; Rn), +the operator T extends boundedly to L2(W; En) and satisfies +∥T ∥L(L2(W;En)) ≤ cn,T ([W]A2[W]A∞[W −1]A∞)1/2 ≤ cn,T [W]3/2 +A2 . +Note that Theorem 6.2 applies to a general Banach space E, but contains the +(difficult) a priori boundedness hypothesis that T ∈ L (L2(Rd; E)). Concrete ex- +amples are available in the class of UMD spaces, treated in detail in [18]. +6.3. Corollary. Let E be a UMD space, and T ∈ L (L2(Rd)) be a scalar-valued +Calderón–Zygmund operator with a Hölder-type modulus of continuity ω(t) = ctδ, +δ ∈ (0, 1] in (5.8). For any W ∈ A2(Rd; Rn), the operator T extends boundedly to +L2(W; En) and satisfies +∥T ∥L(L2(W;En)) ≤ cn,E,T ([W]A2[W]A∞[W −1]A∞)1/2 ≤ cn,E,T [W]3/2 +A2 . +In particular, this estimate holds when T is the classical Hilbert transform. +Proof. We reduce Corollary 6.3 to Theorem 6.2 with the help of the T (1) theorem of +David and Journé [12], and its extension to UMD spaces by Figiel [14]. By the (easy +half of) the David–Journé theorem, the assumptions on T imply that that T satisfies +the so-called weak boundedness property as well as T (1), T ∗(1) ∈ BMO(Rd). Then, +by Figiel’s theorem, an operator satisfying these conditions and the Calderón– +Zygmund kernel assumptions extends boundedly to L2(Rd; E), for any UMD space +E. Thus T satisfies the assumptions, and hence the conclusions, of Theorem 6.2, +and we are done. +□ +These results, even just for the Hilbert transform, and even in their qualitative +form (i.e., just concluding the boundedness of T , without specifying any concrete +bound for the norm), are completely new in the combined setting of matrix weights +and Banach spaces. For E = R and the Hilbert transform T , the qualitative form +of Corollary 6.3 is due to Treil and Volberg [31]. The quantitative form for E = R +was obtained by Nazarov et al. [27], and this is the best that is known at the time +of writing. For scalar-weights, the power 3/2 can be replaced by 1 [16], and the + +14 +T. P. HYTÖNEN +product of [W]A∞ and [W −1]A∞ by their sum [19], but extending these to the +general matrix case consists of the outstanding open “matrix A2 conjecture”. +Turning to the proof of Theorem 6.2, we begin with: +6.4. Remark (Without loss of generality, we assume that E is reflexive). Since +Theorem 6.2 is about the bounded extension of an operator, it suffices to prove an a +priori estimate on a dense subspace of functions ⃗f. In particular, we can assume that +each component fi takes its values in a finite-dimensional subspace of E. Since any +finite-dimensional space is reflexive, we make the standing assumption, without loss +of generality, that E is reflexive. (Note that this is automatic in Corollary 6.3 in any +case, since UMD spaces are reflexive [18, Theorem 4.3.3].) Under this assumption, +we have L1(Q; E)∗ = L∞(Q; E∗) (see [18, Theorems 1.3.10 and 1.3.21]), which is +convenient in view of calculations involving the convex bodies ⟨⟨ ⟩⟩Ł1(Q;E). +6.5. Lemma. +|Q|⟨⟨W 1/2 ⃗f⟩⟩Ł1(3Q;E) · ⟨⟨V 1/2⃗g⟩⟩Ł1(Q;E∗) +≤ +ˆ � +1Q(x) + +3Q +|V 1/2(x)W 1/2(y)|∥⃗f(y)∥En dy +� +∥⃗g(x)∥ ⃗E∗n dx +Proof. Under the standing assumption from Remark 6.4, we evaluate consider a +generic element of the convex body on the left with φ ∈ ¯BL∞(Q;E∗) and ψ ∈ +¯BL∞(Q;E): +|Q| +��� + +3Q +W 1/2(y)⟨⃗f(y), φ(y)⟩ dy · + +Q +V 1/2(x)⟨⃗g(x), ψ(x)⟩ dx +��� += |Q| +��� + +Q + +3Q +V 1/2(x)W 1/2(y)⟨⃗f(y), φ(y)⟩ · ⟨⃗g(x), ψ(x)⟩ dy dx +��� +≤ +ˆ +Q + +3Q +|V 1/2(x)W 1/2(y)|∥⃗f(y)∥En∥⃗g(x)∥ ⃗E∗n dy dx. +□ +Summing over a sparse collection, we obtain +� +Q∈S +|Q|⟨⟨W 1/2 ⃗f⟩⟩Ł1(3Q;E) · ⟨⟨V 1/2⃗g⟩⟩Ł1(Q;E∗) +≤ +ˆ � � +Q∈S +1Q(x) + +3Q +|V 1/2(x)W 1/2(y)|∥⃗f(y)∥En dy +� +∥⃗g(x)∥ ⃗E∗n dx +=: +ˆ +˜L(∥⃗f∥En)(x)∥⃗g(x)∥ ⃗E∗n dx, +(6.6) +where ˜L, here acting on the scalar-valued function y �→ ∥⃗f(y)∥En, is an operator +denoted by the same symbol in [27, (5.8)]. By [27, Lemma 5.6], we have +∥˜L∥L (L2) ≤ C([W, V ]A2[W]A∞[V ]A∞)1/2. +(6.7) +By duality and standard changes of variables, which present no essential differ- +ence in the Banach space -valued setting, an estimate of the form +∥T ⃗f∥L2(V ;En) ≤ N∥⃗f∥L2(V ;En) +is equivalent to +⟨T (W 1/2 ⃗f), V 1/2⃗g⟩ ≤ N∥⃗f∥L2(Rd;En)∥⃗g∥L2(Rd;E∗n). +(6.8) + +SOME REMARKS ON CONVEX BODY DOMINATION +15 +If T is an in Theorem 6.2, it satisfies the (Ł1(3Q; E), Ł1(Q; E∗)) convex body dom- +ination by Example 5.10, which means that the left-hand side of (6.8) is dominated +by the left-hand side of (6.6), and hence, by (6.6) and (6.7), we have +N ≤ cn,T ([W, V ]A2[W]A∞[V ]A∞)1/2. +This is the desired bound, and concludes the proof of Theorem 6.2. +7. Convex domination and generalised commutators +For an operator T and two vector functions ⃗a = (a1, . . . , an) and ⃗b = (b1, . . . , bn), +let us consider the operator +⃗a · T⃗b : f �→ ⃗a · T (⃗bf) = +n +� +i=1 +aiT (bif). +We are mainly interested in the boundedness on Lp(Rd), or a weighted Lp(w), +or between two such spaces, and the case when T is a singular integral operator +bounded on the space. However, we do not require that ai, bi ∈ L∞(Rd), and hence +the pointwise multipliers f �→ bif and g �→ aig, and the compositions f �→ aiT (bif), +may be unbounded operators. Nevertheless, their sum ⃗a · T⃗b may still be bounded, +thanks to cancellation between different terms. +A case that has been much studied in the literature consists of ⃗b = (1, b) and +⃗a = (b, −1), in which case +⃗a · T (⃗bf) = bT f − T (bf) = [b, T ]f +is the commutator of b and T , whose Lp(Rd)-boundedness is characterised by b ∈ +BMO(Rd), the space of functions of bounded mean oscillation, which is strictly +larger than L∞(Rd), and contains in particular functions like b(x) = log |x|. +By dualising with a function g, and denoting by t(f, g) = ⟨T f, g⟩ the bilinear +form of T , we arrive at +⟨⃗a · T (⃗bf), g⟩ = +n +� +i=1 +⟨T (bif), aig⟩ = t(⃗bf,⃗ag), +where the action of the bilinear form is extended to vector-valued functions as +before. To be precise, if t in defined on F × G, we should now require that +f ∈ F⃗b := {f ∈ F : bif ∈ F for all i = 1, . . . , n}, +and g ∈ G⃗a, defined similarly. If F ⊇ L∞ +c (Rd), then clearly F⃗b contains in particular +all f ∈ L∞ +c (Rd) with supp f ⊆ EN := {|⃗b| ≤ N} for any N ∈ N. For a.e. finite- +valued bi, the union � +N∈N EN covers Rd up to a null set, it is immediate that F⃗b +is dense in any Lp(w) with finite p. +7.1. Lemma. Suppose that T satisfies the (X(Q), Y (Q)) convex body domination. +Then for all relevant functions, we have +|⟨⃗a · T (⃗bf), g⟩| ≤ C +� +Q∈S +|Q|⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q). +(7.2) +Proof. This is immediate by applying definition to ⃗f = ⃗bf and ⃗g = ⃗ag. +□ +We take a closer look at the case when X(Q) = Y (Q) = Ł1(γQ). + +16 +T. P. HYTÖNEN +7.3. Lemma. For all s, t ∈ (1, ∞) and all functions in the relevant spaces, we have +⟨⟨⃗bf⟩⟩Ł1(Q) · ⟨⟨⃗ag⟩⟩Ł1(Q) ≤ ∥(x, y) �→ ⃗a(x) ·⃗b(y)∥Ł(s,t) +min (Q×Q)∥f∥Łt′ (Q)∥g∥Łs′(Q), +where +∥F∥Ł(s,t) +min (Q×Q) := + + + + + +� ffl +Q +� ffl +Q |F(x, y)|s dx +�t/s +dy +�1/t +, +if s ≤ t, +� ffl +Q +� ffl +Q |F(x, y)|t dy +�s/t +dx +�1/s +, +if t ≤ s. +Proof. The generic element of ⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q) has the following form, where +φ, ψ ∈ ¯BL∞(Q): + +Q +⃗b(y)f(y)φ(y) dy · + +Q +⃗a(x)g(x)ψ(x) dx += + +Q + +Q +(⃗a(x) ·⃗b(y))f(y)g(x)φ(y)ψ(x) dx dy, +and hence +⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q) ≤ + +Q + +Q +|⃗a(x) ·⃗b(y)||f(y)||g(x)| dx dy +≤ ∥(x, y) �→ a(x) · b(y)∥Z∥(x, y) �→ f(y)g(x)∥Z∗, +for either choice of +(Z, Z∗) ∈ {(Łs +x(Q; Łt +y(Q)), Łs′ +x (Q; Łt′ +y (Q))), (Łt +y(Q; Łs +x(Q)), Łt′ +y (Q; Łs′ +x (Q)))}, +by Hölder’s inequality for mixed-norm Lp spaces. By Fubini’s theorem, we have +∥(x, y) �→ f(x)g(y)∥Z∗ = ∥f∥Łt′ (Q)∥g∥Łs′(Q) +in either case, and hence, taking the minimum over the two choices of Z, we arrive +at the factor +min +Z ∥(x, y) �→ b(x) · a(y)∥Z = ∥(x, y) �→ ⃗b(x) · ⃗a(y)∥Ł(s,t) +min (Q×Q). +□ +7.4. Proposition. Let T be an operator that satisfies the (Ł1(γQ), Ł1(γQ)) convex +body domination. Let ⃗a,⃗b ∈ L1 +loc(Rd)n be functions such that +As,t := sup +Q +∥(x, y) �→ ⃗a(x) ·⃗b(y)∥Ł(s,t) +min (Q×Q) < ∞. +Then ⃗a·T⃗b extends to a bounded operator on Lp(Rd) for all p ∈ (t′, s). In particular, +if As := As,s < ∞ for some s ∈ (2, ∞), then ⃗a · T⃗b extends boundedly to L2(Rd). + +SOME REMARKS ON CONVEX BODY DOMINATION +17 +Proof. Combining Lemmas 7.1 and 7.3, we find that +|⟨⃗aT (⃗bf), g⟩| ≤ C +� +Q∈S +|Q|⟨⟨⃗bf⟩⟩Ł1(γQ) · ⟨⟨⃗ag⟩⟩Ł1(γQ) +≤ C +� +Q∈S +|Q|∥(x, y) �→ a(x) · b(y)∥Ł(s,t) +min (Q×Q)∥f∥Łt′ (Q)∥g∥Łs′(Q) +≤ C +� +Q∈S +|E(Q)| +δ +As,t inf +Q Mt′f inf +Q Ms′g +≤ CAs,t +δ +� +Q∈S +ˆ +E(Q) +Mt′fMs′g ≤ CAs,t +δ +ˆ +Rd Mt′fMs′g +≤ CAs,t +δ +∥Mt′f∥Lp(Rd)∥Ms′g∥Lp′(Rd), +where +∥Mt′f∥Lp(Rd) ≲t,p ∥f∥Lp(Rd), +∥Ms′g∥Lp′(Rd) ≲s,p ∥g∥Lp′(Rd) +for p > t′ and p′ > s′, where the latter is equivalent to p < s. +□ +Let us consider some examples: +7.5. Example (Classical commutators). As we already observed, ⃗a = (b, −1) and +⃗b = (1, b) gives rise to the usual commutator [b, T ]. In this case +⃗a(x) ·⃗b(y) = b(x) − b(y) +and each As,t is equivalent to ∥b∥BMO(Rd) by elementary considerations and the +John–Nirenberg inequality. Thus Proposition 7.4 reproduces the well-known suffi- +cient condition for the boundedness of commutators. +7.6. Example (Iterated commutators). More generally, choosing ⃗a,⃗b so that +⃗a(x) ·⃗b(y) = (b(x) − b(y))k = +k +� +i=0 +�k +i +� +b(x)k−i(−b(y))i, +thus e.g. ai(x) = +�k +i +� +b(x)k−i and bi(y) = (−b(y))i, we reproduce the kth order +commutator +⃗a · T⃗b = Tk,b := [b, Tk−1,b], +T0,b := T, +and As,t is equivalent to ∥b∥k +BMO(Rd) by the John–Nirenberg inequality. +7.7. Example (Iterated commutators with different multipliers). Let us then choose +⃗a,⃗b so that +⃗a(x) ·⃗b(y) = (b1(x) − b1(y))(b2(x) − b2(y)); +without specifying the precise choice of ai(x) and bi(y), it is evident that such +a choice can be easily written down, if desired. (We deliberately use superscript +indices for bi above, since these not be the same as the components bi of ⃗b.) This +reproduces the second order iterated commutator with two different functions, +⃗a · T⃗b = [b1, [b2, T ]]. +It is well-known and classical that bi ∈ BMO(Rd) for both i = 1, 2 is sufficient for +the L2(Rd) boundedness of [b1, [b2, T ]]; however, as recently observed in [17], much + +18 +T. P. HYTÖNEN +weaker sufficient conditions can be given for the pair (b1, b2). Namely, in [17, (1.1)], +it shown that the pair of conditions +Ss := sup +Q +� +Q +|b1(x) − ⟨b1⟩Q|s dx +�1/s� +Q +|b2(y) − ⟨b2⟩Q|s dy +�1/s +< ∞, +Ts := sup +Q +� +Q +|b1(x) − ⟨b1⟩Q|s|b2(x) − ⟨b2⟩Q|s dx +�1/s +< ∞, +is sufficient for the L2(Rd) boundedness of [b1, [b2, T ]] for s > 2. On the other hand, +by Proposition 7.4, another sufficient condition for the same conclusion is As < ∞. +Let us compare the two. Adding and subtracting terms and multiplying out, we +find that +(b1(x) − b1(y))(b2(x) − b2(y)) += [(b1(x) − ⟨b1⟩Q) − (b1(y) − ⟨b1⟩Q)][(b2(x) − ⟨b2⟩Q) − (b2(y) − ⟨b2⟩Q)] += (b1(x) − ⟨b1⟩Q)(b2(x) − ⟨b2⟩Q) + (b1(y) − ⟨b1⟩Q)(b2(y) − ⟨b2⟩Q) +− (b1(x) − ⟨b1⟩Q)(b2(y) − ⟨b2⟩Q) − (b1(y) − ⟨b1⟩Q)(b2(x) − ⟨b2⟩Q). +Taking Łs(Q × Q) and then supremum over Q on both sides, we deduce that +As ≤ 2(Ts + Ss), +so that the new criterion provided by Proposition 7.4 is at least as sharp as that of +[17, (1.1)], and it seems less obvious to make any estimate in the other direction. +Perhaps more importantly, the new condition As < ∞ arises more “naturally” as +an instance of a general principle. +(Let us note that there is a more general criterion [17, Theorem 3.10], where the +Łs norms in Ss and Tt are replaced by more general Orlicz norms. On the other +hand, it is apparent that similar generalisations could be achieved in Proposition +7.4: what we used was the boundedness of the rescaled maximal operators Mt′ on +Lp(Rd) for p > t′, and this could be replaced having an Orlicz maximal operator +MA with the same mapping property. A characterisation of this property in terms +of the so-called Bp condition on the Orlicz function A is a classical result of Pérez +[29]; this very result is used in [17]; see [17, Proposition 3.8].) +Let us finally consider an “exotic” example with no obvious predecessor in the +existing literature. We begin with a lemma: +7.8. Lemma. Suppose that 0 ≤ b ∈ BMO(Rd). If 0 ≤ α, β and α + β ≤ 1, then +B(x, y) := b(x)αb(y)β − b(x)βb(y)α +satisfies +� +Q + +Q +|B(x, y)|p dx dy +�1/p +≤ (2∥b∥BMOp(Rd))α+β. +Proof. Let γ := min(α, β) ∈ [0, 1 +2] and δ := max(α, β) − γ ∈ [0, 1]. Then +|B(x, y)| = b(x)γb(y)γ|b(x)δ − b(y)δ|. +We observe the following elementary inequality: +|uδ − vδ| ≤ +|u − v| +max(u, v)1−δ , +∀u, v ≥ 0, δ ∈ [0, 1]. +(7.9) + +SOME REMARKS ON CONVEX BODY DOMINATION +19 +Indeed, by symmetry and homogeneity, it is enough to consider u = 1 and v ∈ [0, 1], +in which case we are reduced to proving that +1 − vδ ≤ 1 − v, +which is immediate from the fact that v ≤ vδ for v, δ ∈ [0, 1]. +Using (7.9), and noting that δ + 2γ = α + β ∈ [0, 1], it follows that +|B(x, y)| ≤ b(x)γb(y)γ +|b(x) − b(y)| +max(b(x), b(y))1−δ ≤ +|b(x) − b(y)| +max(b(x), b(y))1−δ−2γ += +� |b(x) − b(y)| +max(b(x), b(y)) +�1−δ−2γ +|b(x) − b(y)|δ+2γ ≤ |b(x) − b(y)|α+β, +and hence +� +Q + +Q +|B(x, y)|p dx dy +�1/p +≤ +� +Q + +Q +|b(x) − b(y)|p dx dy +�(α+β)/p +≤ +�� +Q +|b(x) − c|p dx +�1/p ++ +� +Q +|b(y) − c|p dy +�1/p�α+β +for all constants c. +□ +7.10. Corollary. Let T be an operator satisfying (Ł1(γQ), Ł1(γQ)) convex body +domination, let 0 ≤ b ∈ BMO(Rd) and 0 ≤ α, β with α + β ≤ 1. Then +∥bαT (bβf) − bβT (bαf)∥Lp(Rd) ≲p ∥b∥α+β +BMO(Rd)∥f∥Lp(Rd). +Proof. By Proposition 7.4 with s = t, the Lp(Rd) operator norm of f �→ bαT (bβf)− +bβT (bαf) is dominated by +As := sup +Q +∥(x.y) �→ b(x)αb(y)β − b(x)βb(y)α∥Łs(Q×Q) +if p ∈ (s′, s), i.e., if s > max(p, p′). By Lemma 7.8 and the John–Nirenberg inequal- +ity, we have +As ≤ (2∥b∥BMOs(Rd))α+β ≲s ∥b∥α+β +BMO(Rd), +and fixing (say) s = 2 max(p, p′), we obtain a dependence on p only. +□ +7.11. Remark. Aside from the examples already discussed, the generalised commu- +tators ⃗a · T⃗b also arise in the following question studied by Bloom [4, 5]. Suppose +that a matrix weight W is given in the diagonalised form W = U ∗ΛU, where U is +unitary, Λ is diagonal, and the diagonal entries λk of Λ are scalar A2 weights. What +does one need to know about U in order to conclude that W ∈ A2? (According to +[5, Theorem 4.2], the condition that λk ∈ A2 is necessary for W ∈ A2, if in addition +U is assumed to be continuous.) +Let T be the Hilbert transform, or another operator whose boundedness on +the matrix-weighted L2(W) characterises W ∈ A2. +By connecting the L2(W) +boundedness of T to the boundedness of the classical commutators [T, ¯uij] between +the weighted spaces L2(λi) and L2(λk) (sic: the condition involves triplets of indices +(i, j, k)), [4, Theorem 5.1] shows that uij ∈ BMO√ +λi/λk (a weighted BMO space, +nowadays commonly referred to as Bloom-type BMO) is a sufficient condition. In +the special case of 2 × 2 matrices, it is also necessary by [5, Theorem 4.3] but, over +30 years since these contributions, the general case seems to remain open. (The +author is grateful to Amalia Culiuc for bringing this question to his attention [9].) + +20 +T. P. HYTÖNEN +Here is a possible approach to the problem. As is well known, the L2(W) bound- +edness of T is equivalent to the (unweighted) L2 boundedness of +W 1/2T W −1/2 = U ∗Λ1/2UT U ∗Λ−1/2U. +Multiplication by U and U ∗ is isometric on L2, and the L2 boundedness of a matrix +of operators is equivalent to the L2 boundedness of each of the components +(Λ1/2UT U ∗Λ−1/2)ij = +n +� +k=1 +λ1/2 +i +uikT ¯ujkλ−1/2 +j += λ1/2 +i +⃗ui · T ¯⃗ujλ−1/2 +j +, +where i, j = 1, . . . , n and ⃗ui = (uik)n +k=1. These are operators of the form ⃗a · T⃗b that +we have studied here and, up to this point, we kept an exact equivalence with the +original question; the question then would be, whether we can give useful conditions +on the boundedness of these operators. A further equivalent condition is of course +the two-weight boundedness +⃗ui · T ¯⃗uj : L2(λj) → L2(λi), +i, j = 1, . . . , n, +where the spaces are more complicated, but the multipliers are simply rows of the +unitary matrix U. +7.12. Remark. We have concentrated in this section on the application of convex +body domination—an inherently vector-valued theory—to questions of generalised +commutators acting on scalar-valued functions. We have made this choice for two +reasons: to make the case that this vector-valued theory is useful even for such +scalar-valued applications, and not to obscure the relatively simple basic philoso- +phy behind too many technicalities of notation. This said, it is quite plain that +the presented ideas can be immediately generalised to the case of vector-valued +functions ⃗f and ⃗g (in place of scalar f and g) and matrix-valued multipliers A and +B (in place of the vectors ⃗a and ⃗b). In the particular case of the classical-style +commutator [T, B] with a matrix-valued function, this idea has been developed in +[21]. +8. Stopping times and maximal functions involving convex bodies +The aims of this final section are two-fold. Concretely, we establish a convex- +body analogue of a result of Nieraeth [28], which shows that the estimation of +sums over sparse collection that arise in the usual sparse domination is equivalent +to the estimation of certain maximal functions. On the way of achieving this, we +develop some convex-body versions of the typical stopping time arguments involving +averages of scalar-valued functions; these might have some independent interest +elsewhere. +We begin with an estimate of a sum of convex-body “norms” over disjoint subsets. +8.1. Lemma. Let p, q ∈ [1, ∞) and 1 +r := 1 +p + 1 +q . Let Qi ∈ D(Q0) be disjoint cubes. +Then +∞ +� +i=1 +� +⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) +�r ≤ nmax(r,1)+r/2� +⟨⟨f⟩⟩Lp(Q) · ⟨⟨g⟩⟩Lq(Q) +�r. +Note that for p, q ∈ [1, ∞), we have 1 +r = 1 +p + 1 +q ≤ 1 + 1 = 2, and hence +nmax(r,1)+r/2 = +� +nmax(1,1/r)+1/2�r ≤ +� +n5/2�r. + +SOME REMARKS ON CONVEX BODY DOMINATION +21 +Proof. For orientation, let us begin with the proof in the case n = 1, i.e., with ∥ ∥ +in place of ⟨⟨ ⟩⟩ throughout. By Hölder’s inequality with 1 = r +p + r +q, we have +∞ +� +i=1 +� +∥f∥Lp(Qi)∥g∥Lq(Qi) +�r = +∞ +� +i=1 +� +∥f∥p +Lp(Qi) +�r/p� +∥g∥q +Lq(Qi) +�r/q +≤ +� ∞ +� +i=1 +∥f∥p +Lp(Qi) +�r/p� ∞ +� +i=1 +∥g∥q +Lq(Qi) +�r/q +≤ +� +∥f∥p +Lp(Q0) +�r/p� +∥g∥q +Lq(Q0) +�r/q += +� +∥f∥Lp(Q0)∥g∥Lq(Q0) +�r +. +In the general case of the lemma, let +Ai := ⟨⟨⃗f⟩⟩Lp(Qi) = +� ˆ +Qi +φi ⃗f : ∥φi∥Lp′(Qi) ≤ 1 +� +, +Bi := ⟨⟨⃗g⟩⟩Lq(Qi). +Then we observe that +⟨⟨⃗f⟩⟩Lp(Q) = +� ˆ +Q +φ⃗f : ∥φ∥Lp′(Q) ≤ 1 +� +⊇ +� ∞ +� +i=1 +ai +ˆ +Qi +φi ⃗f : ∥φi∥Lp′(Qi) ≤ 1, ∥(ai)∥ℓp′ ≤ 1 +� += +� ∞ +� +i=1 +aiAi : ∥(ai)∥ℓp′ ≤ 1 +� +=: +� +ℓp +Ai =: A, +and similarly +⟨⟨⃗g⟩⟩Lq(Q) ⊇ +� +ℓq +Bi =: B. +Hence, the lemma is reduced to proving that +∞ +� +i=1 +� +Ai · Bi +�r ≤ nmax(r,1)+r/2� +A · B +�r, +A := +� +ℓp +Ai, +B := +� +ℓq +Bi. +Let EA be the John ellipsoid of A, and let RAEA = ¯BRn. +Since Ai · Bi = +RAAi · R−t +A Bi, The claim above is equivalent to a version where each Ai is replaced +by RAAi and each Bi by R−t +A Bi. Hence, without loss of generality, we assume that +EA = ¯BRn to begin with, hence ¯BRn ⊆ A ⊆ √n ¯BRn. Thus +A · B ⊃ ¯BRn · B = [−M, M], +where +M := max{|⃗b| : ⃗b ∈ B}. +On the other hand, if (⃗ej)n +j=1 is some orthonormal basis of Rn, then +Ai · Bi = {⃗a ·⃗b : ⃗a ∈ Ai,⃗b ∈ Bi} += +� +n +� +j=1 +(⃗a · ⃗ej)(⃗b · ⃗ej) : ⃗a ∈ Ai,⃗b ∈ Bi} ⊆ +n +� +j=1 +(Ai · ⃗ej)(Bi · ⃗ej), +or, using the identification of [−s, s] with s, +Ai · Bi ≤ +n +� +j=1 +(Ai · ⃗ej)(Bi · ⃗ej). + +22 +T. P. HYTÖNEN +Thus +(Ai · Bi)r ≤ +n +� +j=1 +� +(Ai · ⃗ej)(Bi · ⃗ej) +�r, +r ∈ (0, 1], +and +� ∞ +� +i=1 +(Ai · Bi)r�1/r +≤ +n +� +j=1 +� ∞ +� +i=1 +� +(Ai · ⃗ej)(Bi · ⃗ej) +�r�1/r +, +r ∈ [1, ∞). +In the sum over i, we use Hölder’s inequality as in the toy model in the beginning: +∞ +� +i=1 +� +(Ai · ⃗ej)(Bi · ⃗ej) +�r = +∞ +� +i=1 +� +(Ai · ⃗ej)p�r/p� +(Bi · ⃗ej)q�r/q +≤ +� ∞ +� +i=1 +(Ai · ⃗ej)p�r/p� ∞ +� +i=1 +(Bi · ⃗ej)q�r/q += sup +�� ∞ +� +i=1 +aiAi · ⃗ej +�1/r� ∞ +� +i=1 +biBi · ⃗ej +�1/r +: ∥(ai)∥ℓp′ ≤ 1, ∥(bi)∥ℓq′ ≤ 1 +� += (A · ⃗ej)r(B · ⃗ej)r +Here +A · ⃗ej ⊆ √n ¯BRn · ⃗ej = [−√n, √n], +A · ⃗ej ≤ √n, +and clearly +B · ⃗ej ≤ M. +Altogether, writing s := max(r, 1), we have +� ∞ +� +i=1 +(Ai · Bi)r�1/s +≤ +n +� +j=1 +� ∞ +� +i=1 +(Ai · ⃗ej)r(Bi · ⃗ej)r�1/s +≤ +n +� +j=1 +� +(A · ⃗ej)r(B · ⃗ej)r�1/s +≤ n[nr/2M r]1/s, +and hence +∞ +� +i=1 +(Ai · Bi)r ≤ nsnr/2M r = nmax(1,r)+r/2(A · B)r, +which remained to be proved. +□ +The following lemma is a convex-body analogue of the basic principle underlying +the simplest stopping time constructions: for a function on a cube Q0, the total +measure of the subcubes, where the average of a function is much bigger than on +the whole Q0, can be at most a fraction of the measure of Q0. +8.2. Lemma. Let A, p, q ∈ [1, ∞) and let Qi ∈ D(Q0) be disjoint cubes such that +⟨⟨⃗f⟩⟩Łp(Qi) · ⟨⟨⃗g⟩⟩Łq(Qi) ≥ A⟨⟨⃗f⟩⟩Łp(Q0) · ⟨⟨⃗g⟩⟩Łq(Q0). +Then +∞ +� +i=1 +|Qi| ≤ nmax(r,1)+r/2 +Ar +|Q0|, +1 +r := 1 +p + 1 +q . + +SOME REMARKS ON CONVEX BODY DOMINATION +23 +Proof. Directly from the definition, it is easy to extend the basic identity ∥f∥Łp(Q) = +|Q|−1/p∥f∥Lp(Q) to convex bodies as +⟨⟨⃗f⟩⟩Łp(Q) = |Q|−1/p⟨⟨⃗f⟩⟩Lp(Q). +(8.3) +From this, the assumption of the lemma can be rewritten as +|Qi|−1/p−1/q⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) ≥ A|Q0|−1/p−1/q⟨⟨⃗f⟩⟩p(Q0) · ⟨⟨⃗g⟩⟩q(Q0), +or, rearranging, +|Qi| ≤ +A−r|Q0| +� +⟨⟨⃗f⟩⟩p(Q0) · ⟨⟨⃗g⟩⟩q(Q0) +�r +� +⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) +�r. +Summing over i and using Lemma 8.1, we obtain the claim. +□ +We now obtain the following proposition, which is a convex body analogue of +a result of Nieraeth [28, Prop. 2.7; especially Eq. (2.7) for m = 1]. It says that +estimating the sums over sparse collections, like those that arise from convex body +domination, is equivalent to estimating related bi-sublinear maximal operators. In +[28, Prop. 2.7], the result is formulated as a set of equivalent conditions for a tuple +of weights. The formulation below has no reference to weights as such, but as soon +as one starts asking questions about the boundedness of either side on spaces like +Ls(W) × Ls′(W ′), the proposition guarantees that one can equally well study this +boundedness for the other side of the equivalence. +8.4. Proposition. For all δ ∈ (0, 1), all dimensions d, n ≥ 1, exponents p, q ∈ +[1, ∞), and functions ⃗f ∈ Lp +loc(Rd)n, ⃗g ∈ Lq +loc(Rd)n, we have the two-sided estimate +sup +S +� +Q∈S +⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q)|Q| ≂ +��� sup +Q∈D +1Q⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q) +��� +L1(Rd), +where the supremum is taken over all δ-sparse collections of dyadic cubes in Rn, +and the implied constants depend only on n, p, q, and δ. +Proof. With ⃗f ∈ Lp +loc(Rd)n and ⃗g ∈ Lq +loc(Rd)n fixed, let us denote +aQ := ⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q). +The estimate ≲ is immediate: From δ-sparseness, we have |Q| ≤ δ−1|E(Q)| for +some disjoint sets E(Q), and hence +� +Q∈S +aQ|Q| ≤ 1 +δ +� +Q∈S +aQ|E(Q)| = 1 +δ +ˆ +Rd +� +Q∈S +aQ1E(Q) ≤ 1 +δ +ˆ +Rd sup +Q∈D +aQ1Q. +The estimate ≳ needs a bit more. By monotone convergence, it is enough to +consider D(Q0) in place of D. Let S0 := {Q0}. For some A > 1 to be chosen and +Q ∈ D(Q0), let S ′(Q) consist of all maximal Q′ ∈ D(Q) such that aQ′ > AaQ. By +maximality, the cubes Q′ ∈ S ′(Q) are disjoint. By Lemma 8.2, we have +� +Q′∈S ′(Q) +|Q′| ≤ nmax(1,r)+r/2 +Ar +|Q| ≤ (1 − δ)|Q|, +1 +r := 1 +p + 1 +q , +provided that A is chosen large enough, depending on n, p, q, and δ. Hence, defining +inductively Sj+1 := � +Q∈Sj S ′(Q) and S := �∞ +j=0 Sj, we find that S is δ-sparse. + +24 +T. P. HYTÖNEN +If Q ∈ D(Q0) and S ∈ S is the minimal stopping cube that contains Q, then +aQ ≤ AaS by the way that the cubes S ∈ S were chosen, hence +sup +Q∈D(Q0) +1QaQ ≤ sup +S∈S +1SAaS ≤ A +� +S∈S +1SaS, +and thus ��� +sup +Q∈D(Q0) +1QaQ +��� +L1(Rd) ≤ A +��� +� +S∈S +1SaS +��� +L1(Rd) = A +� +S∈S +aS|S|. +□ +References +[1] S. Bagchi, S. Hait, L. Roncal, and S. Thangavelu. On the maximal function associated to the +spherical means on the Heisenberg group. New York J. Math., 27:631–675, 2021. +[2] D. Beltran, +J. Roos, +and A. Seeger. Multi-scale sparse domination, +2020. Preprint, +arXiv:2009.00227. +[3] F. Bernicot, D. Frey, and S. Petermichl. Sharp weighted norm estimates beyond Calderón- +Zygmund theory. Anal. PDE, 9(5):1079–1113, 2016. +[4] S. Bloom. A commutator theorem and weighted BMO. Trans. Amer. Math. Soc., 292(1):103– +122, 1985. +[5] S. Bloom. Applications of commutator theory to weighted BMO and matrix analogs of A2. +Illinois J. Math., 33(3):464–487, 1989. +[6] M. Bownik and D. Cruz-Uribe. Extrapolation and factorization of matrix weights, 2022. +Preprint, arXiv:2210.09443. +[7] J. M. Conde-Alonso, A. Culiuc, F. Di Plinio, and Y. Ou. A sparse domination principle for +rough singular integrals. Anal. PDE, 10(5):1255–1284, 2017. +[8] D. Cruz-Uribe, J. Isralowitz, and K. Moen. Two weight bump conditions for matrix weights. +Integral Equations Operator Theory, 90(3):Paper No. 36, 31, 2018. +[9] A. Culiuc. Personal communication, 2022. 11th International Conference on Harmonic Anal- +ysis and Partial Differential Equations, El Escorial, Spain. +[10] A. Culiuc, F. Di Plinio, and Y. Ou. Uniform sparse domination of singular integrals via dyadic +shifts. Math. Res. Lett., 25(1):21–42, 2018. +[11] A. Culiuc, R. Kesler, and M. T. Lacey. Sparse bounds for the discrete cubic Hilbert transform. +Anal. PDE, 12(5):1259–1272, 2019. +[12] G. David and J.-L. Journé. A boundedness criterion for generalized Calderón-Zygmund op- +erators. Ann. of Math. (2), 120(2):371–397, 1984. +[13] F. Di Plinio, T. Hytönen, and K. Li. Sparse bounds for maximal rough singular integrals via +the Fourier transform. Ann. Inst. Fourier (Grenoble), 70(5):1871–1902, 2020. +[14] T. Figiel. Singular integral operators: a martingale approach. In Geometry of Banach spaces +(Strobl, 1989), volume 158 of London Math. Soc. Lecture Note Ser., pages 95–110. Cambridge +Univ. Press, Cambridge, 1990. +[15] T. S. Hänninen and T. Hytönen. The A2 theorem and the local oscillation decomposition for +Banach space valued functions. J. Operator Theory, 72(1):193–218, 2014. +[16] T. Hytönen. The sharp weighted bound for general Calderón-Zygmund operators. Ann. of +Math. (2), 175(3):1473–1506, 2012. +[17] T. Hytönen, K. Li, and T. Oikari. Iterated commutators under a joint condition on the tuple +of multiplying functions. Proc. Amer. Math. Soc., 148(11):4797–4815, 2020. +[18] T. Hytönen, J. v. Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces. Vol. I. +Martingales and Littlewood-Paley theory, volume 63 of Ergebnisse der Mathematik und ihrer +Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics +and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Cham, +2016. +[19] T. Hytönen and C. Pérez. Sharp weighted bounds involving A∞. Anal. PDE, 6(4):777–818, +2013. +[20] J. Isralowitz, S. Pott, and I. P. Rivera-Ríos. Sharp A1 weighted estimates for vector-valued +operators. J. Geom. Anal., 31(3):3085–3116, 2021. +[21] J. Isralowitz, S. Pott, and S. Treil. Commutators in the two scalar and matrix weighted +setting. J. Lond. Math. Soc. (2), 106(1):1–26, 2022. + +SOME REMARKS ON CONVEX BODY DOMINATION +25 +[22] M. T. Lacey. An elementary proof of the A2 bound. Israel J. Math., 217(1):181–195, 2017. +[23] A. K. Lerner. A simple proof of the A2 conjecture. Int. Math. Res. Not. IMRN, (14):3159– +3170, 2013. +[24] A. K. Lerner. On pointwise estimates involving sparse operators. New York J. Math., 22:341– +349, 2016. +[25] A. K. Lerner. A weak type estimate for rough singular integrals. Rev. Mat. Iberoam., +35(5):1583–1602, 2019. +[26] P. A. Muller and I. P. Rivera-Ríos. Quantitative matrix weighted estimates for certain singular +integral operators. J. Math. Anal. Appl., 509(1):Paper No. 125939, 38, 2022. +[27] F. Nazarov, S. Petermichl, S. Treil, and A. Volberg. Convex body domination and weighted +estimates with matrix weights. Adv. Math., 318:279–306, 2017. +[28] Z. Nieraeth. Quantitative estimates and extrapolation for multilinear weight classes. Math. +Ann., 375(1-2):453–507, 2019. +[29] C. Pérez. On sufficient conditions for the boundedness of the Hardy-Littlewood maximal +operator between weighted Lp-spaces with different weights. Proc. London Math. Soc. (3), +71(1):135–157, 1995. +[30] A. Seeger. Singular integral operators with rough convolution kernels. J. Amer. Math. Soc., +9(1):95–105, 1996. +[31] S. Treil and A. Volberg. Wavelets and the angle between past and future. J. Funct. Anal., +143(2):269–308, 1997. +Department of Mathematics and Statistics, P.O.B. 68 (Pietari Kalmin katu 5), +FI-00014 University of Helsinki, Finland +Email address: tuomas.hytonen@helsinki.fi + diff --git a/UdAyT4oBgHgl3EQfuvkh/content/tmp_files/load_file.txt b/UdAyT4oBgHgl3EQfuvkh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b9e7825b0285802b8d62eba8459d9945f34df85e --- /dev/null +++ b/UdAyT4oBgHgl3EQfuvkh/content/tmp_files/load_file.txt @@ -0,0 +1,972 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf,len=971 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='00617v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='FA] 2 Jan 2023 SOME REMARKS ON CONVEX BODY DOMINATION TUOMAS P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN Dedicated, with admiration, to the Ukrainian people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Convex body domination is an important elaboration of the tech- nique of sparse domination that has seen significant development and applica- tions over the past ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In this paper, we present an abstract framework for convex body domination, which also applies to Banach space -valued func- tions, and yields matrix-weighted norm inequalities in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We explore applications to “generalised commutators”, obtaining new examples of bounded operators among linear combinations of compositions of the form aiTbi, where ai, bi are pointwise multipliers and T is a singular integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Introduction The technique of sparse domination was developed to provide a simpler approach, achieved by Lerner [23], to the “A2 conjecture” on sharp weighted norm inequalities for Calderón–Zygmund operators, which was first proved with a different machinery by the author [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' However, beyond this original aim, sparse domination imme- diately led to significant further consequences and has by now been applied to a variety of new questions, of which [1, 2, 3, 7, 11] is only a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The method consists of two main steps that are largely independent of each other and essentially decouple the operator from the space or norm in which it should be estimated: (1) Dominating an operator of interest by a suitable sparse operator/form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2) Estimating the sparse form with respect to relevant norms of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' While sparse domination very efficiently captures the local size of an object un- der consideration, and this is precisely what is needed in many applications, it loses information about directions, which is sometimes relevant when dealing with vector-valued functions, and especially so, matrix-valued weights are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' To extend the method to such questions, Nazarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [27] developed the so-called convex body domination, where the numerical averages featuring in sparse dom- ination are replaced by convex subsets of Rn, thus containing information about different behaviour in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Since its introduction in the context of Calderón–Zygmund operators and matrix A2 weights by [27] (see also [10] for an- other approach but based on the same key idea), convex body domination has been applied to matrix Ap-weight and two-weight bounds by Cruz-Uribe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [8], and ex- tended to commutators of Calderón–Zygmund operators by Isralowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [20, 21] and rough singular integral operators by Di Plinio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [13] and Muller and Rivera- Ríos [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In a recent breakthrough, Bownik and Cruz-Uribe [6] extended the Rubio Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 42B20, 46E40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The author is supported the Academy of Finland via the Finnish Centre of Excellence in Randomness and Structures “FiRST” (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 346314).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 1 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN de Francia algorithm, and its key application to weighted extrapolation, to matrix- valued weights, by further development of the convex body philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The aim of this paper is to further explore this technique, providing extensions, new applications and—hopefully—some additional insight into the abstract under- lying mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We begin by developing a somewhat general framework, but our claims for originality in this regard are relatively mild, as most of the ideas are at least implicit in the previous works in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A certain justification for this framework comes from the observation that it applies almost verbatim to the case of Banach space -valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' To be precise, given a Ba- nach space E, we consider functions taking values in En, and develop a version of convex body domination applicable to weighted norm inequalities involving matrix weights W : Rd → Rn×n, acting on En in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' That is, we make no attempt towards a fully operator-valued theory of weighted norm inequalities in infinite dimensions, yet the results that we obtain are still new even in this more modest generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In particular, if E is a Banach space with the UMD property, the classical Hilbert transform extends boundedly to the matrix-weighted space L2(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' En) of En-valued functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' see Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3 for the result, and Section 6 for the relevant definitions and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A key to this extension is the observa- tion that the convex bodies arising from our framework are still Rn-valued in this generality—and not, for instance, En-valued, as one might have (and this author certainly had) initially expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus the powerful Euclidean machinery, most notably the John ellipsoid theorem, is still available in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As for new applications of the theory, we build on a recent observation from Isralowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [20, 21] that convex body domination of an operator T bootstraps to a domination of its commutators [b, T ] = bT − T b with pointwise multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As we will explore in Section 7, this phenomenon is far more general, and can be used to estimate any operators of the form f �→ n � i=1 aiT (bif), where an operator T satisfying convex body domination is pre- and post-composed with pointwise multipliers ai, bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' From this general principle, we can in particular recover and sharpen a recent sufficient condition [17] for the boundedness of iterated mixed commutators [b1, [b2, T ]] in terms of joint conditions on the pair of functions (b1, b2), but also obtain new examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In contrast to the development of the abstract framework in the first part of the paper, we have not strived for the greatest generality in terms of the applications in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In many cases, it will be clear to an experienced reader that several variants and extensions could be obtained, and some of them will most likely be pursued in forthcoming works, by this author and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Besides the concrete results contained in this paper, our aim is to hint at the many rich directions for the further development of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Norms and convex bodies Let X be a real normed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We denote by ¯BX := {x ∈ X : ∥x∥X ≤ 1} SOME REMARKS ON CONVEX BODY DOMINATION 3 its closed unit ball, and by X∗ the normed dual, which is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For x∗ ∈ X∗, we define, as usual, ∥x∗∥X∗ := sup{|⟨x, x∗⟩| : x ∈ ¯BX}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As a consequence of the Hahn–Banach theorem, we have ∥x∥X = sup{|⟨x, x∗⟩| : x∗ ∈ ¯BX∗} = max{⟨x, x∗⟩ : x∗ ∈ ¯BX∗};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1) in particular, the supremum is reached as a maximum, and we have ∥x∥X = ⟨x, x∗⟩ for some x∗ ∈ ¯BX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For ⃗x = (xi)n i=1 ∈ Xn and x∗ ∈ X∗, we define the Rn-valued pairing ⟨⃗x, x∗⟩ := (⟨xi, x∗⟩)n i=1 ∈ Rn and the set-valued “norm” ⟨⟨⃗x⟩⟩X := {⟨⃗x, x∗⟩ : x∗ ∈ ¯BX∗} ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The notation is adapted from Nazarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [27], who introduced the version with X = Ł1(Q), the space L1(Q) with the normalised norm 1 |Q|∥ ∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The extension to X = Łp(Q) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', Lp(Q) with the normalised norm 1 |Q|1/p ∥ ∥p) is due to Di Plinio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Although our main applications will be concerned with spaces of functions (living on a cube Q), we find it illuminating to develop the basics of the theory on a completely abstract level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Among other things, this point of view will make it clear that there will be essentially no difference in treating a space X = Lp(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) of E-valued functions for an arbitrary Banach space E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' for ⃗f ∈ Xn, the corresponding ⟨⟨⃗f⟩⟩X will still be subsets of Rn and not, say, of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This will allow us to make effortless use of the powerful John ellipsoid theorem from Euclidean geometry, even when working with functions taking values in an infinite- dimensional Banach space!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In other applications, a choice like X = L log L(Q) might also be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For ⃗a ∈ Rn and ⃗x ∈ Xn, we define the X-valued dot product ⃗a · ⃗x := ⃗x · ⃗a := n � i=1 aixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We observe the easy identities ⃗a · ⟨⃗x, x∗⟩ = ⟨⃗a · ⃗x, x∗⟩, ∀⃗a ∈ Rn, ⃗x ∈ Xn, x∗ ∈ X∗, and spanX(⃗x) := span{xi}n i=1 = {⃗a · ⃗x : a ∈ Rn} ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For each ⃗x ∈ Xn, the set ⟨⟨⃗x⟩⟩X ⊂ Rn is convex, compact, and symmetric about the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Symmetry, convexity and boundedness are immediate from the fact that ¯BX∗ has these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For compactness in Rn, it remains to show closedness, so suppose that ⟨⃗x, x∗ k⟩ → ⃗e ∈ Rn as k → ∞, where each x∗ k ∈ ¯BX∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' we need to show that ⃗e ∈ ⟨⟨x⟩⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For each ⃗a ∈ Rn, it follows that |⃗a · ⃗e| = lim k→∞ |⃗a · ⟨⃗x, x∗ k⟩| = lim k→∞ |⟨⃗a · ⃗x, x∗ k⟩| ≤ ∥⃗a · ⃗x∥X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This in turn implies that Λ(⃗a · ⃗x) := ⃗a · ⃗e, ∀⃗a · ⃗x ∈ spanX(⃗x), 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN gives a well-defined linear functional of norm 1 on the subspace spanX(⃗x) ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By the Hahn–Banach theorem, Λ is the restriction of some x∗ ∈ ¯BX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence ⃗a · ⃗e = Λ(⃗a · ⃗x) = ⟨⃗a · ⃗x, x∗⟩ = ⃗a · ⟨⃗x, x∗⟩ ∀⃗a ∈ Rn, and thus limk→∞⟨⃗x, xk⟩ = ⃗e = ⟨⃗x, x∗⟩ ∈ ⟨⟨⃗x⟩⟩X, as we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ For A, B ⊂ Rn, we define the Minkowski dot product A · B := {⃗a ·⃗b : ⃗a ∈ A,⃗b ∈ B} ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If A, B ⊂ Rn are convex, compact and symmetric, so is A · B ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On R, such sets are precisely intervals of the form [−c, c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence we can, and sometimes will, identify A · B = [−c, c] ⊂ R with its right end-point c ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In particular, for ⃗x ∈ Xn and ⃗y ∈ Y n, we will use this identification when dealing with ⟨⟨⃗x⟩⟩X · ⟨⟨⃗y⟩⟩Y = {⟨⃗x, x∗⟩ · ⟨⃗y, y∗⟩ : x∗ ∈ ¯BX∗, y∗ ∈ ¯BY ∗} = � n � i=1 ⟨xi, x∗⟩⟨yi, y∗⟩ : x∗ ∈ ¯BX∗, y∗ ∈ ¯BY ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Bi-linear forms Let X, Y be real normed spaces, and suppose that we have a bilinear from t : X × Y → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We define its extension acting on pairs of vectors (⃗x, ⃗y) ∈ Xn × Y n as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If ⃗e ∈ Rn and x ∈ Xn, we have ⃗x · ⃗e ∈ F by our previous convention about the X-valued dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If (ei)n i=1 is a fixed orthonormal basis of Rn, we then define t(⃗f,⃗g) := n � i=1 t(⃗f · ⃗ei,⃗g · ⃗ei), For x ∈ X, y ∈ Y , and ⃗e, ⃗u ∈ Rn, it follows that t(x⃗e, y⃗u) := n � i=1 t(x⃗e · ⃗ei, y⃗u · ⃗ei) = t(x, y) n � i=1 (⃗e · ⃗ei)(⃗u · ⃗ei) = t(x, y)⃗e · ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If (⃗ui)n i=1 is another orthonormal basis, then n � i=1 t(⃗x · ⃗ui, ⃗y · ⃗ui) = n � i,j,k=1 t(⃗x · ⃗ej, ⃗y · ⃗ek)(⃗ej · ⃗ui)(⃗ek · ⃗ui) = n � j,k=1 t(⃗x · ⃗ej, ⃗y · ⃗ek)(⃗ej · ⃗ek) = n � j=1 t(⃗x · ⃗ej, ⃗y · ⃗ej) =: t(⃗x, ⃗y), so the definition of t(⃗f,⃗g) is independent of the chosen orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If A ∈ Rn×n is a linear transformation of Rn, acting in a natural way on F n, then t(A⃗x, ⃗y) = n � i=1 t(A⃗f · ⃗ei,⃗g · ⃗ei) = n � i=1 t(⃗f · At⃗ei,⃗g · ⃗ei) = n � i,j=1 t(⃗f · ⃗ej,⃗g · ⃗ei)(⃗ej · At⃗ei) = n � i,j=1 t(⃗f · ⃗ej,⃗g · ⃗ei)(A⃗ej · ⃗ei) = n � j=1 t(⃗f · ⃗ej,⃗g · A⃗ej) = t(⃗f, At⃗g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' From norm bounds to convex body bounds The idea of the following lemma lies behind many of the existing convex body domination results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' To isolate the key point, we state it here in an operator-free version, involving functions and therir norms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let X, Y be normed spaces, and ⃗f ∈ Xn,⃗g ∈ Y n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let EK be the John ellipsoid of K := ⟨⟨⃗f⟩⟩X such that EK ⊂ K ⊂ √nEK, and suppose that EK is non-degenerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', of full dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let RK be a linear transformation such that RKEK = ¯BRn, the closed unit ball of Rn, and let (⃗ei)n i=1 be an orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If fi := RK ⃗f · ⃗ei, gi := R−t K ⃗g · ei, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , n, then n � i=1 ∥fi∥X∥gi∥Y ≤ n3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If φ ∈ ¯BX∗, then ⟨φ, RK ⃗f · ⃗ei⟩ = RK⟨φ, ⃗f⟩ · ⃗ei ∈ RK⟨⟨⃗f⟩⟩X · ⃗ei ⊂ RK √nEK · ⃗ei = √n ¯BRn · ⃗ei = √n[−1, 1], and hence ∥fi∥X = ∥RK ⃗f · ⃗ei∥X ≤ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If ψ ∈ ¯BY ∗, then ⟨ψ, R−t K ⃗g · ⃗ei⟩ = R−t K ⟨ψ,⃗g⟩ · ⃗ei ∈ R−t K ⟨⟨⃗g⟩⟩Y · ⃗ei ⊂ [−M, M], M := max{|⃗y| : ⃗y ∈ R−t K ⟨⟨⃗g⟩⟩Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It follows that |⟨ψ, R−t K ⃗g · ⃗ei⟩| ≤ M, and hence ∥gi∥Y = ∥R−t K ⃗g · ⃗ei∥Y ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Combining the estimates, we have n � i=1 ∥fi∥X∥gi∥Y ≤ n � i=1 √nM = n3/2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the other hand, ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y ⊃ EK · ⟨⟨⃗g⟩⟩Y = RKEK · R−t K ⟨⟨⃗g⟩⟩Y = ¯BRn · R−t K ⟨⟨⃗g⟩⟩Y = [−M, M], and hence M ≤ ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y , using the identification of the symmetric interval ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y with its right end- point in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Substituting back, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ The following proposition contains the basic idea of bootstrapping norm bounds to convex body bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Unfortunately, it is a bit too simple for most actual applications, but we include it as an illustrative toy model for the more serious result to be presented after it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let X, Y be normed spaces with subspaces F ⊂ X and G ⊂ Y , and let t : F × G → R be a bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Consider the following conditions: 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN (1) For all (f, g) ∈ F × G, we have |t(f, g)| ≤ C∥f∥X∥g∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2) For all (⃗f,⃗g) ∈ F n × Gn, we have |t(⃗f,⃗g)| ≤ Cn⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For each n ∈ Z+, condition (1) implies condition (2) with Cn = Cn3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Given ⃗f, consider the compact, convex, symmetric set K := ⟨⟨⃗f⟩⟩X, and denote by EK its John ellipsoid such that EK ⊂ K ⊂ √nEK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Case: EK is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let RK be a linear transformation such that RKEK = ¯BRn, the closed unit ball of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let (⃗ei)n i=1 be some orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We then write t(⃗f,⃗g) = t(R−1 K RK ⃗f,⃗g) = t(RK ⃗f, R−t K ⃗g) = n � i=1 t(RK ⃗f · ⃗ei, R−t K ⃗g · ⃗ei) =: n � i=1 t(fi, gi), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3) where fi and gi are as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By assumption (1) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1, it follows that |t(⃗f,⃗g)| ≤ n � i=1 |t(fi, gi)| ≤ n � i=1 C∥fi∥X∥gi∥Y ≤ Cn3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y, and this completes the proof in the case that EK is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Case: EK is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose then that EK is degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' hence H := span K is a strict subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let P denote the orthogonal projection of Rn onto H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For each x∗ ∈ ¯BX∗, we have ⟨⃗f, x∗⟩ ∈ K ⊂ H, hence ⟨⃗f, x∗⟩ = P⟨⃗f, x∗⟩ = ⟨P ⃗f, x∗⟩, and thus ⃗f = P ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It follows that t(⃗f,⃗g) = t(P ⃗f,⃗g) = t(⃗f, P t⃗g) = t(⃗f, P⃗g), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4) and similarly ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y = P⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y = ⟨⟨⃗f⟩⟩X · P t⟨⟨⃗g⟩⟩Y = ⟨⟨⃗f⟩⟩X · ⟨⟨P⃗g⟩⟩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='5) So it is enough to prove the claim with P⃗g in place of ⃗g, and hence we may assume without loss of generality that also ⃗g = P⃗g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' But then we can repeat the argument in the non-degenerate case, but with Rn replaced by its subspace H throughout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' within this subspace, EK ⊂ H is non-degenerate, and the previous case applies to give the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ In the following proposition, condition (1) is a typical intermediate step that is established in the course of proving a sparse domination result for an operator, while condition (2) is its convex body analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The proposition says that (1) in fact implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It is essentially an abstraction (from L1 averages to general dominating norms) of an idea already present in [27, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let X, Y be normed spaces with subspaces F ⊂ X and G ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let Q0 ∈ D, and suppose that there are bilinear forms tQ : F × G → R indexed by all Q ∈ D(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Consider the following conditions: (1) For all (f, g) ∈ F × G, there exist disjoint ˆQk ⊂ Q0 with � k | ˆQk| ≤ ε|Q0| and such that: whenever Qj ⊂ Q0 are disjoint, not strictly contained in any ˆQk, and cover all ˆQk, then |tQ0(f, g) − � j tQj(f, g)| ≤ C∥f∥X∥g∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2) For all (⃗f,⃗g) ∈ F n × Gn, there exist disjoint Qk ⊂ Q0 with � k |Qk| ≤ εn|Q0| and such that |tQ0(⃗f,⃗g) − � k tQk(⃗f,⃗g)| ≤ Cn⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For each n ∈ Z+, condition (1) implies condition (2) with εn = nε and Cn = Cn3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Of course, the condition � k |Qk| ≤ εn|Q0| is only useful for εn < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For a fixed ε and εn = nε, this would only allow us to conclude (2) for boundedly many values of n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' so in order to obtain (2) for all n ∈ N, we need (1) for arbitrarily small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This is seldom a problem in concrete situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2, given ⃗f, we consider the compact, convex, symmetric set K := ⟨⟨⃗f⟩⟩X, and denote by EK its John ellipsoid such that EK ⊂ K ⊂ √nEK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Case: EK is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let RK be a linear transformation such that RKEK = ¯BRn, the closed unit ball of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let (⃗ei)n i=1 be some orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3), we then write tQ0(⃗f,⃗g) = tQ0(R−1 K RK ⃗f,⃗g) = tQ0(RK ⃗f, R−t K ⃗g) = n � i=1 tQ0(RK ⃗f · ⃗ei, R−t K ⃗g · ⃗ei) =: n � i=1 tQ0(fi, gi), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7) where fi and gi are as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It is from this point on that the present proof requires some elaboration compared to the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' According to assumption (1), for each of the pairs of functions fi := RK ⃗f · ⃗ei and gi := R−t K ⃗g · ⃗ei, we can find disjoint ˆQi,k ⊂ Q0 with � k | ˆQi,k| ≤ ε|Q0| and such that: whenever Qj ⊂ Q0 are disjoint, not strictly contained in any ˆQi,k, and cover all ˆQi,k, then |tQ0(fi, gi) − � j tQj(fi, gi)| ≤ C∥fi∥X∥gi∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8) We make the following specific choice of the cubes Qj: Let {Qj}∞ j=1 be the maximal cubes among { ˆQi,k}1≤k<∞ 1≤i≤n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then � j |Qj| ≤ n � i=1 ∞ � k=1 | ˆQi,k| ≤ n � k=1 ε|Q0| = nε|Q0|, 8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8) holds with these Qj for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7), and observing that it also holds with Q0 replaced by Qj, it follows that |tQ0(⃗f,⃗g) − � j tQj(⃗f,⃗g)| ≤ n � i=1 |tQ0(fi, gi) − � j tQj(fi, gi)| ≤ C n � i=1 ∥fi∥X∥gi∥Y ≤ Cn3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y , using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This completes the proof under the assumption that EK is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Case: EK is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This follows the corresponding case in the proof of Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2 almost verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Like there, let H := span K, and let P denote the orthogonal projection of Rn onto H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We then have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4) for each t = tQ, as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' So it is again enough to prove the claim with P⃗g in place of ⃗g, and hence we may assume without loss of generality that also ⃗g = P⃗g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' But then we can repeat the argument in the non-degenerate case, but with Rn replaced by its subspace H throughout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' within this subspace, EK ⊂ H is non-degenerate, and the previous case applies to give the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' From single-scale bounds to global bounds This passage is by now a relatively routine part of the theory, but we include some details for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The following lemma is again stated in an operator-free, and even function-free way, simply as a criterion for dominating a real number by sum over a sparse collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A more concrete situation for applying this criterion is presented afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Consider numbers a ∈ R and aQ, cQ ∈ R indexed by dyadic cubes Q ∈ D, with the following properties: (1) There is a family Q of disjoint dyadic cubes such that a = � Q∈Q aQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2) For some δ ∈ (0, 1) and each Q ∈ D that is contained in some P ∈ Q, there is a family of disjoint Qk ∈ D(Q) such that � k |Qk| ≤ δ|Q|, ���aQ − � k aQk ��� ≤ cQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (3) For some α, C ∈ [1, ∞) and each Q ∈ D that is contained in some P ∈ Q, we have |aQ| ≤ C|Q|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then there is a (1 − δ)-sparse family of dyadic cubes S such that S ⊂ � Q∈Q D(Q), |a| ≤ � S∈S cS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If Q = {Q0} consists of a single cube only, then condition (1) is automatic with a = aQ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let Q ⊂ D be a disjoint collection provided by assumption (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For each P ∈ Q, denote S0(P) := {P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Assuming that a disjoint Sj(P) ⊂ D(P) has already been constructed, for each Q ∈ Sj(P), let S ′(Q) := {Qk}∞ k=1 be the collection provided by assumption (2), and let Sj+1(P) := � Q∈Sj(P ) S ′(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let also S (P) := �∞ j=0 Sj(P), and S := � P ∈Q S (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For Q ∈ S , let E(Q) := Q \\ � R∈S ′(Q) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' From the construction it is clear that these sets E(Q) are pairwise disjoint, and by assumption (2) we have |E(Q)| ≥ (1 − δ)|Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By telescoping, for each P ∈ Q, we have aP = k−1 � j=0 � Q∈Sj(P ) � aQ − � R∈S ′(Q) aR � + � S∈Sk(P ) aS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' and hence, using assumptions (2) and (3), |aP | ≤ k−1 � j=1 � Q∈Sj(P ) cQ + � S∈Sk(P ) C|S|α By an elementary inequality and induction, we have � S∈Sk(P ) |S|α ≤ � � S∈Sk(P ) |S| �α ≤ (δk|P|)α, and hence |aP | ≤ lim k→∞ k−1 � j=1 � Q∈Sj(P ) cQ = � Q∈S (P ) cQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Substituting this into assumption (1), we obtain the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose that t is a bilinear form on L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) × L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H), and moreover bounded with respect to the norm of Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) × Lq(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H) for some exponents with 1/p+1/q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For (⃗f,⃗g) ∈ L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)n ×L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H)n, the numbers a = t(⃗f,⃗g), aQ = t(13Q ⃗f, 1Q⃗g) satisfy assumptions (1) and (3) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1, provided that D is a dyadic system without quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Since D is without quadrants, each Q ∈ D is contained in some (large enough) R ∈ D that contains supp ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus the collection Q of maximal cubes that do not contain supp ⃗f form a cover of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By maximality, it follows that supp ⃗f ⊂ 3Q, and hence ⃗f = 13Q ⃗f for every Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the other hand, any Q with ℓ(Q) < diam(supp ⃗f) cannot contain supp ⃗f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' hence any Q with ℓ(Q) < 1 2 diam(supp ⃗f) cannot be among the maximal cubes Q, and thus every Q ∈ Q will have to satisfy ℓ(Q) ≥ 1 2 diam ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Since ⃗g ∈ L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' F)n, there are only finitely many Q ∈ Q with 1Q⃗g ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence, without any issues of convergence, we can write t(⃗f,⃗g) = t � ⃗f, � Q∈Q 1Q⃗g � = � Q∈Q t(⃗f, 1Q⃗g) = � Q∈Q t(13Q ⃗f, 1Q⃗g), which is condition (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If n = 1, the assumed boundedness directly implies that |t(13Qf, 1Qg)| ≤ C∥13Qf∥Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E)∥1Qg∥Lq(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='F ) ≤ C3d/p∥f∥∞∥g∥∞|Q|1/p+1/q, 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN where α := 1/p + 1/q ≥ 1, as required for condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In general, if (⃗ei)n i=1 is an orthonormal basis of Rn and ⃗f = �n i=1 fi⃗ei and similarly for ⃗g, we have |t(13Q ⃗f, 1Q⃗g)| ≤ n � i=1 |t(13Qfi, 1Qgi)| ≤ Cn3d/p∥⃗f∥∞∥⃗g∥∞|Q|1/p+1/q, using the previous bound in each component and trivial bounds like ∥fi∥∞ ≤ ∥⃗f∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ We are finally ready to state a semi-generic convex body domination principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Condition (1) below is a typical intermediate estimate in a number of sparse dom- ination proofs for different operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The conclusion is that it is already good enough to conclude convex body domination as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let E and H be Banach spaces, and suppose that t is a bilinear form defined on F × G := L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) × L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H) and bounded with respect to the norm of Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) × Lq(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H) for some exponents with 1/p + 1/q ≥ 1, and suppose that (1) for all (f, g) ∈ F × G and all Q ∈ D, there are disjoint ˆQk ⊂ Q with � k | ˆQk| ≤ ε|Q| and such that: whenever Qj ⊂ Q are disjoint, not strictly contained in any ˆQk, and cover all ˆQk, then |t(13Qf, 1Qg) − � j t(13Qjf, 1Qjg)| ≤ c∥f∥X(Q)∥g∥Y (Q)|Q| (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='5) for some norms ∥ ∥X(Q) on L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) and ∥ ∥Y (Q) on L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then for all (⃗f,⃗g) ∈ F n × Gn, there is a (1 − εn)-sparse collection S ⊂ D such that |t(⃗f,⃗g)| ≤ cn � S∈S ⟨⟨⃗f⟩⟩X(S) · ⟨⟨⃗g⟩⟩Y (S)|S|, where εn = nε and cn = cn3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let us begin by considering a fixed cube Q = Q0 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We observe that assumption (1) of the present corollary coincides with condition (1) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6 with tQ(f, g) := t(13Qf, 1Qg), C = c|Q|, X = X(Q), Y = Y (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence the said proposition, applied to each fixed Q = Q0 ∈ D at a time, implies: (2) For all (⃗f,⃗g) ∈ L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)n ×L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H)n and all Q ∈ D, there are disjoint Qk ⊂ Q with � k |Qk| ≤ εn|Q| and such that |t(13Q ⃗f, 1Q⃗g) − � j t(13Qj ⃗f, 1Qj⃗g)| ≤ cn⟨⟨⃗f⟩⟩X(Q) · ⟨⟨g⟩⟩Y (Q)|Q|, where εn = nε and cn = cn3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let us then consider a fixed pair (⃗f,⃗g) ∈ L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)n×L∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' H)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We observe that condition (2) above coincides with condition (2) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1 with the choices aQ = t(13Q ⃗f, 1Q⃗g), cQ = cn⟨⟨⃗f⟩⟩X(Q) · ⟨⟨g⟩⟩Y (Q)|Q|, δ = εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the other hand, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3 shows that these same aQ, together with a := t(⃗f,⃗g), also satisfy conditions (1) and (3) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus, all assumptions, and hence the conclusions, of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1 are valid for the said quantities, and these SOME REMARKS ON CONVEX BODY DOMINATION 11 conclusions agree with the claims of the result that we are proving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The proof is thus complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ To facilitate the discussion of consequences of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4, we give 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose that a pair of normed spaces (X(Q), Y (Q)) is associated to every dyadic cube Q ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We say that a bilinear form t : F ×G → R satisfies the (X(Q), Y (Q)) convex body domination of order n ∈ N if F ⊆ X(Q) and G ⊆ Y (Q) for every Q ∈ D, and if for every (f, g) ∈ F n×Gn, there exists a δn-sparse collection S ⊂ D such that |t(⃗f,⃗g)| ≤ Cn � Q∈S |Q|⟨⟨⃗f⟩⟩X(Q) · ⟨⟨⃗g⟩⟩Y (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We say that t : F × G → R satisfies the (X(Q), Y (Q)) convex body domination if it satisfies this for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We say that an operator T : F → G∗ satisfies these properties if its associated bilinear form t(f, g) := ⟨T f, g⟩ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let us now consider some examples: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Example (Calderón–Zygmund operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let T be a Dini–Calderón–Zygmund operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', T is L2(Rd) bounded and has the representation T f(x) = ˆ Rd K(x, y)f(y) dy, x /∈ supp f, where |K(x, y)| ≤ c|x − y|−d and, for |x − x′| ≤ 1 2|x − y|, |K(x, y) − K(x′, y)| + |K(y, x) − K(y, x′)| ≤ ω �|x − x′| |x − y| � 1 |x − y|d , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8) where ω : [0, 1 2] → [0, ∞) is increasing, subadditive, and satisfies the Dini condition ˆ 1/2 0 ω(t) dt t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then (1) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 holds for t(f, g) = ⟨T f, g⟩ and E = H = R and X(Q) = Ł1(3Q), Y (Q) = Ł1(Q), even in a stronger form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Namely, on the left oif (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='5), we have ��� � T (13Qf), 1Qg � − � j ⟨T (13Qjf), 1Qjg⟩ ��� ≤ ���1QT (13Qf) − � j 1QjT (13Qjf) ��� L∞(Q)∥g∥L1(Q), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='9) and even the L∞ norm here is dominated by ∥f∥Ł1(3Q), as essentially shown in [24, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (Strictly speaking, [24, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4)] is formally slightly weaker, but a straightfor- ward modification of the argument gives the desired version, as observed in [27, Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=') Thus Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 says that a Dini–Calderón–Zygmund operator satisfies (Ł1(3Q), Ł1(Q)) convex body domination, but this was of course already known from [27] by essentially the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Example (Banach space -valued Calderón–Zygmund operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let T be as in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7 but now acting on the Bochner space L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) of Banach space E valued functions, and with an operator-valued kernel K(x, y) ∈ L (E) satisfying the same estimates as above but for the operator norm in place of the absolute value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', ∥K(x, y)∥L (E) ≤ c|x − y|−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It is in general a difficult problem to check the 12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)-boundedness of such an operator, but we now take this as an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For g ∈ L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E∗), we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='9) with L∞(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) and L1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E∗) in place of L∞(Q) and L1(Q), and the same proof of [24, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4)] (with same modifications pointed out in [27, Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4]) shows that the L∞(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) norm is dominated by ∥f∥Ł1(3Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus we find that (1) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 also holds with X(Q) = Ł1(3Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E) and Y (Q) = Ł1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The resulting sparse domination (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', case n = 1 of the conclusion of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4) was known before, first in [15] for a slightly smaller class of kernels, and since [22, discussion on page 193] in the present generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' However, the convex body domination in this Banach space -valued setting is completely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Example (Operators with grand maximal function control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let 1 ≤ q ≤ r and s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose that T is a linear operator T : L∞ c (Rd) → L1 loc(Rd), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='12) that T has weak type (q, q), and that the bi-sublinear maximal operator MT (f, g)(x) := sup Q∋x Q |T (1(3Q)cf)| · |g| maps boundedly MT : Lr ×Ls → Lν,∞, where 1/ν = 1/r+1/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then condition (1) of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 holds for t(f, g) = ⟨T f, g⟩ and E = H = R and X(Q) = Łr(3Q), Y (Q) = Łs(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This result is essentially contained in the proof of [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1], where it appears as an intermediate step towards the sparse domination (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', case n = 1 of the conclusion of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4) for such operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The extension to convex body domination was recently achieved in [26], so Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 only reproduces a known result here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A key example of concrete operators satisfying these assumptions consists of rough homogeneous singular integrals T f(x) = ˆ Rd Ω(y) |y|d f(x − y) dy, where Ω(y) = Ω(y/|y|) is a bounded function with vanishing average over the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10, the abstract result above, involving a priori bounds of T and MT , extends straightforwardly to the Banach space -valued setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' however, verifying these bounds for concrete operators such as the rough homogeneous sin- gular integrals may present a problem in this generality, since the scalar-valued versions depend on deep results of Seeger [30], which so far lack a Banach space valued extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Matrix-weighted inequalities for Banach space -valued operators A matrix weight is a locally integrable function W : Rd → Rn×n that is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' positive definite -valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The space Lp(W) consists of all measurable ⃗f : Rd → Rn such that W 1/p ⃗f ∈ Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rn), and ∥⃗f∥Lp(W) := ∥W 1/p ⃗f∥Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For a Banach space E, we extend this definition in a natural way: The space Lp(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' En) consists of all measurable ⃗f : Rd → En such that W 1/p ⃗f ∈ Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' En), and ∥⃗f∥Lp(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En) := ∥W 1/p ⃗f∥Lp(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Here, at each x ∈ Rd, we define (W 1/p ⃗f)(x) ∈ En as the vector with components (W 1/p ⃗f)i(x) := �n j=1(W 1/p(x))ijfj(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', the matrix multiplication on Rn is extended to En in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 13 We now concentrate on p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For two matrix weights W, V : Rd → Rn×n, we define [W, V ]A2 := sup Q |⟨W⟩1/2 Q ⟨V ⟩1/2 Q |2, [W]A2 := [W, W −1]A2, where we denote the operator norm in Rn×n ≃ L (Rn) simply by | |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We denote by A2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rn) the class of matrix weights W : Rd → Rn×n for which [W]A2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We also define [W]A∞ := sup ⃗e∈Rn[x �→ ⃗e · W(x)⃗e]A∞, where on the right we have A∞ “norms” of some scalar weights, defined as usual by [w]A∞ := sup Q 1 w(Q) ˆ Q M(1Qw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' According to [27, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4], we have [W]A∞ ≤ 4[W]A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1) As a consequence of the Banach space -valued convex body domination from Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10, we obtain: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let E be a Banach space, and T ∈ L (L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)) be a Dini– Calderón–Zygmund operator with L (E)-valued kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For any W ∈ A2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rn), the operator T extends boundedly to L2(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' En) and satisfies ∥T ∥L(L2(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En)) ≤ cn,T ([W]A2[W]A∞[W −1]A∞)1/2 ≤ cn,T [W]3/2 A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Note that Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2 applies to a general Banach space E, but contains the (difficult) a priori boundedness hypothesis that T ∈ L (L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Concrete ex- amples are available in the class of UMD spaces, treated in detail in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let E be a UMD space, and T ∈ L (L2(Rd)) be a scalar-valued Calderón–Zygmund operator with a Hölder-type modulus of continuity ω(t) = ctδ, δ ∈ (0, 1] in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For any W ∈ A2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rn), the operator T extends boundedly to L2(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' En) and satisfies ∥T ∥L(L2(W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En)) ≤ cn,E,T ([W]A2[W]A∞[W −1]A∞)1/2 ≤ cn,E,T [W]3/2 A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In particular, this estimate holds when T is the classical Hilbert transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We reduce Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3 to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2 with the help of the T (1) theorem of David and Journé [12], and its extension to UMD spaces by Figiel [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By the (easy half of) the David–Journé theorem, the assumptions on T imply that that T satisfies the so-called weak boundedness property as well as T (1), T ∗(1) ∈ BMO(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then, by Figiel’s theorem, an operator satisfying these conditions and the Calderón– Zygmund kernel assumptions extends boundedly to L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E), for any UMD space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus T satisfies the assumptions, and hence the conclusions, of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ These results, even just for the Hilbert transform, and even in their qualitative form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', just concluding the boundedness of T , without specifying any concrete bound for the norm), are completely new in the combined setting of matrix weights and Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For E = R and the Hilbert transform T , the qualitative form of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3 is due to Treil and Volberg [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The quantitative form for E = R was obtained by Nazarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [27], and this is the best that is known at the time of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For scalar-weights, the power 3/2 can be replaced by 1 [16], and the 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN product of [W]A∞ and [W −1]A∞ by their sum [19], but extending these to the general matrix case consists of the outstanding open “matrix A2 conjecture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Turning to the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2, we begin with: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Remark (Without loss of generality, we assume that E is reflexive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Since Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2 is about the bounded extension of an operator, it suffices to prove an a priori estimate on a dense subspace of functions ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In particular, we can assume that each component fi takes its values in a finite-dimensional subspace of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Since any finite-dimensional space is reflexive, we make the standing assumption, without loss of generality, that E is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (Note that this is automatic in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3 in any case, since UMD spaces are reflexive [18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=') Under this assumption, we have L1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E)∗ = L∞(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E∗) (see [18, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='21]), which is convenient in view of calculations involving the convex bodies ⟨⟨ ⟩⟩Ł1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' |Q|⟨⟨W 1/2 ⃗f⟩⟩Ł1(3Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E) · ⟨⟨V 1/2⃗g⟩⟩Ł1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E∗) ≤ ˆ � 1Q(x) 3Q |V 1/2(x)W 1/2(y)|∥⃗f(y)∥En dy � ∥⃗g(x)∥ ⃗E∗n dx Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Under the standing assumption from Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4, we evaluate consider a generic element of the convex body on the left with φ ∈ ¯BL∞(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E∗) and ψ ∈ ¯BL∞(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E): |Q| ��� 3Q W 1/2(y)⟨⃗f(y), φ(y)⟩ dy · Q V 1/2(x)⟨⃗g(x), ψ(x)⟩ dx ��� = |Q| ��� Q 3Q V 1/2(x)W 1/2(y)⟨⃗f(y), φ(y)⟩ · ⟨⃗g(x), ψ(x)⟩ dy dx ��� ≤ ˆ Q 3Q |V 1/2(x)W 1/2(y)|∥⃗f(y)∥En∥⃗g(x)∥ ⃗E∗n dy dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ Summing over a sparse collection, we obtain � Q∈S |Q|⟨⟨W 1/2 ⃗f⟩⟩Ł1(3Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E) · ⟨⟨V 1/2⃗g⟩⟩Ł1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E∗) ≤ ˆ � � Q∈S 1Q(x) 3Q |V 1/2(x)W 1/2(y)|∥⃗f(y)∥En dy � ∥⃗g(x)∥ ⃗E∗n dx =: ˆ ˜L(∥⃗f∥En)(x)∥⃗g(x)∥ ⃗E∗n dx, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6) where ˜L, here acting on the scalar-valued function y �→ ∥⃗f(y)∥En, is an operator denoted by the same symbol in [27, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By [27, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6], we have ∥˜L∥L (L2) ≤ C([W, V ]A2[W]A∞[V ]A∞)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7) By duality and standard changes of variables, which present no essential differ- ence in the Banach space -valued setting, an estimate of the form ∥T ⃗f∥L2(V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En) ≤ N∥⃗f∥L2(V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En) is equivalent to ⟨T (W 1/2 ⃗f), V 1/2⃗g⟩ ≤ N∥⃗f∥L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='En)∥⃗g∥L2(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='E∗n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8) SOME REMARKS ON CONVEX BODY DOMINATION 15 If T is an in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2, it satisfies the (Ł1(3Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E), Ł1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' E∗)) convex body dom- ination by Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10, which means that the left-hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8) is dominated by the left-hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6), and hence, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7), we have N ≤ cn,T ([W, V ]A2[W]A∞[V ]A∞)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This is the desired bound, and concludes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Convex domination and generalised commutators For an operator T and two vector functions ⃗a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , an) and ⃗b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , bn), let us consider the operator ⃗a · T⃗b : f �→ ⃗a · T (⃗bf) = n � i=1 aiT (bif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We are mainly interested in the boundedness on Lp(Rd), or a weighted Lp(w), or between two such spaces, and the case when T is a singular integral operator bounded on the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' However, we do not require that ai, bi ∈ L∞(Rd), and hence the pointwise multipliers f �→ bif and g �→ aig, and the compositions f �→ aiT (bif), may be unbounded operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Nevertheless, their sum ⃗a · T⃗b may still be bounded, thanks to cancellation between different terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A case that has been much studied in the literature consists of ⃗b = (1, b) and ⃗a = (b, −1), in which case ⃗a · T (⃗bf) = bT f − T (bf) = [b, T ]f is the commutator of b and T , whose Lp(Rd)-boundedness is characterised by b ∈ BMO(Rd), the space of functions of bounded mean oscillation, which is strictly larger than L∞(Rd), and contains in particular functions like b(x) = log |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By dualising with a function g, and denoting by t(f, g) = ⟨T f, g⟩ the bilinear form of T , we arrive at ⟨⃗a · T (⃗bf), g⟩ = n � i=1 ⟨T (bif), aig⟩ = t(⃗bf,⃗ag), where the action of the bilinear form is extended to vector-valued functions as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' To be precise, if t in defined on F × G, we should now require that f ∈ F⃗b := {f ∈ F : bif ∈ F for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , n}, and g ∈ G⃗a, defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If F ⊇ L∞ c (Rd), then clearly F⃗b contains in particular all f ∈ L∞ c (Rd) with supp f ⊆ EN := {|⃗b| ≤ N} for any N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' finite- valued bi, the union � N∈N EN covers Rd up to a null set, it is immediate that F⃗b is dense in any Lp(w) with finite p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose that T satisfies the (X(Q), Y (Q)) convex body domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then for all relevant functions, we have |⟨⃗a · T (⃗bf), g⟩| ≤ C � Q∈S |Q|⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This is immediate by applying definition to ⃗f = ⃗bf and ⃗g = ⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ We take a closer look at the case when X(Q) = Y (Q) = Ł1(γQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 16 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For all s, t ∈ (1, ∞) and all functions in the relevant spaces, we have ⟨⟨⃗bf⟩⟩Ł1(Q) · ⟨⟨⃗ag⟩⟩Ł1(Q) ≤ ∥(x, y) �→ ⃗a(x) ·⃗b(y)∥Ł(s,t) min (Q×Q)∥f∥Łt′ (Q)∥g∥Łs′(Q), where ∥F∥Ł(s,t) min (Q×Q) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � ffl Q � ffl Q |F(x, y)|s dx �t/s dy �1/t , if s ≤ t, � ffl Q � ffl Q |F(x, y)|t dy �s/t dx �1/s , if t ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The generic element of ⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q) has the following form, where φ, ψ ∈ ¯BL∞(Q): Q ⃗b(y)f(y)φ(y) dy · Q ⃗a(x)g(x)ψ(x) dx = Q Q (⃗a(x) ·⃗b(y))f(y)g(x)φ(y)ψ(x) dx dy, and hence ⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q) ≤ Q Q |⃗a(x) ·⃗b(y)||f(y)||g(x)| dx dy ≤ ∥(x, y) �→ a(x) · b(y)∥Z∥(x, y) �→ f(y)g(x)∥Z∗, for either choice of (Z, Z∗) ∈ {(Łs x(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Łt y(Q)), Łs′ x (Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Łt′ y (Q))), (Łt y(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Łs x(Q)), Łt′ y (Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Łs′ x (Q)))}, by Hölder’s inequality for mixed-norm Lp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By Fubini’s theorem, we have ∥(x, y) �→ f(x)g(y)∥Z∗ = ∥f∥Łt′ (Q)∥g∥Łs′(Q) in either case, and hence, taking the minimum over the two choices of Z, we arrive at the factor min Z ∥(x, y) �→ b(x) · a(y)∥Z = ∥(x, y) �→ ⃗b(x) · ⃗a(y)∥Ł(s,t) min (Q×Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let T be an operator that satisfies the (Ł1(γQ), Ł1(γQ)) convex body domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let ⃗a,⃗b ∈ L1 loc(Rd)n be functions such that As,t := sup Q ∥(x, y) �→ ⃗a(x) ·⃗b(y)∥Ł(s,t) min (Q×Q) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then ⃗a·T⃗b extends to a bounded operator on Lp(Rd) for all p ∈ (t′, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In particular, if As := As,s < ∞ for some s ∈ (2, ∞), then ⃗a · T⃗b extends boundedly to L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Combining Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3, we find that |⟨⃗aT (⃗bf), g⟩| ≤ C � Q∈S |Q|⟨⟨⃗bf⟩⟩Ł1(γQ) · ⟨⟨⃗ag⟩⟩Ł1(γQ) ≤ C � Q∈S |Q|∥(x, y) �→ a(x) · b(y)∥Ł(s,t) min (Q×Q)∥f∥Łt′ (Q)∥g∥Łs′(Q) ≤ C � Q∈S |E(Q)| δ As,t inf Q Mt′f inf Q Ms′g ≤ CAs,t δ � Q∈S ˆ E(Q) Mt′fMs′g ≤ CAs,t δ ˆ Rd Mt′fMs′g ≤ CAs,t δ ∥Mt′f∥Lp(Rd)∥Ms′g∥Lp′(Rd), where ∥Mt′f∥Lp(Rd) ≲t,p ∥f∥Lp(Rd), ∥Ms′g∥Lp′(Rd) ≲s,p ∥g∥Lp′(Rd) for p > t′ and p′ > s′, where the latter is equivalent to p < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ Let us consider some examples: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Example (Classical commutators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As we already observed, ⃗a = (b, −1) and ⃗b = (1, b) gives rise to the usual commutator [b, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In this case ⃗a(x) ·⃗b(y) = b(x) − b(y) and each As,t is equivalent to ∥b∥BMO(Rd) by elementary considerations and the John–Nirenberg inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 reproduces the well-known suffi- cient condition for the boundedness of commutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Example (Iterated commutators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' More generally, choosing ⃗a,⃗b so that ⃗a(x) ·⃗b(y) = (b(x) − b(y))k = k � i=0 �k i � b(x)k−i(−b(y))i, thus e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' ai(x) = �k i � b(x)k−i and bi(y) = (−b(y))i, we reproduce the kth order commutator ⃗a · T⃗b = Tk,b := [b, Tk−1,b], T0,b := T, and As,t is equivalent to ∥b∥k BMO(Rd) by the John–Nirenberg inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Example (Iterated commutators with different multipliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let us then choose ⃗a,⃗b so that ⃗a(x) ·⃗b(y) = (b1(x) − b1(y))(b2(x) − b2(y));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' without specifying the precise choice of ai(x) and bi(y), it is evident that such a choice can be easily written down, if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (We deliberately use superscript indices for bi above, since these not be the same as the components bi of ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=') This reproduces the second order iterated commutator with two different functions, ⃗a · T⃗b = [b1, [b2, T ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It is well-known and classical that bi ∈ BMO(Rd) for both i = 1, 2 is sufficient for the L2(Rd) boundedness of [b1, [b2, T ]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' however, as recently observed in [17], much 18 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN weaker sufficient conditions can be given for the pair (b1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Namely, in [17, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1)], it shown that the pair of conditions Ss := sup Q � Q |b1(x) − ⟨b1⟩Q|s dx �1/s� Q |b2(y) − ⟨b2⟩Q|s dy �1/s < ∞, Ts := sup Q � Q |b1(x) − ⟨b1⟩Q|s|b2(x) − ⟨b2⟩Q|s dx �1/s < ∞, is sufficient for the L2(Rd) boundedness of [b1, [b2, T ]] for s > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the other hand, by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4, another sufficient condition for the same conclusion is As < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let us compare the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Adding and subtracting terms and multiplying out, we find that (b1(x) − b1(y))(b2(x) − b2(y)) = [(b1(x) − ⟨b1⟩Q) − (b1(y) − ⟨b1⟩Q)][(b2(x) − ⟨b2⟩Q) − (b2(y) − ⟨b2⟩Q)] = (b1(x) − ⟨b1⟩Q)(b2(x) − ⟨b2⟩Q) + (b1(y) − ⟨b1⟩Q)(b2(y) − ⟨b2⟩Q) − (b1(x) − ⟨b1⟩Q)(b2(y) − ⟨b2⟩Q) − (b1(y) − ⟨b1⟩Q)(b2(x) − ⟨b2⟩Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Taking Łs(Q × Q) and then supremum over Q on both sides, we deduce that As ≤ 2(Ts + Ss), so that the new criterion provided by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 is at least as sharp as that of [17, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1)], and it seems less obvious to make any estimate in the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Perhaps more importantly, the new condition As < ∞ arises more “naturally” as an instance of a general principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (Let us note that there is a more general criterion [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10], where the Łs norms in Ss and Tt are replaced by more general Orlicz norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the other hand, it is apparent that similar generalisations could be achieved in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4: what we used was the boundedness of the rescaled maximal operators Mt′ on Lp(Rd) for p > t′, and this could be replaced having an Orlicz maximal operator MA with the same mapping property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A characterisation of this property in terms of the so-called Bp condition on the Orlicz function A is a classical result of Pérez [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' this very result is used in [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' see [17, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=') Let us finally consider an “exotic” example with no obvious predecessor in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We begin with a lemma: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose that 0 ≤ b ∈ BMO(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' If 0 ≤ α, β and α + β ≤ 1, then B(x, y) := b(x)αb(y)β − b(x)βb(y)α satisfies � Q Q |B(x, y)|p dx dy �1/p ≤ (2∥b∥BMOp(Rd))α+β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let γ := min(α, β) ∈ [0, 1 2] and δ := max(α, β) − γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then |B(x, y)| = b(x)γb(y)γ|b(x)δ − b(y)δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We observe the following elementary inequality: |uδ − vδ| ≤ |u − v| max(u, v)1−δ , ∀u, v ≥ 0, δ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='9) SOME REMARKS ON CONVEX BODY DOMINATION 19 Indeed, by symmetry and homogeneity, it is enough to consider u = 1 and v ∈ [0, 1], in which case we are reduced to proving that 1 − vδ ≤ 1 − v, which is immediate from the fact that v ≤ vδ for v, δ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='9), and noting that δ + 2γ = α + β ∈ [0, 1], it follows that |B(x, y)| ≤ b(x)γb(y)γ |b(x) − b(y)| max(b(x), b(y))1−δ ≤ |b(x) − b(y)| max(b(x), b(y))1−δ−2γ = � |b(x) − b(y)| max(b(x), b(y)) �1−δ−2γ |b(x) − b(y)|δ+2γ ≤ |b(x) − b(y)|α+β, and hence � Q Q |B(x, y)|p dx dy �1/p ≤ � Q Q |b(x) − b(y)|p dx dy �(α+β)/p ≤ �� Q |b(x) − c|p dx �1/p + � Q |b(y) − c|p dy �1/p�α+β for all constants c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let T be an operator satisfying (Ł1(γQ), Ł1(γQ)) convex body domination, let 0 ≤ b ∈ BMO(Rd) and 0 ≤ α, β with α + β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then ∥bαT (bβf) − bβT (bαf)∥Lp(Rd) ≲p ∥b∥α+β BMO(Rd)∥f∥Lp(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4 with s = t, the Lp(Rd) operator norm of f �→ bαT (bβf)− bβT (bαf) is dominated by As := sup Q ∥(x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='y) �→ b(x)αb(y)β − b(x)βb(y)α∥Łs(Q×Q) if p ∈ (s′, s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', if s > max(p, p′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='8 and the John–Nirenberg inequal- ity, we have As ≤ (2∥b∥BMOs(Rd))α+β ≲s ∥b∥α+β BMO(Rd), and fixing (say) s = 2 max(p, p′), we obtain a dependence on p only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Aside from the examples already discussed, the generalised commu- tators ⃗a · T⃗b also arise in the following question studied by Bloom [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Suppose that a matrix weight W is given in the diagonalised form W = U ∗ΛU, where U is unitary, Λ is diagonal, and the diagonal entries λk of Λ are scalar A2 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' What does one need to know about U in order to conclude that W ∈ A2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (According to [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2], the condition that λk ∈ A2 is necessary for W ∈ A2, if in addition U is assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=') Let T be the Hilbert transform, or another operator whose boundedness on the matrix-weighted L2(W) characterises W ∈ A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By connecting the L2(W) boundedness of T to the boundedness of the classical commutators [T, ¯uij] between the weighted spaces L2(λi) and L2(λk) (sic: the condition involves triplets of indices (i, j, k)), [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1] shows that uij ∈ BMO√ λi/λk (a weighted BMO space, nowadays commonly referred to as Bloom-type BMO) is a sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In the special case of 2 × 2 matrices, it is also necessary by [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3] but, over 30 years since these contributions, the general case seems to remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (The author is grateful to Amalia Culiuc for bringing this question to his attention [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=') 20 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN Here is a possible approach to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' As is well known, the L2(W) bound- edness of T is equivalent to the (unweighted) L2 boundedness of W 1/2T W −1/2 = U ∗Λ1/2UT U ∗Λ−1/2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Multiplication by U and U ∗ is isometric on L2, and the L2 boundedness of a matrix of operators is equivalent to the L2 boundedness of each of the components (Λ1/2UT U ∗Λ−1/2)ij = n � k=1 λ1/2 i uikT ¯ujkλ−1/2 j = λ1/2 i ⃗ui · T ¯⃗ujλ−1/2 j , where i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , n and ⃗ui = (uik)n k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' These are operators of the form ⃗a · T⃗b that we have studied here and, up to this point, we kept an exact equivalence with the original question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' the question then would be, whether we can give useful conditions on the boundedness of these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A further equivalent condition is of course the two-weight boundedness ⃗ui · T ¯⃗uj : L2(λj) → L2(λi), i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' , n, where the spaces are more complicated, but the multipliers are simply rows of the unitary matrix U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We have concentrated in this section on the application of convex body domination—an inherently vector-valued theory—to questions of generalised commutators acting on scalar-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We have made this choice for two reasons: to make the case that this vector-valued theory is useful even for such scalar-valued applications, and not to obscure the relatively simple basic philoso- phy behind too many technicalities of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' This said, it is quite plain that the presented ideas can be immediately generalised to the case of vector-valued functions ⃗f and ⃗g (in place of scalar f and g) and matrix-valued multipliers A and B (in place of the vectors ⃗a and ⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In the particular case of the classical-style commutator [T, B] with a matrix-valued function, this idea has been developed in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Stopping times and maximal functions involving convex bodies The aims of this final section are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Concretely, we establish a convex- body analogue of a result of Nieraeth [28], which shows that the estimation of sums over sparse collection that arise in the usual sparse domination is equivalent to the estimation of certain maximal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the way of achieving this, we develop some convex-body versions of the typical stopping time arguments involving averages of scalar-valued functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' these might have some independent interest elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' We begin with an estimate of a sum of convex-body “norms” over disjoint subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let p, q ∈ [1, ∞) and 1 r := 1 p + 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let Qi ∈ D(Q0) be disjoint cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then ∞ � i=1 � ⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) �r ≤ nmax(r,1)+r/2� ⟨⟨f⟩⟩Lp(Q) · ⟨⟨g⟩⟩Lq(Q) �r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Note that for p, q ∈ [1, ∞), we have 1 r = 1 p + 1 q ≤ 1 + 1 = 2, and hence nmax(r,1)+r/2 = � nmax(1,1/r)+1/2�r ≤ � n5/2�r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For orientation, let us begin with the proof in the case n = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', with ∥ ∥ in place of ⟨⟨ ⟩⟩ throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By Hölder’s inequality with 1 = r p + r q, we have ∞ � i=1 � ∥f∥Lp(Qi)∥g∥Lq(Qi) �r = ∞ � i=1 � ∥f∥p Lp(Qi) �r/p� ∥g∥q Lq(Qi) �r/q ≤ � ∞ � i=1 ∥f∥p Lp(Qi) �r/p� ∞ � i=1 ∥g∥q Lq(Qi) �r/q ≤ � ∥f∥p Lp(Q0) �r/p� ∥g∥q Lq(Q0) �r/q = � ∥f∥Lp(Q0)∥g∥Lq(Q0) �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In the general case of the lemma, let Ai := ⟨⟨⃗f⟩⟩Lp(Qi) = � ˆ Qi φi ⃗f : ∥φi∥Lp′(Qi) ≤ 1 � , Bi := ⟨⟨⃗g⟩⟩Lq(Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then we observe that ⟨⟨⃗f⟩⟩Lp(Q) = � ˆ Q φ⃗f : ∥φ∥Lp′(Q) ≤ 1 � ⊇ � ∞ � i=1 ai ˆ Qi φi ⃗f : ∥φi∥Lp′(Qi) ≤ 1, ∥(ai)∥ℓp′ ≤ 1 � = � ∞ � i=1 aiAi : ∥(ai)∥ℓp′ ≤ 1 � =: � ℓp Ai =: A, and similarly ⟨⟨⃗g⟩⟩Lq(Q) ⊇ � ℓq Bi =: B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence, the lemma is reduced to proving that ∞ � i=1 � Ai · Bi �r ≤ nmax(r,1)+r/2� A · B �r, A := � ℓp Ai, B := � ℓq Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let EA be the John ellipsoid of A, and let RAEA = ¯BRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Since Ai · Bi = RAAi · R−t A Bi, The claim above is equivalent to a version where each Ai is replaced by RAAi and each Bi by R−t A Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence, without loss of generality, we assume that EA = ¯BRn to begin with, hence ¯BRn ⊆ A ⊆ √n ¯BRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thus A · B ⊃ ¯BRn · B = [−M, M], where M := max{|⃗b| : ⃗b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the other hand, if (⃗ej)n j=1 is some orthonormal basis of Rn, then Ai · Bi = {⃗a ·⃗b : ⃗a ∈ Ai,⃗b ∈ Bi} = � n � j=1 (⃗a · ⃗ej)(⃗b · ⃗ej) : ⃗a ∈ Ai,⃗b ∈ Bi} ⊆ n � j=1 (Ai · ⃗ej)(Bi · ⃗ej), or, using the identification of [−s, s] with s, Ai · Bi ≤ n � j=1 (Ai · ⃗ej)(Bi · ⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 22 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN Thus (Ai · Bi)r ≤ n � j=1 � (Ai · ⃗ej)(Bi · ⃗ej) �r, r ∈ (0, 1], and � ∞ � i=1 (Ai · Bi)r�1/r ≤ n � j=1 � ∞ � i=1 � (Ai · ⃗ej)(Bi · ⃗ej) �r�1/r , r ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In the sum over i, we use Hölder’s inequality as in the toy model in the beginning: ∞ � i=1 � (Ai · ⃗ej)(Bi · ⃗ej) �r = ∞ � i=1 � (Ai · ⃗ej)p�r/p� (Bi · ⃗ej)q�r/q ≤ � ∞ � i=1 (Ai · ⃗ej)p�r/p� ∞ � i=1 (Bi · ⃗ej)q�r/q = sup �� ∞ � i=1 aiAi · ⃗ej �1/r� ∞ � i=1 biBi · ⃗ej �1/r : ∥(ai)∥ℓp′ ≤ 1, ∥(bi)∥ℓq′ ≤ 1 � = (A · ⃗ej)r(B · ⃗ej)r Here A · ⃗ej ⊆ √n ¯BRn · ⃗ej = [−√n, √n], A · ⃗ej ≤ √n, and clearly B · ⃗ej ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Altogether, writing s := max(r, 1), we have � ∞ � i=1 (Ai · Bi)r�1/s ≤ n � j=1 � ∞ � i=1 (Ai · ⃗ej)r(Bi · ⃗ej)r�1/s ≤ n � j=1 � (A · ⃗ej)r(B · ⃗ej)r�1/s ≤ n[nr/2M r]1/s, and hence ∞ � i=1 (Ai · Bi)r ≤ nsnr/2M r = nmax(1,r)+r/2(A · B)r, which remained to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ The following lemma is a convex-body analogue of the basic principle underlying the simplest stopping time constructions: for a function on a cube Q0, the total measure of the subcubes, where the average of a function is much bigger than on the whole Q0, can be at most a fraction of the measure of Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let A, p, q ∈ [1, ∞) and let Qi ∈ D(Q0) be disjoint cubes such that ⟨⟨⃗f⟩⟩Łp(Qi) · ⟨⟨⃗g⟩⟩Łq(Qi) ≥ A⟨⟨⃗f⟩⟩Łp(Q0) · ⟨⟨⃗g⟩⟩Łq(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Then ∞ � i=1 |Qi| ≤ nmax(r,1)+r/2 Ar |Q0|, 1 r := 1 p + 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Directly from the definition, it is easy to extend the basic identity ∥f∥Łp(Q) = |Q|−1/p∥f∥Lp(Q) to convex bodies as ⟨⟨⃗f⟩⟩Łp(Q) = |Q|−1/p⟨⟨⃗f⟩⟩Lp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='3) From this, the assumption of the lemma can be rewritten as |Qi|−1/p−1/q⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) ≥ A|Q0|−1/p−1/q⟨⟨⃗f⟩⟩p(Q0) · ⟨⟨⃗g⟩⟩q(Q0), or, rearranging, |Qi| ≤ A−r|Q0| � ⟨⟨⃗f⟩⟩p(Q0) · ⟨⟨⃗g⟩⟩q(Q0) �r � ⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) �r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Summing over i and using Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='1, we obtain the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ We now obtain the following proposition, which is a convex body analogue of a result of Nieraeth [28, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' especially Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7) for m = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' It says that estimating the sums over sparse collections, like those that arise from convex body domination, is equivalent to estimating related bi-sublinear maximal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In [28, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='7], the result is formulated as a set of equivalent conditions for a tuple of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The formulation below has no reference to weights as such, but as soon as one starts asking questions about the boundedness of either side on spaces like Ls(W) × Ls′(W ′), the proposition guarantees that one can equally well study this boundedness for the other side of the equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For all δ ∈ (0, 1), all dimensions d, n ≥ 1, exponents p, q ∈ [1, ∞), and functions ⃗f ∈ Lp loc(Rd)n, ⃗g ∈ Lq loc(Rd)n, we have the two-sided estimate sup S � Q∈S ⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q)|Q| ≂ ��� sup Q∈D 1Q⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q) ��� L1(Rd), where the supremum is taken over all δ-sparse collections of dyadic cubes in Rn, and the implied constants depend only on n, p, q, and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' With ⃗f ∈ Lp loc(Rd)n and ⃗g ∈ Lq loc(Rd)n fixed, let us denote aQ := ⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The estimate ≲ is immediate: From δ-sparseness, we have |Q| ≤ δ−1|E(Q)| for some disjoint sets E(Q), and hence � Q∈S aQ|Q| ≤ 1 δ � Q∈S aQ|E(Q)| = 1 δ ˆ Rd � Q∈S aQ1E(Q) ≤ 1 δ ˆ Rd sup Q∈D aQ1Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The estimate ≳ needs a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By monotone convergence, it is enough to consider D(Q0) in place of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Let S0 := {Q0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' For some A > 1 to be chosen and Q ∈ D(Q0), let S ′(Q) consist of all maximal Q′ ∈ D(Q) such that aQ′ > AaQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By maximality, the cubes Q′ ∈ S ′(Q) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='2, we have � Q′∈S ′(Q) |Q′| ≤ nmax(1,r)+r/2 Ar |Q| ≤ (1 − δ)|Q|, 1 r := 1 p + 1 q , provided that A is chosen large enough, depending on n, p, q, and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hence, defining inductively Sj+1 := � Q∈Sj S ′(Q) and S := �∞ j=0 Sj, we find that S is δ-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 24 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' HYTÖNEN If Q ∈ D(Q0) and S ∈ S is the minimal stopping cube that contains Q, then aQ ≤ AaS by the way that the cubes S ∈ S were chosen, hence sup Q∈D(Q0) 1QaQ ≤ sup S∈S 1SAaS ≤ A � S∈S 1SaS, and thus ��� sup Q∈D(Q0) 1QaQ ��� L1(Rd) ≤ A ��� � S∈S 1SaS ��� L1(Rd) = A � S∈S aS|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' □ References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Bagchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hait, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Roncal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Thangavelu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On the maximal function associated to the spherical means on the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' New York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 27:631–675, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Beltran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Roos, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Seeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Multi-scale sparse domination, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Preprint, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='00227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Bernicot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Frey, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Petermichl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Sharp weighted norm estimates beyond Calderón- Zygmund theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' PDE, 9(5):1079–1113, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Bloom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A commutator theorem and weighted BMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 292(1):103– 122, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Bloom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Applications of commutator theory to weighted BMO and matrix analogs of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Illinois J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 33(3):464–487, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Bownik and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Cruz-Uribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Extrapolation and factorization of matrix weights, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Preprint, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='09443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Conde-Alonso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Culiuc, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Di Plinio, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A sparse domination principle for rough singular integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' PDE, 10(5):1255–1284, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Cruz-Uribe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Isralowitz, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Moen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Two weight bump conditions for matrix weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Integral Equations Operator Theory, 90(3):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 36, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Culiuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Personal communication, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 11th International Conference on Harmonic Anal- ysis and Partial Differential Equations, El Escorial, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Culiuc, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Di Plinio, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Uniform sparse domination of singular integrals via dyadic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 25(1):21–42, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Culiuc, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Kesler, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lacey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Sparse bounds for the discrete cubic Hilbert transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' PDE, 12(5):1259–1272, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' David and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Journé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A boundedness criterion for generalized Calderón-Zygmund op- erators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2), 120(2):371–397, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Di Plinio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hytönen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Sparse bounds for maximal rough singular integrals via the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Fourier (Grenoble), 70(5):1871–1902, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Figiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Singular integral operators: a martingale approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' In Geometry of Banach spaces (Strobl, 1989), volume 158 of London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', pages 95–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Press, Cambridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hänninen and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hytönen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The A2 theorem and the local oscillation decomposition for Banach space valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Operator Theory, 72(1):193–218, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hytönen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' The sharp weighted bound for general Calderón-Zygmund operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2), 175(3):1473–1506, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hytönen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Li, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Oikari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Iterated commutators under a joint condition on the tuple of multiplying functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 148(11):4797–4815, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hytönen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Neerven, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Veraar, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Weis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Analysis in Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Martingales and Littlewood-Paley theory, volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Folge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 3rd Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A Series of Modern Surveys in Mathematics].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Springer, Cham, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Hytönen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Pérez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Sharp weighted bounds involving A∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' PDE, 6(4):777–818, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Isralowitz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Pott, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rivera-Ríos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Sharp A1 weighted estimates for vector-valued operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 31(3):3085–3116, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Isralowitz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Pott, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Treil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Commutators in the two scalar and matrix weighted setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (2), 106(1):1–26, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' SOME REMARKS ON CONVEX BODY DOMINATION 25 [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lacey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' An elementary proof of the A2 bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 217(1):181–195, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lerner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A simple proof of the A2 conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' IMRN, (14):3159– 3170, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lerner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On pointwise estimates involving sparse operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' New York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 22:341– 349, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Lerner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A weak type estimate for rough singular integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 35(5):1583–1602, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Muller and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Rivera-Ríos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Quantitative matrix weighted estimates for certain singular integral operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 509(1):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 125939, 38, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [27] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Nazarov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Petermichl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Treil, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Volberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Convex body domination and weighted estimates with matrix weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 318:279–306, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [28] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Nieraeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Quantitative estimates and extrapolation for multilinear weight classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 375(1-2):453–507, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Pérez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted Lp-spaces with different weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' (3), 71(1):135–157, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Seeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Singular integral operators with rough convolution kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 9(1):95–105, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Treil and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Volberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Wavelets and the angle between past and future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=', 143(2):269–308, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' Department of Mathematics and Statistics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content=' 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland Email address: tuomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='hytonen@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} +page_content='fi' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdAyT4oBgHgl3EQfuvkh/content/2301.00617v1.pdf'} diff --git a/UdE1T4oBgHgl3EQfIgMu/content/tmp_files/2301.02939v1.pdf.txt b/UdE1T4oBgHgl3EQfIgMu/content/tmp_files/2301.02939v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2c8bf73cc447f8eae491933977c4b4f94599acb --- /dev/null +++ b/UdE1T4oBgHgl3EQfIgMu/content/tmp_files/2301.02939v1.pdf.txt @@ -0,0 +1,1334 @@ +Study of the long-range transverse field Ising model with fermionic Gaussian states +Michael P. Kaicher,1, ∗ Davide Vodola,2, † and Simon B. J¨ager3 +1Departamento de F´ısica Te´orica, Universidad Complutense, 28040 Madrid, Spain +2Dipartimento di Fisica e Astronomia, Universit`a di Bologna, I-40129, Bologna, Italy +3Department of Physics and Research Center OPTIMAS, +University of Kaiserslautern-Landau, D-67663 Kaiserslautern, Germany +(Dated: January 10, 2023) +We numerically study the one-dimensional long-range Transverse Field Ising Model (TFIM) in the +antiferromagnetic (AFM) regime at zero temperature using Generalized Hartree-Fock (GHF) theory. +The spin-spin interaction extends to all spins in the lattice and decays as 1/rα, where r denotes the +distance between two spins and α is a tunable exponent. We map the spin operators to Majorana +operators and approximate the ground state of the Hamiltonian with a Fermionic Gaussian State +(FGS). Using this approximation, we calculate the ground state energy and the entanglement entropy +which allows us to map the phase diagram for different values of α. In addition, we compute the +scaling behavior of the entanglement entropy with the system size to determine the central charge +at criticality for the case of α > 1. For α < 1 we find a logarithmic divergence of the entanglement +entropy even far away from the critical point, a feature of systems with long-range interactions. +We provide a detailed comparison of our results to outcomes of Density Matrix Renormalization +Group (DMRG) and the Linked Cluster Expansion (LCE) methods. In particular, we find excellent +agreement of GHF with DMRG and LCE in the weak long-range regime α ≥ 1, and qualitative +agreement with DMRG in the strong-long range regime α ≤ 1. Our results highlight the power of +the computationally efficient GHF method in simulating interacting quantum systems. +I. +INTRODUCTION +Quantum phase transitions describe the behavior of +quantum many-body systems at zero temperature when +tuning a non-thermal control parameter, such as an ap- +plied magnetic field. +The phase transition appears as +a result of competing phases that describe the ground +state at the corresponding parameter and typically lead +to a fundamental change in the nature of the correlation +present in the ground state. Quantum many-body sys- +tems can undergo a quantum phase transition and their +study has lead to the discovery of many exotic collective +phenomena such as superconducting ground states [1], +long-range topological order [2], and anyonic statistics[3]. +Close to the critical point, the properties of many differ- +ent physical systems can be classified by a universality +class which is independent of the system size and only +depends on the underlying dimensions and symmetries of +the problem. In this situation, one can in many instances +describe the many-body problem by an interacting spin +system [4]. +One of the paradigmatic microscopic models display- +ing a quantum phase transition is the Transverse Field +Ising Model (TFIM) at zero temperature [5]. This model +is exactly solvable in the limit of short-range, nearest- +neighbour interactions. +However, the solution of this +problem is much harder if one considers beyond nearest- +∗ Correspondence to: +michael.p.kaicher(at)gmail.com; Now at: +BASF Digital Solutions, Next Generation Computing, Pfalz- +grafenstr. 1, D-67056, Ludwigshafen, Germany +† Now at: BASF Digital Solutions, Next Generation Computing, +Pfalzgrafenstr. 1, D-67056, Ludwigshafen, Germany +neighbour or even long-range interactions [6–10]. Long- +range interacting systems can host exotic states of quan- +tum matter and are therefore of large scientific interest. +Recent advances have made effective long-range spin- +interactions experimentally accessible [11–15]. +In such +systems, the effective interaction extends to all spins in +the lattice and decays as a power law 1/rα, where r is +the distance of the spins in the lattice and α is a tunable +algebraic exponent. In the experiments one can realize +0 ≤ α ≤ 3 which allows one to experimentally probe the +regime of long-range interactions in spin systems [11]. +In order to analyze the properties of a quantum many- +body system, it is important to study large system sizes, +which is in our case the number of spins N. The exponen- +tial scaling of the Hilbert space dimension with N makes +the ad-hoc diagonalization of such many-body problems +illusive. Consequently, one demands numerical methods +which are able to capture the qualitative behavior of the +many-body system with a computational cost that dis- +plays a low scaling with N. A range of many-body meth- +ods of varying computational complexity have been ap- +plied to study finite size long-range quantum many-body +systems, including Quantum Monte Carlo (QMC) [16], +stochastic series expansion QMC [17], a combination of +QMC and renormalization group methods [18], Lanczos +exact diagonalization [19], and Density Matrix Renor- +malization Group (DRMG) [6, 8]. Recently, a method +to study short-range quantum-lattice models in the ther- +modynamic limit, the Linked-Cluster Expansion (LCE), +has been extended to allow for the study of long-range +systems for α > 1 [9, 10]. +In this work, we add Generalized Hartree-Fock (GHF) +theory to this mix of methods. +GHF is a mean-field +method which aims to approximate the ground state of +arXiv:2301.02939v1 [quant-ph] 7 Jan 2023 + +2 +an interacting quantum system as a free electron gas [20], +where the latter describes a class of variational func- +tions known as Fermionic Gaussian States (FGS). Due +to its mean-field nature, GHF is a method with very +low computational cost, where the most-demanding com- +pute operation—the evaluation of the Pfaffian Pf(A) of +a M × M matrix A—scales at most as O(M 3) [21]. +Even though FGS describe ground or thermal states of +quadratic fermionic Hamiltonians [22], they have been +applied to various areas of quantum many-body physics +with great success, most notably as ab-initio methods to +obtain approximate ground states in electronic structure +problems and to condensed matter systems [20, 23, 24]. +In this paper, in order to find the FGS which best ap- +proximates the ground state of the long-range TFIM, +we employ two physically-motivated methods which have +been described in Ref. [23]. The first one (ITE) derives +the ground state using Imaginary Time Evolution. The +second one (ZT) uses a self-consistent equation for the +FGS ground state covariance matrix. Using these two +methods we calculate the ground state energy and the +entanglement entropy. +By comparison of these results +with the ones obtained from DMRG and ZT we will show +that GHF is able to capture the qualitative and quantita- +tive behavior of the long-range TFIM. This highlights the +ability of GHF in predicting physically relevant material +properties at computationally low cost. +This work is structured as follows. In Section II we +discuss the GHF theory which we then apply to the +TFIM model described in +II A. The introduced meth- +ods are used in Section III where we numerically study +the ground state energy and the entanglement entropy. +We conclude by summarizing our findings in Section IV +and providing an outlook for future work. +II. +THEORY +A. +Long-range transverse field Ising model +In this work we consider the TFIM Hamiltonian de- +scribing a system of N spins with open boundary condi- +tions +ˆH = +N +� +p=1 +hpˆσz +p + +N +� +p 0 +or θ ∈ (0, π). +Because the Hamiltonian is symmetric +under the simultaneous transformations ˆσz +p �→ −ˆσz +p and +θ → π − θ we can restrict our study to θ ∈ (0, π/2]. +In a next step, we map the TFIM Hamiltonian onto a +fermionic Hamiltonian. To this end, we use the Jordan- +Wigner transformation ˆσ+ +p = ˆc† +peiπ �p−1 +q=1 ˆc† +qˆcq and ˆσ− +p = +ˆcpe−iπ �p−1 +q=1 ˆc† +qˆcq [25]. Here, we used ˆσ± +p = [ˆσx +p ± iˆσy +p]/2 +and introduced the fermionic raising and lowering opera- +tors ˆc† +p, ˆcp, respectively. The latter obey the canonical an- +ticommutation relations {ˆcp, ˆcq} = 0 and {ˆcp, ˆc† +q} = δp,q, +where δp,q is the Kronecker delta and { ˆA, ˆB} = ˆA ˆB+ ˆB ˆA +denotes the anticommutator of two operators ˆA, ˆB. In- +stead of analyzing the problem in the basis of the 2 × N +fermionic operators ˆcp, ˆc† +p we represent the Hamiltonian +in 2N Majorana operators ˆa2p−1 = ˆc† +p + ˆcp and ˆa2p = +i(ˆc† +p−ˆcp). The latter posses the anticommutation relation +{ˆal, ˆam} = 2δl,m (l, m = 1, 2, . . . , 2N) and the Hamilto- +nian (1) in the Majorana representation is given by +ˆH = − i +N +� +p=1 +hpˆa2p−1ˆa2p + +N +� +p 1 as +the weak long-range regime, while the special case α = 1 +is marginal. +B. +Fermionic Gaussian States +The formal definition of a FGS is given by [22], +ˆρGS =tr +� +e−β ˆ +HGS�−1 +e−β ˆ +HGS, +(3) +where ˆHGS = +i +4ˆaT Gˆa is a Hermitian operator, β ∈ R, +ˆa = (ˆa1, ˆa2, . . . , ˆa2N)T is a column vector of Majorana +operators, and G is a (2N × 2N) real-valued and anti- +symmetric matrix. FGS are fully described by the real + +3 +and anti-symmetric covariance matrix Γ with entries +Γlm = i +2tr (ˆρGS[ˆal, ˆam]) , +(4) +l, m ∈ {1, 2, . . . , 2N}, and where [ ˆA, ˆB] = +ˆA ˆB − ˆB ˆA +denotes the commutator of two operators ˆA, ˆB. While +Eqs. (3)-(4) describe both pure and mixed FGS, we only +focus on pure FGS in this work, since we are interested +in the ground state. +Pure FGS are characterized by +Γ2 = −12N (1k denotes the (k × k)-identity matrix), +and eigenvalues of the covariance matrix are given by +λ ∈ {−1, 1}. All information contained in the density +matrix (3) of a FGS is also contained in its covariance +matrix (4). The expectation value of a single tensor prod- +uct of Majorana or fermionic operators can be computed +efficiently through Wick’s theorem [22, 28], +tr (ˆρGSˆai1ˆai2 · · · ˆai2m) =(−i)mPf +� +Γ|i1i2...i2m +� +, +(5) +where i1 ̸= i2 ̸= . . . ̸= i2m for ik ∈ {1, . . . , 2N} and k = +1, . . . , 2N. The matrix Γ|i1i2...i2m denotes a (2m × 2m)- +submatrix of Γ with the corresponding rows and columns +i1, i2, . . . , i2m, and Pf(A) denotes the Pfaffian of a skew- +symmetric matrix A. +C. +Approximating the ground state with a +fermionic Gaussian State +Using Wick’s theorem (5), we are able to compute the +energy expectation value +E(Γ) =tr +� +ˆρGS ˆH +� +, +(6) +which results in +E(Γ) = − +N +� +p=1 +hp +2 (Γ2p−1,2p − Γ2p,2p−1) ++ +N +� +p d = 1, and the +strong long-range interactions, α < d = 1. +For weak long-range interactions and a non-vanishing +energy gap we expect also an area law scaling, implying +that SN/2 is independent of N. For the case of a vanish- +ing gap one also finds a logarithmic divergence [33, 35–37] +following Eq. (12). +For strong long-range interactions in the AFM-TFIM +we expect instead a logarithmic divergence of the entan- +glement entropy, where SN/2 obeys Eq. (12) and one can +find c ̸= 0 even in presence of a non-vanishing gap [38– +41]. In this regime c is strictly speaking not a central +charge but because of the same functional dependence of +SN/2 in Eq. (12), we also denote c as the effective central +charge. +III. +RESULTS +A. +Phase diagram +In this section, we show that a computationally in- +expensive GHF mean-field approach can reproduce the +phase diagram of the AFM-TFIM for a wide range of +values α, both in the weak and strong long-range regime, +and is able to locate the point of the phase transition +for α ≥ 1 in excellent agreement with state-of-the-art +numerical methods. +As a first benchmark and in the same spirit of Ref. [6] +we map the phase diagram by calculating the entangle- +ment entropy for a wide range of values of α, from weak to +strong long-range interactions, and for θ ∈ (0, π/2). The +values of SN/2 [Eq. (11)] computed with the ZT GHF +method are visible in Fig. 1 for N = 100. For θ = 0 the +interactions vanish and SN/2 = 0 for all values of α. This +FIG. 1. +We plot the entanglement entropy SN/2 from the +covariance matrix obtained through the ZT algorithm for a +system size N = 100, α ∈ {0.30, 0.50, 0.75, 1.00, . . . , 3.00}, +and θ ∈ (0, π/2). Black squares represent the quantum critical +points θ∞ +c /π in the thermodynamic limit, which are listed in +Tab. I, while the dashed line serves as a guide to the eye. +represents the phase where all spins are uncorrelated and +align with the external magnetic field. However, when +θ and therefore the AFM interactions are increased, the +minimization of the interaction energy competes with the +external magnetic field. This is accompanied by an in- +crease of SN/2. Dependent on α, there is a critical value +θc(α) beyond which the spins favor an AFM order. This +transition is highlighted in Fig. 1 by a sharp rise of SN/2. +Our findings are in qualitative agreement with the ones +obtained in Ref. [6] from DMRG calculations. To com- +pare our results also quantitatively, we will now focus on +the weak and strong long-range interactions cases sepa- +rately. +B. +Weak long-range interactions +1. +Comparison of GHF and DMRG +For weak long-range interactions, α ≥ 1, we show the +ground state energy and the entanglement entropy in +Fig. 2(a) and Fig. 2(b), respectively. +The solid lines represent the results obtained from +the GHF theory while hollow markers represent the re- +sults obtained from DMRG simulations. Both simulation +methods predict a rather smooth behavior of the energy +in Fig. 2(a). For larger values of α ≥ 1.5 we find a max- +imum and a decrease beyond the maximum point. The +GHF and DMRG simulations agree perfectly. +The entanglement entropy, visible in Fig. 2(b), shows +for all values and both simulations methods a very quick +increase and a pronounced singularity. The latter is an +indicator for the phase transition point. +Beyond this + +1.4 +3.0 +1.2 +2.5 +1.0 +2.0 - +0.8 +SN/2 +a +1.5 +0.6 +1.0 +0.4 +0.2 +0.5 +0.1 +0.2 +0.3 +0.4 +0/ π5 +(a) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +θ/π +−100 +−95 +−90 +−85 +−80 +−75 +−70 +Energy +ZT, α=1.0 +ZT, α=1.25 +ZT, α=1.5 +ZT, α=1.75 +ZT, α=2.0 +ZT, α=2.25 +ZT, α=2.5 +ZT, α=2.75 +ZT, α=3.0 +DMRG +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +θ/π +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +SN/2 +ZT, α=1.0 +ZT, α=1.25 +ZT, α=1.5 +ZT, α=1.75 +ZT, α=2.0 +ZT, α=2.25 +ZT, α=2.5 +ZT, α=2.75 +ZT, α=3.0 +DMRG +FIG. 2. +For a system of size N = 100 and exponents +α ∈ [1, 3], we plot (a) the energy E and (b) the entanglement +entropy SN/2 (bottom), as defined in Eqs. (6) and (11), ob- +tained from the covariance matrix of the ZT algorithm (solid +lines) and compare it to DMRG (hollow markers). +point we find again a decrease of the entanglement en- +tropy. Both methods, GHF and DMRG, are in very good +agreement. +2. +Threshold and central charge +In order to find a value for the threshold at N → ∞, +we are performing a finite-size scaling. +For this we +carry out analogue simulations for a range of smaller +system sizes N ∈ {20, 30, . . . , 100}. +Then we find nu- +merically the maximum of the entanglement entropy +of the half chain Smax = SN/2(θmax) and the corre- +sponding value θmax. +The latter is found using the +0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 +1/N +0.26 +0.28 +0.30 +0.32 +0.34 +0.36 +0.38 +θmax/π +α =1.0, θ∞ +c /π =0.3534(4) +α =2.0, θ∞ +c /π =0.3013(2) +α =3.0, θ∞ +c /π =0.2760(2) +FIG. 3. +Example for the fit of Eq. (13) to the value of θmax +obtained by maximization of the entanglement entropy S with +FGS. The thresholds θ∞ +c /π are shown for the respective cases +α ∈ {1, 2, 3}, see Tab. I for more details. +optimizer scipy.optimize.fminbound() which is pre- +implemented in python. +For every value θ examined +by the optimizer we find the optimal FGS for the cor- +responding Hamiltonian. Optimizing SN/2 over θ can be +achieved as FGS provide a way for calculating SN/2 poly- +nomially in N, see Eq. (11). We then use the following +finite-size scaling law [42] +θmax(N) = θ∞ +c + a +N , +(13) +where θ∞ +c +is the threshold at N → ∞ and a is a fitting +parameter which determines the finite-size scaling. Fit- +ting Eq. (13) to the numerically obtained data of θmax +reveals the θ∞ +c +in the thermodynamic limit. In Fig. 3 we +provide examples for the fits that are used to calculate +θ∞ +c . We perform these fits for various values of α and the +results for the threshold are collected in Tab. I. In addi- +tion, we have plotted the results of θmax +c +in Fig 1 as black +squares which mark the sudden spike of the entanglement +entropy. In Tab. I we compare the results obtained from +the GHF theory with the ones obtained from LCE calcu- +lations [10], DMRG data of Ref. [6] (labeled DMRG) and +Ref. [8] (labeled DMRG*). We find in general very good +agreement of the thresholds obtained from the different +methods. +Besides the threshold θ∞ +c +we can also extract the scal- +ing of the maximum entropy Smax = S(θmax). At the +critical point we use the scaling law [34] given by Eq. (12). +We fit Eq. (12) to the maximum values Smax as displayed +in Fig. 4(a). From these fits we extract the central charge +c, which is shown in Fig. 4(b) as function of α. The cen- +tral charge is always above the result c = 1/2 expected +from the short-range TFIM. We also compare our results +to different DMRG results of Ref. [6, 8]. We find that +the central charges obtained from FGS are systematically +smaller than the values provided by Ref. [6] and larger + +6 +than the DMRG results of Ref. [8]. The central charge c is +monotonically decreasing in the weak long-range regime, +but drops at the onset of the strong long-range regime +at α = 1. In conclusion, we found that the results of +the GHF method are in good qualitative and quantita- +tive agreement with state-of-the-art numerical methods +for weak long-range interactions. +C. +Strong long-range interactions +1. +Comparison of GHF and DMRG +We will now shift our focus to the regime of strong +long-range interactions, α < 1. We first plot the ground +state energy and the entanglement entropy in Fig. 5(a) +and Fig. 5(b) for three different values of α < 1 of size +N = 100. +In Fig. 5(a) we obtain for all three values +of α a monotonously increasing energy with θ. This is +different to the case of weak long-range interactions (see +Fig. 2(a)) where we have observed a maximum close to +the threshold at least for sufficiently large α ≥ 1.5. We +compare our results obtained from FGS also with the +ones obtained from DMRG results. Here, we find that +DMRG always predicts a lower ground state energy. The +discrepancy of the two methods is even more striking +in the entanglement entropy visible in Fig. 5(b). Here, +while we still observe very good agreement for α = 0.75 +we found clear deviations for α = 0.3. The DMRG results +predict tendentially a larger entanglement entropy than +the FGS. This is an indicator that FGS are less well- +suited for the description of the TFIM for very small α, +i.e. very strong long-range interactions. +α +θ∞ +c /π FGS +LCE +DMRG DMRG* +1.00 +0.3534(4) - +0.3509 +- +1.25 +0.3357(1) 0.35(5) +- +- +1.50 +0.3218(1) 0.3213(5) +0.3226 +- +1.75 +0.3106(1) - +- +- +2.00 +0.3013(2) 0.3026(8) +0.3027 +0.3021 +2.25 +0.2932(2) 0.294(4) +- +- +2.50 +0.2865(1) 0.2871(11) +- +- +2.75 +0.2807(2) - +- +- +3.00 +0.2760(2) 0.27722(25) 0.2782 +- +TABLE I. The critical points θ∞ +c /π obtained from Eq. (13) +with FGS and ZT, in comparison to LCE [10], and DMRG +[6], DMRG*[8] results. The values are obtained for various +exponents α and for simulations up to N = 100 spins. The +error indicated in the FGS column in round brackets is the +standard deviation for the intersect of a linear regression fit +of θmax/π as a function of 1/N. +(a) +3.0 +3.2 +3.4 +3.6 +3.8 +4.0 +4.2 +4.4 +4.6 +log(N) +0.72 +0.74 +0.76 +0.78 +0.80 +SN/2 +α =1.0, c =0.5455(40) +α =2.0, c =0.5428(72) +α =3.0, c =0.5269(73) +(b) +1.00 +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +α +0.50 +0.52 +0.54 +0.56 +0.58 +0.60 +c +ZT +FGS +DMRG +DMRG* +FIG. 4. +(a) Extracting the central charge. Using the ZT +algorithm for various α, here exemplified by α ∈ {1, 2, 3}, we +plot the entanglement entropy SN/2 against log(N). For each +α we perform a linear regression fit, neglecting the system +sizes N ∈ {20, 30, 40} to mitigate finite size effects. (b) Cen- +tral charge c obtained from finite-size scaling up to system +size N = 100 of FGS evolutions through the ZT algorithm +(blue squares) for the AFM long-range TFIM. For compari- +son, DMRG results from finite-size scaling of system sizes of +up to N = 100 from Ref. [6] (’DMRG’, orange square) and +[8] (’DMRG*’, green triangles) are included. The red hori- +zontal line represents the value c = 1/2 which describes the +Ising universality class. +Error bars represent the standard +deviation from the linear regression fit. +2. +Violations to the area law +We will now analyze the scaling of the entanglement +entropy with the system size. For this we calculate the +entanglement entropy for various parameters θ and α and +for different numbers of spins N ∈ {40, 50, . . . , 100}. We + +7 +(a) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +θ/π +−100 +−90 +−80 +−70 +−60 +Energy +ITE, α=0.3 +ITE, α=0.5 +ITE, α=0.75 +DMRG +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +θ/π +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +SN/2 +ITE, α=0.3 +ITE, α=0.5 +ITE, α=0.75 +DMRG +FIG. 5. +(a) Energy and (b) entanglement entropy obtained +from the covariance matrix of the ITE algorithm (solid lines) +and DMRG (empty markers) simulations for N = 100 and +α ∈ {0.3, 0.5, 0.75}. +then fit the coefficients c and B using Eq. (12) to the +obtained values of the entanglement entropy. +The ob- +tained values of c are shown in Fig. 6. At this point we +remark that the effective central charge c is calculated far +away from the threshold in a phase with a non-vanishing +energy gap [6]. +For the values α < 1, we find c = 0 only at θ = 0. For +increasing θ we find a sharp increase of c. For α = 0.3 +and α = 0.5 we find a maximum and then a decrease +again for larger values of θ. A qualitatively similar be- +havior has also been observed in Ref. [6]. This has been +seen as a violation to the area law since this logarithmic +divergence does not originate from a closing gap in the +spectrum of the system [6]. We therefore conclude that +the FGS are able to predict this feature, although the +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +θ/π +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +0.045 +c/6 +α =0.3 +α =0.5 +α =0.75 +α =1 +FIG. 6. +Violations to the area law: The effective central +charge c [Eq. (12)] calculated from finite scaling of system +sizes N ∈ {40, 50, . . . , 100} for 50 different values deep in the +gapped region θ ∈ (0, π/4) for the GHF ITE algorithm. Error +bars for the standard deviation are also included, but too +small to be visible. +quantitative values deviate from the ones obtained from +DMRG results. +IV. +SUMMARY AND OUTLOOK +This work presents an extensive study of the AFM +long-range TFIM in both the weak and strong long-range +regime using generalized Hartree Fock theory, a mean- +field method with low computational cost. We validate +our results by comparing the computed energy and entan- +glement entropy to DMRG. We plot the phase diagram +and provide estimates for the location of the critical point +of the second order phase transition through finite-size +scaling for α ∈ [1, 3] and find that they are in excel- +lent agreement with both LCE calculations of Ref. [10] +and DMRG simulations of Refs. [6, 8]. At the critical +point, we compute the central charge c of the underly- +ing conformal field theory for α ∈ {0.3, 0.5, 1}, and find +c > 1/2 for all values of α. +In the strong long-range +regime we still found qualitative agreement between FGS +and DMRG calculations. Hereby we found larger quan- +titative deviations for smaller values of α. Remarkably, +GHF can predict the logarithmic violations to the area +law in the AFM-TFIM which has previously been stud- +ied with DRMG. Based on these findings, we conclude +that FGS provide a numerically inexpensive alternative +to study the AFM long-range TFIM and that our results +are in good agreement with DMRG, the current state- +of-the-art numerical method for one-dimensional lattice +systems. +All simulations were carried out using a standard lap- +top computer. +Since the dimensionality of the system +only appears in the Hamiltonian elements hpq and Jpq, + +8 +it is straightforward to apply FGS to the two- and +three-dimensional TFIM. Therefore, it would be inter- +esting to compare FGS simulations with methods that +can be applied to the two-dimensional AFM-TFIM [10]. +Moreover, while we have focused on the AFM regime, +FGS can readily be applied to the ferromagnetic regime +θ ∈ (−π, 0). In this work we have focused on the en- +tanglement entropy, however, pair correlation functions +and the entanglement spectrum can be extracted from +the covariance matrix as well. +FGS can also be used +to study dynamics under the evolution of the TFIM, +with equations of motion similar to Eq. (8) [23, 24]. In +particular, studying the dynamics of the entropy after a +quench would offer the possibility to verify the breaking +of conformal symmetry in the regime α < 1 [43]. From +a numerical standpoint, more efficient calculations of the +central quantities such as Eqs. (7) could lead to dramatic +computational speedups. As a possible pathway, it would +be interesting to see if sum-identities for Pfaffians such as +provided in Refs. [44–46] could be applied to the TFIM +Hamiltonian. Finally, one could study if different spin- +to-fermion mappings [47–49], each resulting in a different +form of H when expressed in fermionic operators, have +an effect on the FGS simulations. +ACKNOWLEDGMENTS +The authors thank Kai Phillip Schmidt for providing +the data for the LCE calculations from Ref.[10] and Luca +Tagliacozzo for providing the DMRG data from Ref. [6]. +The authors also thank Giovanna Morigi for insightful +discussions. M.K. thanks Miguel ´Angel Mart´ın-Delgado +and Frank Wilhelm-Mauch for helpful discussions and +support. S.B.J. acknowledges support from the Research +Centers of the Deutsche Forschungsgemeinschaft (DFG): +Projects A4 and A5 in SFB/Transregio 185: “OSCAR.” +[1] M. Tinkham, Introduction to Superconductivity, 2nd ed. +(Dover Publications, 2004). +[2] F. D. M. Haldane, Rev. Mod. Phys. 89, 040502 (2017). +[3] A. Stern, Annals of Physics 323, 204 (2008), january Spe- +cial Issue 2008. +[4] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cam- +bridge University Press, 2011). +[5] R. J. Elliott, P. Pfeuty, and C. Wood, Phys. Rev. Lett. +25, 443 (1970). +[6] T. Koffel, M. Lewenstein, and L. Tagliacozzo, Phys. Rev. +Lett. 109, 267203 (2012). +[7] M. Knap, A. Kantian, T. Giamarchi, I. Bloch, M. D. +Lukin, and E. Demler, Phys. Rev. Lett. 111, 147205 +(2013). +[8] D. Vodola, L. Lepori, E. Ercolessi, and G. Pupillo, New +Journal of Physics 18, 015001 (2015). +[9] S. Fey and K. P. Schmidt, Phys. Rev. B 94, 075156 +(2016). +[10] S. Fey, S. C. Kapfer, and K. P. Schmidt, Phys. Rev. Lett. +122, 017203 (2019). +[11] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, +J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, +Nature 484, 489 (2012). +[12] C. Schneider, D. Porras, and T. Schaetz, Reports on +Progress in Physics 75, 024401 (2012). +[13] A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras, +and T. Schaetz, Nature Physics 4, 757 (2008). +[14] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, +R. Blatt, and C. F. Roos, Nature 511, 202 (2014). +[15] A. Bermudez, T. Schaetz, and M. B. Plenio, Phys. Rev. +Lett. 110, 110502 (2013). +[16] S. Humeniuk, Phys. Rev. B 93, 104412 (2016). +[17] J. A. Koziol, A. Langheld, S. C. Kapfer, and K. P. +Schmidt, Phys. Rev. B 103, 245135 (2021). +[18] N. Laflorencie, I. Affleck, and M. Berciu, Journal of +Statistical Mechanics: +Theory and Experiment 2005, +P12001 (2005). +[19] A. W. Sandvik, Phys. Rev. Lett. 104, 137204 (2010). +[20] V. Bach, E. H. Lieb, and J. P. Solovej, Journal of Statis- +tical Physics 76, 3 (1994). +[21] M. Wimmer, ACM Trans. Math. Softw. 38, 1 (2012). +[22] S. Bravyi, Quantum Info. Comput. 5, 216–238 (2005). +[23] C. V. Kraus and J. I. Cirac, New Journal of Physics 12, +113004 (2010). +[24] T. Shi, E. Demler, and J. Ignacio Cirac, Annals of Physics +390, 245 (2018). +[25] P. Jordan and E. Wigner, Zeitschrift f¨ur Physik 47, 631 +(1928). +[26] M. E. Fisher, S.-k. Ma, and B. G. Nickel, Phys. Rev. Lett. +29, 917 (1972). +[27] N. Defenu, A. Trombettoni, and A. Codello, Phys. Rev. +E 92, 052113 (2015). +[28] G. C. Wick, Phys. Rev. 80, 268 (1950). +[29] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. +Mod. Phys. 80, 517 (2008). +[30] R. +Horodecki, +P. +Horodecki, +M. +Horodecki, +and +K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). +[31] I. Peschel, Journal of Physics A: Mathematical and Gen- +eral 36, L205 (2003). +[32] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. +Rev. Lett. 90, 227902 (2003). +[33] J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. +82, 277 (2010). +[34] P. Calabrese and J. Cardy, Journal of Statistical Mechan- +ics: Theory and Experiment 2004, P06002 (2004). +[35] B.-Q. Jin and V. E. Korepin, Journal of Statistical +Physics 116, 79 (2004). +[36] A. R. Its, B.-Q. Jin, and V. E. Korepin, Journal of +Physics A: Mathematical and General 38, 2975 (2005). +[37] J. P. Keating and F. Mezzadri, Phys. Rev. Lett. 94, +050501 (2005). +[38] J. Eisert and T. J. Osborne, Phys. Rev. Lett. 97, 150404 +(2006). +[39] D. Peter, S. M¨uller, S. Wessel, and H. P. B¨uchler, Phys. +Rev. Lett. 109, 025303 (2012). +[40] D. Vodola, L. Lepori, E. Ercolessi, A. V. Gorshkov, and +G. Pupillo, Phys. Rev. Lett. 113, 156402 (2014). +[41] F. Ares, J. G. Esteve, F. Falceto, and A. R. de Queiroz, + +9 +Phys. Rev. A 92, 042334 (2015). +[42] S. Nishimoto, Phys. Rev. B 84, 195108 (2011). +[43] J. Schachenmayer, B. P. Lanyon, C. F. Roos, and A. J. +Daley, Phys. Rev. X 3, 031015 (2013). +[44] M. Ishikawa and M. Wakayama, Linear and Multilinear +Algebra 39, 285 (1995). +[45] M. Ishikawa, M. Wakayama, et al., in Combinatorial +Methods in Representation Theory (Mathematical Soci- +ety of Japan, 2000) pp. 133–142. +[46] M. Ishikawa and M. Wakayama, Journal of Combinato- +rial Theory, Series A 113, 113 (2006). +[47] S. B. Bravyi and A. Y. Kitaev, Annals of Physics 298, +210 (2002). +[48] A. Tranter, S. Sofia, J. Seeley, M. Kaicher, J. McClean, +R. Babbush, P. V. Coveney, F. Mintert, F. Wilhelm, and +P. J. Love, International Journal of Quantum Chemistry +115, 1431 (2015). +[49] Z. Jiang, A. Kalev, W. Mruczkiewicz, and H. Neven, +Quantum 4, 276 (2020). +[50] R. M. Wilcox, Journal of Mathematical Physics 8, 962 +(1967). +Appendix A: Derivation of the equations of motion for the ITE algorithm +In this section, we will derive Eq. (8) which describes the imaginary-time evolution of a FGS. Eq. (8) was also +shown in Ref. [23] for fourth-order polynomials of fermionic opertors and for an even more general case in Ref. [24]. +We start by writing down the imaginary-time time evolution for the pure FGS ˆρGS = |ΨGS⟩ ⟨ΨGS| determined by +d +dτ |ΨGS⟩ = − +� +ˆH − ⟨ΨGS| ˆH|ΨGS⟩ +� +|ΨGS⟩ . +(A1) +The pure FGS can be generated by a Gaussian transformation +|ΨGS⟩ = ˆUGS |vac⟩ , +(A2) +where |vac⟩ denotes the fermionic vacuum and +ˆUGS(ξ) =e +i +4 ˆaT ξˆa +(A3) +describes the generator of a pure FGS [22]. Here, ξ denotes a (2n × 2n) anti-symmetric and Hermitian matrix (the +matrix elements ξkl = −ξlk are purely imaginary). To calculate the covariance matrix Γ we use +Γ = −UξΥUT +ξ , +(A4) +where +Υ = +N +� +p=1 +� +0 +1 +−1 0 +� +. +(A5) +is the covariance of the vacuum state and where we employed the transformation +ˆU † +GS(ξ)ˆa ˆUGS(ξ) = Uξˆa, +(A6) +with +Uξ = eiξ. +(A7) +Now, the idea is that we derive from Eq. (A1) a differential equation for Uξ which can then be used to calculate Γ +using Eq. (A4). For this we treat the left- and right-hand side of Eq. (A1) separately and rewrite it as +ˆUGS ˆL |vac⟩ = ˆUGS ˆR |vac⟩ . +(A8) +We then expand the operators ˆL and ˆR up to second order in terms of normal-ordered monomials of the Majorana +operators and apply them to the vacuum state. +a. +Left-hand side of Eq. (A1): +The operator ˆL is defined as +ˆL = ˆU † +GS +� +d ˆUGS(ξ) +dτ +� +. +(A9) + +10 +The derivative of the unitary transformation is given by +d ˆUGS(ξ) +dτ +=UGS(ξ) +� i +4ˆaT UT +ξ +dUξ +dτ ˆa +� +. +(A10) +Here, we have used Eq. (A3), the identity [50] +de ˆ +J(τ) +dτ += +� 1 +0 +due(1−u) ˆ +J(τ) � +dτ ˆJ(τ) +� +eu ˆ +J(τ), +(A11) +and the orthogonality property UξUT +ξ = 1. For the normal-ordered expression we therefore find +ˆL = ˆL0 + ˆL2, +(A12) +with +ˆL0 = i +4tr +�dUξ +dτ UT +ξ Γ +� +, +(A13) +ˆL2 =1 +4 : ˆaT UT +ξ +dUξ +dτ ˆa :, +(A14) +where : ˆA : denotes the elementwise normal-ordering of ˆA. +b. +Right-hand side of Eq. (A4): +The definition of ˆR is +ˆR = − ˆU † +GS +� +ˆH − ⟨ΨGS| ˆH |ΨGS⟩ +� +ˆUGS. +(A15) +We can now use a modification of Wicks theorem to calculate +ˆU † +GS ˆH ˆUGS = ⟨ΨGS| ˆH |ΨGS⟩ + i +4 : ˆaT UT +ξ H(mf)Uξa : + ˜Q, +(A16) +where ˜Q collects all normally ordered monomials of quartic order or higher. We derive this expression in Appendix B. +Inserting Eq. (A16) into Eq. (A15), we find for ˆR the following expression +ˆR = ˆR2 − ˜Q, +(A17) +with +ˆR2 = − i +4 : ˆaT UT +ξ H(mf)Uξa : . +(A18) +c. +Comparing left- and right-hand side: +We now require to match ˆL |vac⟩ and ˆR |vac⟩ up to second order. This +is a consequence of our restriction to FGS. Therefore, we find two equations +ˆL0 |vac⟩ =0, +(A19) +ˆL2 |vac⟩ = ˆR2 |vac⟩ , +(A20) +from which we wish to derive the equations of motion of the ITE. We first consider Eq. (A20), +: ˆaT UT +ξ +dUξ +dτ ˆa : |vac⟩ = − i : ˆaT UT +ξ H(mf)Uξˆa : |vac⟩ . +(A21) +For any normal-ordered polynomial of fermionic operators applied to the vacuum state, the only terms that do not +vanish are polynomials which exclusively contain fermionic creation operators. Therefore, we define the vector +r = ˆc† ⊗ +� +1 +i +� +, +(A22) +where ˆc† = (ˆc† +1, . . . , ˆc† +N), and rewrite Eq. (A21) in terms of fermionic creation and annihilation operators, which leads +to +ˆrT UT +ξ +dUξ +dτ ˆr |vac⟩ = − iˆrT UT +ξ H(mf)Uξˆr |vac⟩ , +(A23) + +11 +where the normal-ordering “: :” may now be dropped. We would now like to compare the matrices of Eq. (A23). +However, before doing so, we need to take into account the symmetry operations which leave the operator ˆr invariant. +For this, we first rewrite Υ defined in Eq. (A5) as Υ = 1N ⊗ +� 0 +1 +−1 0 +� +. Thus, the symmetry operations on the operators +are given by −iΥˆr = ˆr and iˆrT Υ = ˆrT . The real-valued skew-symmetric solution for dUξ +dτ +which satisfies Eq. (A19) +is then given by +dUξ +dτ += −1 +2ΓH(mf)Uξ − 1 +2H(mf)UξΥ. +(A24) +For the derivative of the covariance matrix Γ we find then +dΓ +dτ = − dUξ +dτ ΥUT +ξ − UξΥ +dUT +ξ +dτ += −H(mf) − ΓH(mf)Γ, +(A25) +which is identical to Eq. (8) using Γ2 = −1. +Appendix B: Best quadratic approximation +In this section we will show Eq. (A16), which can be derived using Wick’s theorem. In particular, we will derive +this formula for an arbitrary Hamiltonian which is a sum of even products of Majorana operators. By the application +of normal-ordering onto a polynomial ˆp(k) of order k we understand the sum of normal-ordered monomials ˆm(l) of +order l ≤ k, in other words : ˆp(k) := �k +l=0 : ˆm(l) :. Therefore, it is sufficient to show the relation (A16) for arbitrary +even products of Majorana operators. Without loss of generality we number the Majorana operators from 1, ..., 2n +with n ∈ N. Using normal-ordering and Wicks theorem we can write the product ˆA = ˆa1ˆa2 . . . ˆa2n (a monomial of +Majorana operators) in the following way +ˆA = ⟨vac| ˆA |vac⟩ + +� +iF +Co +WWvalues. Then, when computing the loss function, we choose the smaller of the two. This +will ensure that we do not overestimate the target Q value. +• Delayed policy update: Unlike DDPG, we added a delay to the actor-network parameter +update in TD3. While the parameters of the critic networks are updated at each step of the +episode, the actor-network (policy network) is delayed and updated after every two steps. +• Target policy smoothing: In DDPG, the algorithm generates distinct target values for +identical actions. As a result, the target’s variance would be high. Thus, we reduce variance +by adding noise to the target action. + +This section provides additional information about Twin Delayed Deep Deterministic Policy +Gradients (TD3). In TD3, we employ six artificial neural networks, four of which are critic +networks and two of which are two-actor networks. +• The main critic neural network parameters are denoted by 𝜃1 and 𝜃2 +• The two target critic neural network parameters are represented by 𝜃1′and 𝜃2′ +• The main actor-network parameter is denoted by 𝜙 +• The target actor-network parameter is denoted by 𝜙′ + +Initially, we must initialize the two main critic network parameters, 𝜃1 and 𝜃2, and the main critic +network parameter 𝜙 with random values. Since the target network parameter is only a copy of the +main network parameter, the two target critic network parameter 𝜃1′ and 𝜃2′ by copying 𝜃1 and +𝜃2, are simply initialized, respectively. Similarly, we initialize the target actor parameter 𝜙′, by +just copying the main actor-network parameter 𝜙 . Also, the replay buffer is initialized too. Now, +we select action a using the actor-network: +( ) +a +s + + += +. Rather than selecting the action directly, +some noise  is added to ensure exploration when ~ +(0, +) +N + + +. Accordingly, the output action is +expressed as follows: +( ) +a +s + + += ++ (16). +Then, after performing action a, we move to the next state s and obtain reward r . This transition +information is stored in the replay buffer. During the next step, we randomly sample a minibatch +of K transition ( , , , ) +s a r s from the replay buffer. The K transition will be used for updating both +the critic and actor network. +The loss function of the critic network is expressed as follows: + +2 +1 +( +) +( +( , +)) +1,2 +j +j +i +i +i +J +y +Q +s a +for j +k + + += + +− += + +(17) + +In the preceding equation, the following applies: +The action +ia is the action produced by the actor-network as follows: + + + + + + + + + + + + + +( ) +ia +s + + += + +(18) + +𝑦𝑖 is the target value of the critic, that is +1,2 +min +( , ) +j +i +i +j +y +r +Q +s a + + += + + += ++ + +(19) + +, and the action a is the action produced by the target critic network: +( ) +~ ( +(0, +), +, +) +i +a +s +where +N +c +c + + + + + += ++ + +− ++ + +(20) + +After computing the loss of the critic network, the gradients +( +) +j +j +J + + + + is computed, and then the +critic network parameters will be updated using the gradient descent method. + + + + + + + + + +( +) +1,2 +j +j +j +j +J +for j + + + + + += +−  += + +(21) + +Now, the actor network must be updated. The objective function of the actor-network is as follows: + + + + + + + + + + +1 +( ) +( , ) +i +i +i +J +Q +s a +k + + =  + +(22) + +It must be noted that in Eq. 22, we only use state (si) from the sampled K transitions ( , , , ) +s a r s . +The action a is selected by the actor-network. +( ) +ia +s + + += + +(23) + + In order to maximize the objective function, the gradient of the objective function +( ) +J + + + + is +computed, and the parameters of the network are updated using the gradient ascent: + + + + + + + + + + + +( ) +J + + + + + += ++  + +(24) + +We delay the update rather than updating the actor-parameters networks at each time step of the +episode. If the time step of the episode is denoted by t and d denotes the number of time steps +which we want to delay the update, it can be described as follows: + +1- If t mod d=0, then: +1. Compute the gradient of the objective function +( ) +J + + + + +2. Update the actor-network parameter using the gradient ascent method (24). +Finally, the parameters of the target critic network 𝜃1′ and 𝜃2′and the parameters of the target +actor-network 𝜙′ will be updated via soft replacement: + + + + + + + + +{𝜃𝑗 +′ = 𝜏𝜃𝑗 + (1 − 𝜏)𝜃𝑗 +′ 𝑓𝑜𝑟 𝑗 = 1,2 +𝜙′ = 𝜏𝜙𝑗 + (1 − 𝜏)𝜙′ + +(25) + +There is a small change in updating the parameters of the target networks. As with the actor- +network parameter, we delay updating it for d steps; similarly, we update the target network +parameters for every d step; in this case, we can say: + +1- If t mod d=0, then: +1. Compute the gradient of the objective function 𝛻𝜙𝐽(𝜙) and update the actor- +network parameter using gradient ascent. +𝜙 = 𝜙 + 𝛼𝛻𝜙𝐽(𝜙) +(26) + + +2. Update the target critic network parameter and target actor-network parameter as +(1 +) +1,2 +j +j +j for j + + +  + + += ++ +− += + +(27) +and +(1 +) + + +  + + += ++ +− + +(28) +, respectively. + +The previous steps for several episodes must be repeated to improve the policy. The following +pseudocode is prepared to understand better how TD3 works: + + + + +Fig. 3. TD3 Algorithm pseudocode + +2.2. TD3 application in closed-loop vibration control for a semi-active suspension system +The main inputs for vehicle dynamics are road disturbance profile and damping force produced by +the MR device. The outputs are body (sprung mass) acceleration and suspension working space +(SWS). The RL-Agent inputs for implementing a controller for a one-quarter suspension system +are body acceleration, denoted by 𝑞, and a reward function, which is as follows: + + + + + + + + + +𝑟 = {0 𝑖𝑓 𝑞 = 𝑞𝑔𝑜𝑎𝑙 = 0 +−𝑘𝑞2 𝑖𝑓 𝑞 ≠ 0 + + +(29) + +Where k is a hyperparameter that specifies the intensity coefficient for agent punishment, the agent +punishments experienced in the replay buffer are also exerted in the RL-agent. The damper’s input +voltage must be applied to the coil via the RL-agent output (action). TD3 analyzes its performance +by measuring body acceleration and fine-tuning its action to road profile disturbances. The + +Twin Delayed Deep Deterministic Policy Gradient Algorithm: +Initialize critic networks Qe1, Qe2, and actor network Ts +withrandomparameters01.02.Φ +Initializetargetnetworks←1%←2,'←Φ +2 +InitializereplaybufferB +for t=1 toTdo +3 +Selectactionwithexplorationnoisea~π(s)+E +e~N(O,o)and observe reward r and new states +4 +Store transition tuple (s, a,r,s in B +Samplemini-batchofNtransitions(s,a,r,sfromB +a ← π(s) + E, E ～ clip(N(O,), -C,c) +5 +y ← r +mini=1.2Qe(s',a) +Update critics i ← argming. N-1 (y-Qe, (s, a))2 +iftmoddthen +7 +Update by the deterministic policy gradient: +VJ(d) = N-i VaQe (s,a)lg +元(s)V(s) +Updatetargetnetworks: +8 +, ← T+ (1 - T)O +Φ← +(1-) +end if +end forsurrounding environment consists of a vehicle suspension system equipped with an MR-damper +and a road profile. The agent is a neural network that constructs the controller part. The +hyperparameters listed in Table 3 pertain to TD3 agents used in closed-loop suspension control. +Figures 5 and 6 detail the neural networks used in the TD3 algorithm. +The hidden size is 400 and 300 neurons in each layer for the actor and critic network, respectively. +The optimization method is Adam for both actor and critic networks. The discount factor 𝛾 is 0.8. +The input voltage varies from 0 to 3 V, and the damping force varies from -1.5 to 1.5 KN. +Table 3. Neural Networks Parameters in TD3 +Network +Learning Rate +Optimizer +Delay for update +Actor +0.002 +Adam +2 +Critic +0.002 +Adam +1 +Actor Target +0.006 +- +2 +Critic Target +0.006 +- +2 + + + +Fig. 4. Closed-loop semi-active suspension system block diagram with DRL Agent + + + +Road +Profile +qt +F +Vehicle +MR Damper +Model +Deep +Reinforcement +Learning +action by policy μ,voltage +Suspension Working Space(SWws) +Agent (TD3) +Reward Function: r, = -kq +3+(IB II'D))+= + +Replay Buffer +Fig. 5. Actor-critic architecture in TD3 Agent + + +Agent +Critic Network +Actor Network +Qe (stat) +Random +△Js TD error update +△Jμ DPG update +noise +εE N(O,o) +at +Vehicle +0 +Critic 2 +Critic 1 +at +Model +' Target +target2 +target1 +Behavior +target +policy +Compared target y + +Mini batch +qt llqgoal , at, qt+1 lqgoal , rt +ReplayBuffer +Parameter +update +Data flow +neural network +Global Memory +HER +Fig. 6. The details of actor-critic networks in semi-active suspension control + +3. Results and Discussion +Vertical body acceleration (BA), Dynamic Tire Load (DTL), and Suspension Working space are +three primary performance criteria used in the suspension design of a vehicle to improve ride +comfort and road holding, preventing the suspension system from bottoming out excessively. To +achieve the objectives mentioned above, we should reduce BA or SWS to improve ride comfort +and DTL to improve road holding and minimize suspension space distance. The BA has been +chosen as the objective function in this article. +Two types of controllers for semi-active suspension are investigated in this section: the RL-based +TD3 and the PID controller; their gains are tuned using the Particle Swarm Optimization (PSO) +algorithm from [17], as well as an uncontrolled system (MR-Passive) with no applied voltage to +the damper’s coil. +A particular type of bump-road excitation was chosen due to its resemblance to actual road profiles. +Additionally, the bump-road profile demonstrates the characteristics of transient response. The +road-bump profile has been proposed in [27] as: + + +Critic +0;=1,2 +Ao VeQe,(quHe(qt Il qgoal) +Actor Φ +qt,at +Qe i-1, (qt, at) ++ μp (qt+1 Il qgoal) +a, = μp(qt+1 Il qgoat) + +qt Il qgoal +A; (t - Qe: (qt.at)? +Replay Buffer +Yt = r + ymini=12Qe(qt+1, at) +(qt Il qgoal , t, qt+1, Il qgoal , rt)qt+: +Qg i=12 (q+1 +at) +Hp(qt+1 Il qgoal ) +qt+1 I qgoal +at +Critic target e't=1,2 +Actor target $' +at = μs'(qt+1 I qgoat) + s + + + + + 𝑋𝑟 = {𝑎 {1 − 𝑐𝑜𝑠( 𝜔𝑟(𝑡 − 0.5))}, 𝑓𝑜𝑟0.5 ≤ 𝑡 ≤ 0.5 + +𝑑𝑏 +𝑉𝑐 +0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 + +(1) + + +Where a denotes half of the bump amplitude, +2 +c +r +b +V +d + + += + , db is the bump width, and Vc denotes +the vehicle velocity. In this research, 𝑎 = 0.035𝑚, 𝑑𝑏 = 0.8𝑚, 𝑉𝑐 = 0.856 +𝑚 +𝑠 , derived from [27]. + + +Fig. 7. Road-bump profile + +Table 4 compares the results from the TD3 controller to an uncontrolled system (applied zero +voltage), and Table 5 compares the results from the PSO PID controller to the TD3. +The results demonstrate unequivocally that the DRL-based controller (TD3) algorithm +outperforms PSO-tuned PID. TD3 is excellent at dissipating vibrations caused by bump excitation. +Additionally, it reduces settling time and enhances road holding and ride comfort. +DRL TD3 significantly reduces body acceleration, suspension working space, and Dynamic Tire +Load compared to an uncontrolled suspension system by 35.8%, 68.5%, and 33.6%, respectively. +Simultaneously, PSO-PID reduces those criteria by 32.2%, 50%, and 12.4%, respectively, +compared to MR passive (uncontrolled suspension). + + +0.07 +0.06 +0.05 + (m) +0.04 +Road +0.03 +0.02 +0.01 +0 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time (sec) +Fig. 7. Body Acceleration (BA) + + + +Fig. 8. Dynamic Tire Load (DTL) + +..Uncontrolled +3 +-PSO-PID +DRL-TD3 +Acceleration(m/s) +Body, +-2 +-3 +4 +Q +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time (sec)1500 +uncontrolled +-PSO-PID +1000 +DRL-TD3 +Dynamic Tyre Load (N) +500 +-500 +-1000 +-1500 +- +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time (sec) +Fig. 9. Suspension Working Space (SWS) + +Table 4. RMS Values for the suspension system criteria with TD3 algorithm and Uncontrolled +Suspension System +RMS-values +Body +Acceleration(𝒎/𝒔𝟐) +Suspension Working +Space(𝒎) +Dynamic Tire Load (𝑵) +DRL-TD3 + +0.97 + +0.0084 + +381.32 + +Uncontrolled + +1.5179 + +0.0267 + +537.7 + + +Table 5. Comparison of DRL-TD3 with PSO-PID +Controller Type +Body +Acceleration(𝒎/𝒔𝟐) +Suspension +Working +Space(𝒎) +Dynamic Tire Load (𝑵) +DRL-TD3 +35.8 % +68.53 % +33.6% +PSO-PID +22.4% +46.1% +11.9% +Improvement +13.4% +22.43% +21.7% + +4. Conclusion +The paper proposes the DRL-based TD3 algorithm for vibration mitigation in a semi-active +suspension system equipped with an MR damper. Three major criteria, BA, SWS, and DTL, +were considered and investigated when evaluating the proposed controller. Instead of using two +distinct controllers for damper control and force estimation, a single universal controller based on +real-time learning was used. The time-domain results were analyzed, and the effectiveness of the + +0.06 + uncontrolled +PSO-PID +DRL-TD3 +0.04 +0.02 +(w) s +swS +-0.02 +-0.04 +-0.06 +1 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time (sec)DRL-based controller was compared to the optimal PSO-PID controller. Finally, the results +demonstrate that the TD3 algorithm outperforms the PSO-tuned PID controller. +Declaration of competing interest +The authors declare that they have no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. + +References +[1] +A. H. F. Lam and W. H. Liao, “Semi-active control of automotive suspension systems with +magnetorheological dampers,” Int. J. Veh. Des., vol. 33, no. 1–3, pp. 50–75, 2003, doi: +10.1504/ijvd.2003.003652. +[2] +M. A. Karkoub and M. Zribi, “Active/semi-active suspension control using magnetorheological +actuators,” Int. J. Syst. Sci., vol. 37, no. 1, pp. 35–44, Jan. 2006, doi: 10.1080/00207720500436344. +[3] +S. J. Dyke, B. F. Spencer, M. K. Sain, and J. D. Carlson, “Modeling and control of +magnetorheological dampers for seismic response reduction,” Smart Mater. Struct., vol. 5, no. 5, +pp. 565–575, Oct. 1996, doi: 10.1088/0964-1726/5/5/006. +[4] +K. A. Bani-Hani and M. A. Sheban, “Semi-active neuro-control for base-isolation system using +magnetorheological (MR) dampers,” Earthq. Eng. Struct. Dyn., vol. 35, no. 9, pp. 1119–1144, Jul. +2006, doi: 10.1002/eqe.574. +[5] +W. H. Liao and D. H. Wang, “Semiactive Vibration Control of Train Suspension Systems via +Magnetorheological Dampers,” J. Intell. Mater. Syst. Struct., vol. 14, no. 3, pp. 161–172, Mar. 2003, +doi: 10.1177/1045389X03014003004. +[6] +H. Du, K. Yim Sze, and J. Lam, “Semi-active H∞ control of vehicle suspension with +magnetorheological dampers,” J. Sound Vib., vol. 283, no. 3–5, pp. 981–996, May 2005, doi: +10.1016/j.jsv.2004.05.030. +[7] +S. B. Choi, H. S. Lee, and Y. P. Park, “H∞ control performance of a full-vehicle suspension featuring +magnetorheological dampers,” Veh. Syst. Dyn., vol. 38, no. 5, pp. 341–360, Nov. 2002. +[8] +S. B. Choi and K. G. Sung, “Vibration control of magnetorheological damper system subjected to +parameter variations,” Int. J. Veh. Des., vol. 46, no. 1, pp. 94–110, 2008, doi: +10.1504/IJVD.2008.017071. +[9] +H.-S. Lee and S.-B. Choi, “Control and Response Characteristics of a Magnetorheological Fluid +Damper for Passenger Vehicles,” J. Intell. Mater. Syst. Struct., vol. 11, no. 1, pp. 80–87, Jan. 2000, +doi: 10.1106/412A-2GMA-BTUL-MALT. +[10] +M. Ahmadian and C. A. Pare, “A Quarter-Car Experimental Analysis of Alternative Semiactive +Control Methods,” J. Intell. Mater. Syst. Struct., vol. 11, no. 8, pp. 604–612, Aug. 2000, doi: +10.1106/MR3W-5D8W-0LPL-WGUQ. +[11] +Y. Shen, M. F. Golnaraghi, and G. R. Heppler, “Semi-active Vibration Control Schemes for +Suspension Systems Using Magnetorheological Dampers,” J. Vib. Control, vol. 12, no. 1, pp. 3–24, +Jan. 2006, doi: 10.1177/1077546306059853. +[12] +D. L. Guo, H. Y. Hu, and J. Q. Yi, “Neural Network Control for a Semi-Active Vehicle Suspension + +with a Magnetorheological Damper,” J. Vib. Control, vol. 10, no. 3, pp. 461–471, Mar. 2004, doi: +10.1177/1077546304038968. +[13] +M. Zribi and M. Karkoub, “Robust Control of a Car Suspension System Using Magnetorheological +Dampers,” J. Vib. Control, vol. 10, no. 4, pp. 507–524, Apr. 2004, doi: 10.1177/1077546303039697. +[14] +A. Elsawaf, F. Ashida, and S. I. Sakata, “Optimum structure design of a multilayer piezo-composite +disk for control of thermal stress,” J. Therm. Stress., vol. 35, no. 9, pp. 805–819, Sep. 2012, doi: +10.1080/01495739.2012.689233. +[15] +H. A. Metered, “Application of Nonparametric Magnetorheological Damper Model in Vehicle +Semi-active Suspension System,” SAE Int. J. Passeng. Cars - Mech. Syst., vol. 5, no. 1, pp. 715– +726, Apr. 2012, doi: 10.4271/2012-01-0977. +[16] +S. Gad, H. Metered, A. Bassuiny, and A. M. Abdel Ghany, “Vibration control of semi-active MR +seat suspension for commercial vehicles using genetic PID controller,” in Mechanisms and Machine +Science, 2015, vol. 23, pp. 721–732, doi: 10.1007/978-3-319-09918-7_64. +[17] +H. Metered, A. Elsawaf, T. Vampola, and Z. Sika, “Vibration Control of MR-Damped Vehicle +Suspension System Using PID Controller Tuned by Particle Swarm Optimization,” SAE Int. J. +Passeng. Cars - Mech. Syst., vol. 8, no. 2, pp. 426–435, Jul. 2015, doi: 10.4271/2015-01-0622. +[18] +J. Yao, S. Taheri, S. Tian, Z. Zhang, and L. Shen, “A novel semi-active suspension design based on +decoupling skyhook control,” J. Vibroengineering, vol. 16, no. 3, 2014. +[19] +Y. YU, C. ZHOU, L. ZHAO, Y. XING, … P. S.-J. of S., and undefined 2017, “Design of LQG +controller for vehicle active suspension system based on alternate iteration,” en.cnki.com.cn, +Accessed: Apr. 18, 2021. [Online]. Available: https://en.cnki.com.cn/Article_en/CJFDTotal- +SDGY201704009.htm. +[20] +D. Wang, H. W.-C. M. Engineering, and undefined 2017, “Control method of vehicle semi active +suspensions based on variable universe fuzzy control,” en.cnki.com.cn, Accessed: Apr. 18, 2021. +[Online]. Available: https://en.cnki.com.cn/Article_en/CJFDTotal-ZGJX201703019.htm. +[21] +J. L. Yao, W. K. Shi, J. Q. Zheng, and H. P. Zhou, “Development of a sliding mode controller for +semi-active vehicle suspensions,” JVC/Journal Vib. Control, vol. 19, no. 8, pp. 1152–1160, Jun. +2013, doi: 10.1177/1077546312441045. +[22] +M. Canale, M. Milanese, and C. Novara, “Semi-active suspension control using ‘fast’ model- +predictive techniques,” IEEE Trans. Control Syst. Technol., vol. 14, no. 6, pp. 1034–1046, Nov. +2006, doi: 10.1109/TCST.2006.880196. +[23] +D. Wang, D. Zhao, M. Gong, and B. Yang, “Research on Robust Model Predictive Control for +Electro-Hydraulic Servo Active Suspension Systems,” IEEE Access, vol. 6, pp. 3231–3240, Dec. +2017, doi: 10.1109/ACCESS.2017.2787663. +[24] +“Neural network control method of automotive semi-active air suspension--《Journal of Traffic and +Transportation +Engineering》2006年04期.” +https://en.cnki.com.cn/Article_en/CJFDTotal- +JYGC200604014.htm (accessed Apr. 18, 2021). +[25] +Y. Qin, J. J. Rath, C. Hu, C. Sentouh, and R. Wang, “Adaptive nonlinear active suspension control +based on a robust road classifier with a modified super-twisting algorithm,” Nonlinear Dyn., vol. +97, no. 4, pp. 2425–2442, Sep. 2019, doi: 10.1007/s11071-019-05138-8. +[26] +L. Ming, L. Yibin, R. Xuewen, Z. Shuaishuai, and Y. Yanfang, “Semi-active suspension control +based on deep reinforcement learning,” IEEE Access, vol. 8, pp. 9978–9986, 2020, doi: + +10.1109/ACCESS.2020.2964116. +[27] +S. B. Choi, Y. T. Choi, and D. W. Park, “A sliding mode control of a full-car electrorheological +suspension system via hardware in-the-loop simulation,” J. Dyn. Syst. Meas. Control. Trans. ASME, +vol. 122, no. 1, pp. 114–121, Mar. 2000, doi: 10.1115/1.482435. +[28] +C. Y. Lai and W. H. Liao, “Vibration Control of a Suspension System via a Magnetorheological +Fluid Damper,” J. Vib. Control, vol. 8, no. 4, pp. 527–547, Apr. 2002, doi: +10.1177/107754602023712. + + + diff --git a/UtE0T4oBgHgl3EQf2gK9/content/tmp_files/load_file.txt b/UtE0T4oBgHgl3EQf2gK9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdd4b8d33fe7a9cbccce6f8e25b28995d3b25798 --- /dev/null +++ b/UtE0T4oBgHgl3EQf2gK9/content/tmp_files/load_file.txt @@ -0,0 +1,654 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf,len=653 +page_content='A Deep Reinforcement Learning-Based Controller for Magnetorheological-Damped Vehicle Suspension AmirReza BabaAhmadi1 , Masoud ShariatPanahi2 , Moosa Ayati3 1,2,3 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran Abstract This paper proposes a novel approach to controller design for an MR-damped vehicle suspension system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' This approach is predicated on the premise that the optimal control strategy can be “learned” through real-world or simulated experiments utilizing a reinforcement learning algorithm with continuous states/actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The sensor data is fed into a Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm, which generates the actuation voltage required for the MR damper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The resulting suspension working space (displacement), sprung mass acceleration, and dynamic tire load are calculated using a quarter vehicle model incorporating the modified Bouc-Wen MR damper model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Deep RL’s reward function is based on sprung mass acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The proposed approach outperforms traditional suspension control strategies regarding ride comfort and stability, as demonstrated by multiple simulated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Keywords: Magnetorheological-damped suspension, ride comfort, deep reinforcement learning 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Introduction As one of the most critical components of the vehicle, the suspension system significantly improves ride comfort and road holding, preventing damage and reducing passenger fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Suspension systems are classified as passive, active, or semi-active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Due to the fixed and unchanging nature of passive suspension parameters, passive suspension cannot guarantee ride comfort and stability when the environment or suspension parameters vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Thus, active and semi- active suspension systems with tunable parameters can compensate for the limitations mentioned above of passive suspension systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The uncertainty inherent in the suspension system’s road roughness and parameter variation in real applications is unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' MR fluid dampers are adaptive controllable devices that have garnered significant interest due to their simplicity, low power consumption, and high capacity force, among other characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' MR dampers have found widespread application in a variety of industries, including the automotive industry [1][2], civil structures [3][4], and railway vehicles [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' MR dampers are semi- active devices because we can simply apply voltage to their coils rather than using a mechanism for the damper to produce damping force directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' As a result, two distinct controller types are required to regulate semi-active suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' First, the system controller calculates the required damping force to ensure ride comfort and road holding simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The system controller inputs are derived from the suspension system’s state feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Second, the damper controller determines the voltage that should be applied to the MR-damper for its current force to track the force specified by the system controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Numerous control techniques for semi-active suspension systems have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 𝐻∞ control [6], [7], [8], skyhook control [9], [10], [11], adaptive control based on neural networks [12], robust control [13], LQG control [14], [15], and optimal PID control [16][17] are some of the studies that have been conducted to develop an efficient controlling system for improving ride comfort and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' According to previous research, PID controllers have a simple structure but are ineffective in semi-active suspension systems with uncertain parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Skyhook suspension control is a classic semi-active suspension control method described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' It features simple structures, straightforward implementation, and acceptable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' On the other hand, the time delay significantly affects suspension performance, resulting in instability and wheel jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The authors of [19] designed an optimal LQG controller, but the control parameters were calculated by ignoring uncertain factors in the system modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' When the system’s parameters change sufficiently, the system becomes unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The authors of [20] aimed to address the issue of the blind design of the fuzzy controller, where this was, in fact, a PID controller, and a PID was designed using rule description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Indeed, developing a proper fuzzy control system necessitates extensive knowledge of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The sliding mode controller in [21] shows good performance and robust behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' On the other hand, chattering is a fatal flaw in sliding mode control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Chattering may result in instability due to the controlled system’s activated high-frequency modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Although the adaptive controller in [12] improves ride comfort, it performs poorly in road holding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Reference [22] proposed a fast model prediction controller (FMPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The authors of [23] proposed a robust model prediction controller (RMPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Both of the aforementioned predictive controllers utilized road models in advance of designing controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The primary goal of developing a predictive controller is to provide highly accurate information from the system, which is impossible when driving on various roads with high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' In [24], it is demonstrated that neural-networked-based controllers can solve complex and nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' However, like many other supervised learning algorithms, neural networks require a large number of labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' When developing the control effort, only the current state is considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' future states are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' These critical issues impose constraints on the use of neural network controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Reference [25] proposes a novel nonlinear adaptive smart controller based on the classification of road profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The primary disadvantage of this method is that a new classifier must be constructed when the control strategy changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' With the advent of Deep Learning (DL) techniques that overcome several of the significant limitations of classical machine learning algorithms, some researchers attempted to apply DL algorithms to the suspension control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The authors of [26] propose a DDPG-based algorithm for a particular type of suspension system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The primary drawback of this algorithm is its inability to locate the globally optimal solution to a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Two types of controllers are required to control a semi-active suspension system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A system controller analyzes the system’s feedback samples and predicts the desired damping force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The damper controller receives the system controller’s predicted force and suspension system displacement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' it then predicts the applied voltage exerted on damper coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The paper proposes an improved DRL controller that combines a system controller and a damper controller to provide the best ride comfort and stability possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Notably, MR-based suspension systems operate in two modes: open-loop mode (without the use of a controller) operates with a constant damping coefficient, similar to passive suspension systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Both the system and damper controllers work in a closed-loop system to adjust the damping force in response to road conditions by tuning the required damper’s voltage input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' MR-damped suspension model A simplified quarter-vehicle model with a semi-active suspension is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, where mb represents the vehicle’s body mass, mw represents the wheel mass, and xb and xw denote body displacement and wheel displacement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The road profile is denoted by xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The spring stiffness of the suspension system is ks, and the tire spring stiffness is denoted by kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' We exclude tire damping from this study due to its negligible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Table 1 contains parameters taken from [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Newton’s second law is applied to the quarter-model of the vehicle to derive the following equations: 𝑚𝑏𝑋̈𝑏 + 𝐾𝑠(𝑋𝑏 − 𝑋𝑤) + 𝑓 = 0 (1) 𝑚𝑤𝑋̈𝑤 − 𝐾𝑠(𝑋𝑏 − 𝑋𝑤) + 𝐾𝑡(𝑋𝑤 − 𝑋𝑟) − 𝑓 = 0 (2) m, oassive m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='w semi-activeFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A quarter-car model with two operational modes for a semi-active suspension system (passive and semi-active) Where the following equation gives the damping force generated by the MR device: 𝑓 = {𝐶𝑠(𝑋̇𝑏 − 𝑋̇𝑤) for passive suspension 𝑓 𝑀𝑅 for semi − active suspension (3) Cs is a coefficient describing the MR passive suspension mode of operation in an MR-damper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The state-space representation of the semi-active suspension system is prepared as follows [17]: 𝑤̇ = 𝐴𝑤 + 𝐵𝑓𝑀𝑅 + 𝐷𝑥𝑟 (4) 𝑤 = [𝑥𝑏 𝑥𝑤 𝑥̇𝑏 𝑥̇𝑤]𝑇 (5) 𝐴= [ 0 0 0 0 1 0 0 1 − 𝐾𝑠 𝑚𝑠 𝐾𝑠 𝑚𝑠 𝐾𝑠 𝑚𝑤 − 𝐾𝑠+𝐾𝑡 𝑚𝑤 0 0 0 0] (6) 𝐵 = [0 0 − 1 𝑚𝑠 1 𝑚𝑤] 𝑇 (7) 𝐷 =[0 0 0 𝐾𝑡 𝑚𝑤] 𝑇 (8) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Semi active suspension parameters [27] Value (Unit) Symbol Parameter 375(kg) 𝑚𝑏 Vehicle body mass 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 (kg) 𝑚𝑤 Vehicle wheel mass 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='58 (KN/m) 𝑘𝑠 Suspension stiffness 772 (Ns/m) 𝐶𝑠 Damping coefficient 200 (KN/m) 𝑘𝑡 Tire stiffness Magnetorheological damper force is denoted by fmr depending on the time series of the external voltage to its magnetic coil V and relative displacement x=xb-xw, which is known as suspension working space (SWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' fmr is computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 9, the modified Bouc-Wen model, derived from [28] and incorporated into the suspension system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 𝐹(𝑡) = 𝑐1𝑦̇ + 𝑘1(𝑥 − 𝑥0) (9) 𝑧̇ = −𝛾|𝑥̇ − 𝑦̇||𝑧||𝑧|𝑛−1 − 𝛽(𝑥̇ − 𝑦̇)|𝑧|𝑛 + 𝐴(𝑥̇ − 𝑦̇) (10) 𝑦̇ = 1 𝑐1+𝑐0 {𝛼𝑧 + 𝑘0(𝑥 − 𝑦) + 𝑐0𝑥̇} (11) 𝛼 = 𝑎𝑎 + 𝑎𝑏𝑢 (12) 𝑐1 = 𝑐1𝑎 + 𝑐1𝑏𝑢 (13) 𝑐0 = 𝑐0𝑎 + 𝑐0𝑏𝑢 (14) 𝑢̇ = −𝜂(𝑢 − 𝑣) (15) The internal movement of the MR-damper is denoted as y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' u is the output signal from a first-order filter, and 𝑧 is a parameter that guarantees the hysteretic behavior of MR fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Accumulator stiffness is denoted by kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' C0 and C1 represent viscous damping at high and low velocity for the MR-damper, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The stiffness regulator at high damper velocity is denoted by K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' MR damper accumulator simulation is performed via X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The scale and shape of the hysteresis behavior of the MR-damper parameters for the simulation are 𝛾, 𝛽, 𝛿 and 𝜂, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Modified Bouc-Wen model for the MR-damper Parameters for the modified Bouc-Wen MR-damper model are presented in Table 2 (adapted from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=') Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Modified Bouc-Wen parameters for the MR-damper [28] Parameter Value Parameter Value 𝑐0𝑎 784 ( 1) Nsm − 𝛼𝑏 38430 ( 1) ( 1) NsV m − − 𝑐0𝑏 1803 ( 1) ( 1) NsV m − − 𝛽 2059020 2 m− 𝑐1𝑎 14649 ( 1) Nsm − 𝛾 136320 2 m− 𝑐1𝑏 34622 ( 1) ( 1) NsV m − − 𝜂 190 1 s− 𝑘1 840 ( 1) Nm − A 58 𝑘0 3610 ( 1) Nm − n 2 𝛼𝑎 12441 ( 1) Nm − 𝑥0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='245 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control Strategy based on DRL-TD3 As previously stated, the Twin Delayed Deep Deterministic Policy Gradient (TD3) was used as the universal controller for the MR-damped suspension system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' TD3 is identical to DDPG but incorporates three additional features to address DDPG issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' • Clipped double Q learning: In TD3, we compute the Q value using two critic networks and the target value using two target networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' We used only one critic and one target network in DDPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' In TD3, we use two target critic networks to compute two target Q Bouc-Wen ko >F Co WWvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Then, when computing the loss function, we choose the smaller of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' This will ensure that we do not overestimate the target Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' • Delayed policy update: Unlike DDPG, we added a delay to the actor-network parameter update in TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' While the parameters of the critic networks are updated at each step of the episode, the actor-network (policy network) is delayed and updated after every two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' • Target policy smoothing: In DDPG, the algorithm generates distinct target values for identical actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' As a result, the target’s variance would be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Thus, we reduce variance by adding noise to the target action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' This section provides additional information about Twin Delayed Deep Deterministic Policy Gradients (TD3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' In TD3, we employ six artificial neural networks, four of which are critic networks and two of which are two-actor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' • The main critic neural network parameters are denoted by 𝜃1 and 𝜃2 • The two target critic neural network parameters are represented by 𝜃1′and 𝜃2′ • The main actor-network parameter is denoted by 𝜙 • The target actor-network parameter is denoted by 𝜙′ Initially, we must initialize the two main critic network parameters, 𝜃1 and 𝜃2, and the main critic network parameter 𝜙 with random values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Since the target network parameter is only a copy of the main network parameter, the two target critic network parameter 𝜃1′ and 𝜃2′ by copying 𝜃1 and 𝜃2, are simply initialized, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Similarly, we initialize the target actor parameter 𝜙′, by just copying the main actor-network parameter 𝜙 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Also, the replay buffer is initialized too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Now, we select action a using the actor-network: ( ) a s \uf066 \uf06d = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Rather than selecting the action directly, some noise \uf0ce is added to ensure exploration when ~ (0, ) N \uf073 \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Accordingly, the output action is expressed as follows: ( ) a s \uf066 \uf06d = +\uf0ce (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Then, after performing action a, we move to the next state s\uf0a2 and obtain reward r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' This transition information is stored in the replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' During the next step, we randomly sample a minibatch of K transition ( , , , ) s a r s\uf0a2 from the replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The K transition will be used for updating both the critic and actor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The loss function of the critic network is expressed as follows: 2 1 ( ) ( ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' )) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2 j j i i i J y Q s a for j k \uf071 \uf071 = \uf053 − = (17) In the preceding equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' the following applies: The action ia is the action produced by the actor-network as follows: ( ) ia s \uf066 \uf06d = (18) 𝑦𝑖 is the target value of the critic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' that is 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2 min ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ) j i i j y r Q s a \uf071 \uf067 = \uf0a2 \uf0a2 = + (19) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' and the action a is the action produced by the target critic network: ( ) ~ ( (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ) i a s where N c c \uf066 \uf06d \uf073 \uf0a2 \uf0a2 = +\uf0ce \uf0ce − + (20) After computing the loss of the critic network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' the gradients ( ) j j J \uf071 \uf071 \uf0d1 is computed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' and then the critic network parameters will be updated using the gradient descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ( ) 1,2 j j j j J for j \uf071 \uf071 \uf071 \uf061 \uf071 = − \uf0d1 = (21) Now, the actor network must be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The objective function of the actor-network is as follows: 1 ( ) ( , ) i i i J Q s a k \uf071 \uf066 = \uf0e5 (22) It must be noted that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 22, we only use state (si) from the sampled K transitions ( , , , ) s a r s\uf0a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The action a is selected by the actor-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ( ) ia s \uf066 \uf06d = (23) In order to maximize the objective function, the gradient of the objective function ( ) J \uf066 \uf066 \uf0d1 is computed, and the parameters of the network are updated using the gradient ascent: ( ) J \uf066 \uf066 \uf066 \uf061 \uf066 = + \uf0d1 (24) We delay the update rather than updating the actor-parameters networks at each time step of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' If the time step of the episode is denoted by t and d denotes the number of time steps which we want to delay the update, it can be described as follows: 1- If t mod d=0, then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Compute the gradient of the objective function ( ) J \uf066 \uf066 \uf0d1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Update the actor-network parameter using the gradient ascent method (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Finally, the parameters of the target critic network 𝜃1′ and 𝜃2′and the parameters of the target actor-network 𝜙′ will be updated via soft replacement: {𝜃𝑗 ′ = 𝜏𝜃𝑗 + (1 − 𝜏)𝜃𝑗 ′ 𝑓𝑜𝑟 𝑗 = 1,2 𝜙′ = 𝜏𝜙𝑗 + (1 − 𝜏)𝜙′ (25) There is a small change in updating the parameters of the target networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' As with the actor- network parameter, we delay updating it for d steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' similarly, we update the target network parameters for every d step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' in this case, we can say: 1- If t mod d=0, then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Compute the gradient of the objective function 𝛻𝜙𝐽(𝜙) and update the actor- network parameter using gradient ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 𝜙 = 𝜙 + 𝛼𝛻𝜙𝐽(𝜙) (26) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Update the target critic network parameter and target actor-network parameter as (1 ) 1,2 j j j for j \uf071 \uf074\uf071 \uf074 \uf071 \uf0a2 \uf0a2 = + − = (27) and (1 ) \uf066 \uf074\uf066 \uf074 \uf066 \uf0a2 \uf0a2 = + − (28) , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The previous steps for several episodes must be repeated to improve the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The following pseudocode is prepared to understand better how TD3 works: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' TD3 Algorithm pseudocode 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' TD3 application in closed-loop vibration control for a semi-active suspension system The main inputs for vehicle dynamics are road disturbance profile and damping force produced by the MR device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The outputs are body (sprung mass) acceleration and suspension working space (SWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The RL-Agent inputs for implementing a controller for a one-quarter suspension system are body acceleration, denoted by 𝑞, and a reward function, which is as follows: 𝑟 = {0 𝑖𝑓 𝑞 = 𝑞𝑔𝑜𝑎𝑙 = 0 −𝑘𝑞2 𝑖𝑓 𝑞 ≠ 0 (29) Where k is a hyperparameter that specifies the intensity coefficient for agent punishment, the agent punishments experienced in the replay buffer are also exerted in the RL-agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The damper’s input voltage must be applied to the coil via the RL-agent output (action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' TD3 analyzes its performance by measuring body acceleration and fine-tuning its action to road profile disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The Twin Delayed Deep Deterministic Policy Gradient Algorithm: Initialize critic networks Qe1, Qe2, and actor network Ts withrandomparameters01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content="Φ Initializetargetnetworks←1%←2,'←Φ 2 InitializereplaybufferB for t=1 toTdo 3 Selectactionwithexplorationnoisea~π(s)+E e~N(O,o)and observe reward r and new states 4 Store transition tuple (s, a,r,s in B Samplemini-batchofNtransitions(s,a,r,sfromB a ← π(s) + E, E ～ clip(N(O,), -C,c) 5 y ← r +mini=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content="2Qe(s',a) Update critics i ← argming." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' N-1 (y-Qe, (s, a))2 iftmoddthen 7 Update by the deterministic policy gradient: VJ(d) = N-i VaQe (s,a)lg 元(s)V(s) Updatetargetnetworks: 8 , ← T+ (1 - T)O Φ← +(1-) end if end forsurrounding environment consists of a vehicle suspension system equipped with an MR-damper and a road profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The agent is a neural network that constructs the controller part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The hyperparameters listed in Table 3 pertain to TD3 agents used in closed-loop suspension control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Figures 5 and 6 detail the neural networks used in the TD3 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The hidden size is 400 and 300 neurons in each layer for the actor and critic network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The optimization method is Adam for both actor and critic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The discount factor 𝛾 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The input voltage varies from 0 to 3 V, and the damping force varies from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 KN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Neural Networks Parameters in TD3 Network Learning Rate Optimizer Delay for update Actor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='002 Adam 2 Critic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='002 Adam 1 Actor Target 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='006 - 2 Critic Target 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='006 - 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=" Closed-loop semi-active suspension system block diagram with DRL Agent Road Profile qt F Vehicle MR Damper Model Deep Reinforcement Learning action by policy μ,voltage Suspension Working Space(SWws) Agent (TD3) Reward Function: r, = -kq 3+(IB II'D))+= + Replay Buffer Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=" Actor-critic architecture in TD3 Agent Agent Critic Network Actor Network Qe (stat) Random △Js TD error update △Jμ DPG update noise εE N(O,o) at Vehicle 0 Critic 2 Critic 1 at Model ' Target target2 target1 Behavior target policy Compared target y + Mini batch qt llqgoal , at, qt+1 lqgoal , rt ReplayBuffer Parameter update Data flow neural network Global Memory HER Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The details of actor-critic networks in semi-active suspension control 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Results and Discussion Vertical body acceleration (BA), Dynamic Tire Load (DTL), and Suspension Working space are three primary performance criteria used in the suspension design of a vehicle to improve ride comfort and road holding, preventing the suspension system from bottoming out excessively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' To achieve the objectives mentioned above, we should reduce BA or SWS to improve ride comfort and DTL to improve road holding and minimize suspension space distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The BA has been chosen as the objective function in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Two types of controllers for semi-active suspension are investigated in this section: the RL-based TD3 and the PID controller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' their gains are tuned using the Particle Swarm Optimization (PSO) algorithm from [17], as well as an uncontrolled system (MR-Passive) with no applied voltage to the damper’s coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A particular type of bump-road excitation was chosen due to its resemblance to actual road profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Additionally, the bump-road profile demonstrates the characteristics of transient response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The road-bump profile has been proposed in [27] as: Critic 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='=1,2 Ao VeQe,(quHe(qt Il qgoal) Actor Φ qt,at Qe i-1, (qt, at) + μp (qt+1 Il qgoal) a, = μp(qt+1 Il qgoat) + qt Il qgoal A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' (t - Qe: (qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='at)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=" Replay Buffer Yt = r + ymini=12Qe(qt+1, at) (qt Il qgoal , t, qt+1, Il qgoal , rt)qt+: Qg i=12 (q+1 +at) Hp(qt+1 Il qgoal ) qt+1 I qgoal at Critic target e't=1,2 Actor target $' at = μs'(qt+1 I qgoat) + s 𝑋𝑟 = {𝑎 {1 − 𝑐𝑜𝑠( 𝜔𝑟(𝑡 − 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5))}, 𝑓𝑜𝑟0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 ≤ 𝑡 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 + 𝑑𝑏 𝑉𝑐 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (1) Where a denotes half of the bump amplitude, 2 c r b V d \uf077 \uf070 = , db is the bump width, and Vc denotes the vehicle velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' In this research, 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='035𝑚, 𝑑𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='8𝑚, 𝑉𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='856 𝑚 𝑠 , derived from [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Road bump profile Table 4 compares the results from the TD3 controller to an uncontrolled system (applied zero voltage), and Table 5 compares the results from the PSO PID controller to the TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The results demonstrate unequivocally that the DRL-based controller (TD3) algorithm outperforms PSO-tuned PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' TD3 is excellent at dissipating vibrations caused by bump excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Additionally, it reduces settling time and enhances road holding and ride comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' DRL TD3 significantly reduces body acceleration, suspension working space, and Dynamic Tire Load compared to an uncontrolled suspension system by 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='8%, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5%, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='6%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Simultaneously, PSO-PID reduces those criteria by 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2%, 50%, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='4%, respectively, compared to MR passive (uncontrolled suspension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='05 (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='04 Road 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='01 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 5 Time (sec) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Body Acceleration (BA) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Dynamic Tire Load (DTL) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='.Uncontrolled 3 -PSO-PID DRL-TD3 Acceleration(m/s) Body, -2 -3 4 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 5 Time (sec)1500 uncontrolled -PSO-PID 1000 DRL-TD3 Dynamic Tyre Load (N) 500 -500 -1000 -1500 - 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 5 Time (sec) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Suspension Working Space (SWS) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' RMS Values for the suspension system criteria with TD3 algorithm and Uncontrolled Suspension System RMS-values Body Acceleration(𝒎/𝒔𝟐) Suspension Working Space(𝒎) Dynamic Tire Load (𝑵) DRL-TD3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='0084 381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='32 Uncontrolled 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='0267 537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='7 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Comparison of DRL-TD3 with PSO-PID Controller Type Body Acceleration(𝒎/𝒔𝟐) Suspension Working Space(𝒎) Dynamic Tire Load (𝑵) DRL-TD3 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='8 % 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='53 % 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='6% PSO-PID 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='4% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='9% Improvement 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='4% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='43% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='7% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Conclusion The paper proposes the DRL-based TD3 algorithm for vibration mitigation in a semi-active suspension system equipped with an MR damper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Three major criteria, BA, SWS, and DTL, were considered and investigated when evaluating the proposed controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Instead of using two distinct controllers for damper control and force estimation, a single universal controller based on real-time learning was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' The time-domain results were analyzed, and the effectiveness of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='06 uncontrolled PSO-PID DRL-TD3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='02 (w) s swS -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='06 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='5 5 Time (sec)DRL-based controller was compared to the optimal PSO-PID controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Finally, the results demonstrate that the TD3 algorithm outperforms the PSO-tuned PID controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Lam and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Liao, “Semi-active control of automotive suspension systems with magnetorheological dampers,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1–3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 50–75, 2003, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1504/ijvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='003652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Karkoub and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Zribi, “Active/semi-active suspension control using magnetorheological actuators,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 35–44, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2006, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1080/00207720500436344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Dyke, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Spencer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sain, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Carlson, “Modeling and control of magnetorheological dampers for seismic response reduction,” Smart Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 565–575, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1996, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1088/0964-1726/5/5/006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Bani-Hani and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sheban, “Semi-active neuro-control for base-isolation system using magnetorheological (MR) dampers,” Earthq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1119–1144, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2006, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1002/eqe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Liao and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Wang, “Semiactive Vibration Control of Train Suspension Systems via Magnetorheological Dampers,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 161–172, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2003, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1177/1045389X03014003004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Du, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yim Sze, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Lam, “Semi-active H∞ control of vehicle suspension with magnetorheological dampers,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sound Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 283, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3–5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 981–996, May 2005, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='jsv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Choi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Lee, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Park, “H∞ control performance of a full-vehicle suspension featuring magnetorheological dampers,” Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 341–360, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Choi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sung, “Vibration control of magnetorheological damper system subjected to parameter variations,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Veh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 94–110, 2008, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1504/IJVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='017071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Lee and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Choi, “Control and Response Characteristics of a Magnetorheological Fluid Damper for Passenger Vehicles,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 80–87, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2000, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1106/412A-2GMA-BTUL-MALT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Ahmadian and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Pare, “A Quarter-Car Experimental Analysis of Alternative Semiactive Control Methods,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 604–612, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2000, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1106/MR3W-5D8W-0LPL-WGUQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Shen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Golnaraghi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Heppler, “Semi-active Vibration Control Schemes for Suspension Systems Using Magnetorheological Dampers,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3–24, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2006, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1177/1077546306059853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Guo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Hu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yi, “Neural Network Control for a Semi-Active Vehicle Suspension with a Magnetorheological Damper,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 461–471, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2004, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1177/1077546304038968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Zribi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Karkoub, “Robust Control of a Car Suspension System Using Magnetorheological Dampers,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 507–524, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2004, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1177/1077546303039697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Elsawaf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Ashida, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sakata, “Optimum structure design of a multilayer piezo-composite disk for control of thermal stress,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Therm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 805–819, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2012, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1080/01495739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='689233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Metered, “Application of Nonparametric Magnetorheological Damper Model in Vehicle Semi-active Suspension System,” SAE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Passeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Cars - Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 715– 726, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2012, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='4271/2012-01-0977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Gad, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Metered, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Bassuiny, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Abdel Ghany, “Vibration control of semi-active MR seat suspension for commercial vehicles using genetic PID controller,” in Mechanisms and Machine Science, 2015, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 23, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 721–732, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1007/978-3-319-09918-7_64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Metered, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Elsawaf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Vampola, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sika, “Vibration Control of MR-Damped Vehicle Suspension System Using PID Controller Tuned by Particle Swarm Optimization,” SAE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Passeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Cars - Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 426–435, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2015, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='4271/2015-01-0622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Taheri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Tian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Zhang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Shen, “A novel semi-active suspension design based on decoupling skyhook control,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Vibroengineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' YU, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ZHOU, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ZHAO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' XING, … P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', and undefined 2017, “Design of LQG controller for vehicle active suspension system based on alternate iteration,” en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cn, Accessed: Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 18, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Available: https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cn/Article_en/CJFDTotal- SDGY201704009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Engineering, and undefined 2017, “Control method of vehicle semi active suspensions based on variable universe fuzzy control,” en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cn, Accessed: Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 18, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Available: https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cn/Article_en/CJFDTotal-ZGJX201703019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Zheng, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Zhou, “Development of a sliding mode controller for semi-active vehicle suspensions,” JVC/Journal Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1152–1160, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2013, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1177/1077546312441045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Canale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Milanese, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Novara, “Semi-active suspension control using ‘fast’ model- predictive techniques,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1034–1046, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2006, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1109/TCST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='880196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Zhao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Gong, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yang, “Research on Robust Model Predictive Control for Electro-Hydraulic Servo Active Suspension Systems,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 3231–3240, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2017, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2787663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [24] “Neural network control method of automotive semi-active air suspension--《Journal of Traffic and Transportation Engineering》2006年04期.” https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cnki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='cn/Article_en/CJFDTotal- JYGC200604014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='htm (accessed Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 18, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Qin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Rath, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Hu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Sentouh, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Wang, “Adaptive nonlinear active suspension control based on a robust road classifier with a modified super-twisting algorithm,” Nonlinear Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 97, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2425–2442, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2019, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1007/s11071-019-05138-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Ming, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yibin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Xuewen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Shuaishuai, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Yanfang, “Semi-active suspension control based on deep reinforcement learning,” IEEE Access, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 9978–9986, 2020, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='2964116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Choi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Choi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Park, “A sliding mode control of a full-car electrorheological suspension system via hardware in-the-loop simulation,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' ASME, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 122, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 114–121, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2000, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1115/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='482435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Lai and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Liao, “Vibration Control of a Suspension System via a Magnetorheological Fluid Damper,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 527–547, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content=' 2002, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} +page_content='1177/107754602023712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE0T4oBgHgl3EQf2gK9/content/2301.02714v1.pdf'} diff --git a/VdE0T4oBgHgl3EQfVgAw/content/tmp_files/2301.02264v1.pdf.txt b/VdE0T4oBgHgl3EQfVgAw/content/tmp_files/2301.02264v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4eb9b471e01bec002fe5c67cb022f124870fbe7 --- /dev/null +++ b/VdE0T4oBgHgl3EQfVgAw/content/tmp_files/2301.02264v1.pdf.txt @@ -0,0 +1,2200 @@ +arXiv:2301.02264v1 [hep-th] 5 Jan 2023 +MITP/23-001 +On ε-factorised bases and pure Feynman integrals +Hjalte Frellesvig a and Stefan Weinzierl b +a Niels Bohr International Academy, University of Copenhagen, +Blegdamsvej 17, 2100 København, Denmark +b PRISMA Cluster of Excellence, Institut für Physik, +Johannes Gutenberg-Universität Mainz, +D - 55099 Mainz, Germany +Abstract +We investigate ε-factorised differential equations and purity for Feynman integrals. We are +in particular interested in Feynman integrals beyond the ones which evaluate to multiple +polylogarithms. We show that a ε-factorised differential equation does not necessarily lead +to pure Feynman integrals. We also point out that a proposed definition of purity works +locally, but not globally. + +1 +Introduction +The concepts of ε-factorised differential equations [1], purity and uniform transcendental weight, +simple poles and constant leading singularities [2–4] play a crucial role in modern techniques for +computing Feynman integrals. These concepts are well understood for Feynman integrals which +evaluate to multiple polylogarithms. +However, as soon as we leave this class of function not everything is as clear as we want +it. This is already the case for the simplest Feynman integrals beyond the class of multiple +polylogarithms, the ones which are associated to an elliptic curve. It is therefore timely and +appropriate to clarify several issues. The points which we discuss can be exemplified by the +simplest elliptic Feynman integral, the two-loop sunrise integral with equal non-zero masses. +We start with ε-factorised differential equations. A ε-factorised differential equation together +with boundary values at a given point allows for a systematic solution in terms of iterated inte- +grals to any order in the dimensional regularisation parameter ε. But do these iterated integrals +have additional nice properties like a definition of transcendental weight or integrands with sim- +ple poles only? In this paper we show that the general answer is no, but there might be bases of +master integrals which have more of the nice properties than others. +This occurs already for the sunrise integral: We know two bases of master integrals, which +put the the associated differential equation into an ε-factorised form. The construction of either +basis generalises to more complicated integrals, so it is worth examining the two bases in detail. +The first basis is constructed along the lines of an analysis of the maximal cut [5, 6] and/or +along the lines of prescriptive unitarity [7, 8]. Concretely this basis is constructed by the re- +quirement that the period matrix on the maximal cut is proportional to the unit matrix [9]. For +the sunrise integral we present a cleaned-up basis along these lines. Throughout this paper we +denote this basis by ⃗K. +The second basis is constructed from Picard-Fuchs operators and leads to a differential equa- +tion with modular forms [10]. For the sunrise integral we consider the basis given in [11]. This +approach generalises nicely to more complicated Feynman integrals [12–17]. Throughout this +paper we denote this basis by ⃗J. +In this paper we work out the relation between the two bases. The first question we address +is the following: Do these bases define master integrals of uniform weight? In principle, this +requires a definition of transcendental weight for elliptic Feynman integrals. Let us first be +agnostic to a full and complete definition of transcendental weight. We only make the minimal +assumption that the definition of transcendental weight in the elliptic case should be compatible +with the restriction of the kinematic space to a sub-space. With this assumption we may restrict +to a point in kinematic space where the elliptic curve degenerates. The master integrals reduce +to multiple polylogarithms, for which the definition of transcendental weight is unambiguous. +Choosing this point as the boundary point for the integration of the differential equation forces +the boundary constants (given by special values of multiple polylogarithms) to be of uniform +weight (in the classical sense for multiple polylogarithms). In this way we may detect master +integrals of non-uniform weight. +It turns out that basis ⃗K (constructed by the requirement that the period matrix on the maximal +cut is proportional to the unit matrix) has boundary constants of non-uniform weight. Hence it is +2 + +not a basis of uniform weight if we require that the notion of uniform weight is compatible with +restrictions in the kinematic space. +The second question which we address in this paper is the relation between functions of +uniform weight and logarithmic singularities. Functions of uniform weight are also called pure +functions. In order to answer this question we have to adopt a definition of transcendental weight +for elliptic Feynman integrals. A generalisation of weight, which can be applied to the elliptic +case, has been defined in ref. [18]: Functions which satisfy a differential equation without any +homogeneous term are called unipotent. Unipotent functions, whose total differential involves +only pure functions and one-forms with at most simple poles are called pure. Adopting this +definition, we investigate if basis ⃗J (i.e. the modular form basis) for the sunrise integral is of +uniform weight in this sense. We find that this is the case locally, but not globally. The argument +which we present applies not only to the specific example of the equal mass sunrise integral, but +to a wide range of elliptic Feynman integrals expressible in terms of the elliptic polylogarithms +�Γ [19]. +This paper is organised as follows: In section 2 we start with a toy example, showing that +an ε-factorised differential equation alone does not guarantee a solution of uniform weight. The +boundary values need to be of uniform weight as well. The toy example is entirely within the +class of multiple polylogarithms. In section 3 we introduce the standard example of an elliptic +Feynman integral: the two-loop sunrise integral with equal non-zero masses. We introduce the +notation which we will use in later sections of this paper. +In section 4 we investigate the first question: Are the known bases, which put the differential +equation into an ε-factorised form also of uniform weight? In sub-section 4.1 we introduce three +bases⃗I, ⃗J and ⃗K for the sunrise integral. The first one⃗I is a pre-canonical basis and serves only +in intermediate steps. The basis ⃗J is the one appearing in [11], while the basis ⃗K is the one +appearing in [9]. The associated differential equations are given in sub-section 4.2. For the bases +⃗J and ⃗K, the differential equations are in ε-factorised form. In sub-section 4.3 we discuss the +period matrix on the maximal cut for the bases ⃗J and ⃗K. By construction, the period matrix for +the basis ⃗K is proportional to the unit matrix. In sub-section 4.4 we present the solutions for +the master integrals for the bases ⃗J and ⃗K. We then look at the values at p2 = 0. At this point +the elliptic curve degenerates and both solutions are given in terms of special values of multiple +polylogarithms. We find that the basis ⃗K is not of uniform weight. +In section 5 we investigate the second question: What is the relation between purity and +simple poles? We start in sub-section 5.1 with recapitulating the definition of purity from the +literature. We then show in sub-section 5.2 that this definition does fit the modular form basis +locally, but not globally. In sub-section 5.3 we demonstrate that our argument extends to Feyn- +man integrals expressible in terms of elliptic polylogarithms �Γ. The problem is the behaviour at +the finite cusps. However, modular transformations, which we discuss in sub-section 5.4, allow +us to cover the kinematic space with coordinate charts such that in each coordinate chart the +requirement from the definition of purity holds locally. Our conclusions are given in section 6. +In appendix A we present the q-expansions of the modular forms and Eisenstein series appearing +in the main text. In appendix B we give the boundary constants for the sunrise integral. +3 + +2 +A toy example +We start with a simple toy example, showing that an ε-factorised differential equation alone does +not guarantee a solution of uniform weight. The boundary values need to be of uniform weight +as well. +Consider the two functions F1(x) and F2(x) +F1 (x) += +eεln(x) += +1+εln(x)+ 1 +2ε2(ln(x))2 +O +� +ε3� +, +F2 (x) += +(1+2ε)eεln(x) += +1+ε[2+ln(x)]+ε2 +� +2ln(x)+ 1 +2 (ln(x))2 +� ++O +� +ε3� +. +(1) +F1(x) is of uniform weight (where we count algebraic numbers to be of weight zero, ln(x) to be +of weight one, and ε to be of weight minus one), while F2(x) is not. However, both function +satisfy the ε-factorised differential equation +d +dxFi (x) += +ε +xFi (x), +i ∈ {1,2}. +(2) +The general solution of eq. (2) as a power series in ε reads +Fi (x) = C(0) +i ++ +� +C(1) +i ++C(0) +i +ln(x) +� +ε+ +� +C(2) +i ++C(1) +i +ln(x)+ 1 +2C(0) +i +(ln(x))2 +� +ε2 +O +� +ε3� +, +(3) +with boundary values C(j) +i +. For F1(x) the boundary values are +C(0) +1 += 1, +C(j) +1 += 0 for j ≥ 1. +(4) +For F2(x) the boundary values are +C(0) +2 += 1, +C(1) +2 += 2, +C(j) +2 += 0 for j ≥ 2. +(5) +For a solution of uniform weight we must have that any non-zero boundary valueC(j) +i +is of weight +j. This is the case for F1(x), but not for F2(x): The boundary value C(1) +2 +is of weight zero, for a +solution of uniform weight it is supposed to be of weight one. +From this simple example we see that a ε-factorised differential equation alone does not +guarantee a solution of uniform weight, we must in addition require that the boundary values +C(j) +i +ε j are of uniform weight as well. +3 +Feynman integrals and elliptic curves +In this section we introduce the standard example of an elliptic Feynman integral: the two-loop +sunrise integral with equal non-zero masses. This section also serves to set up the notation. +4 + +We consider the family of Feynman integrals +Iν1ν2ν3 (D,x) += +e2εγE � +m2�ν123−D � dDk1 +iπ +D +2 +dDk2 +iπ +D +2 +1 +� +−q2 +1 +m2�ν1 � +−q2 +2 +m2�ν2 � +−q2 +3 +m2�ν3 , +(6) +with x = p2/m2, ν123 = ν1 +ν2 +ν3 and q1 = k1, q2 = k2 −k1, q3 = −k2 − p. Below we will set +D = 2−2ε. +The elliptic curve associated to this Feynman integral can be obtained from the maximal cut +and is given by a quartic polynomial P(u,v) = 0: +E +: +v2 −u(u+4) +� +u2 +2(1+x)u+(1−x)2� += 0. +(7) +We denote the roots of the quartic polynomial by +u1 = −4, +u2 = − +� +1+√x +�2 , +u3 = − +� +1−√x +�2 , +u4 = 0. +(8) +For 0 < x < 1 the roots are real and ordered as +u1 < u2 < u3 < u4. +(9) +We set +U1 = (u3 −u2)(u4 −u1) = 16√x, +U2 = (u2 −u1)(u4 −u3) = +� +1−√x +�3� +3+√x +� +, +U3 = (u3 −u1)(u4 −u2) = +� +1+√x +�3� +3−√x +� +. +(10) +We define the modulus and the complementary modulus of the elliptic curve E by +k2 = U1 +U3 += +16√x +(1+√x)3 (3−√x) +, +¯k2 = 1−k2 = U2 +U3 += (1−√x)3(3+√x) +(1+√x)3(3−√x) +. +(11) +Our standard choice for the periods and quasi-periods is +ψ1 = 4K (k) +U +1 +2 +3 +, +ψ2 = 4iK +�¯k +� +U +1 +2 +3 +, +φ1 = 4[K (k)−E (k)] +U +1 +2 +3 +, +φ2 = 4iE +�¯k +� +U +1 +2 +3 +. +(12) +The geometric interpretation is as follows: For simplicity we assume that the roots u1-u4 are +real and ordered as in eq. (9). The square root v can be taken as a single-valued and continuous +function on C\([u1,u2]∪[u3,u4]) +v += +√u−u1 +√u−u2 +√u3 −u√u4 −u, +(13) +5 + +u1 +u2 +u3 +u4 +γ1 +γ2 +Figure 1: Branch cuts and cycles for the computation of the periods of an elliptic curve. +where √x denotes the standard square root with a branch cut along the negative real axis. For the +ordering as in eq. (9) v is positive for u ∈]u2,u3[. It is purely imaginary with positive imaginary +part just below the branch cut [u3,u4]. Let γ1 and γ2 be two cycles which generate the homology +group H1(E,Z). This is shown in fig. 1. We choose γ1 and γ2 such that their intersection number +is (γ1,γ2) = +1. Note that the intersection number is anti-symmetric: (γ2,γ1) = −1. The periods +are alternatively given by +ψ1 = +� +γ1 +du +v += 2 +u3 +� +u2 +du +v , +ψ2 = +� +γ2 +du +v += 2 +u3 +� +u4 +du +v . +(14) +In the last expression the square root is evaluated below the cut [u3,u4]. Similar formulae can be +given for the quasi-periods. +The derivatives of the periods and quasi-periods are given for i ∈ {1,2} by +d +dxψi += +−1 +2ψi +d +dx (lnU2)+ 1 +2φi +d +dx +� +ln U2 +U1 +� +, +d +dxφi += +−1 +2ψi +d +dx +� +ln U2 +U3 +� ++ 1 +2φi +d +dx +� +ln U2 +U2 +3 +� +. +(15) +In particular we may use these relations to replace φi by dψi +dx or vice versa. Explicitly we have +3 +� +1+√x +�2 φi += +4√x +� +2+√x +� +ψi −4x +� +1−√x +�� +3+√x +� d +dxψi. +(16) +Replacing φi by dψi +dx is often advantageous to eliminate the square root √x. In the following we +will often write ∂x for d +dx. The Legendre relation reads +ψ1φ2 −φ1ψ2 += +8πi +(1+√x)3 (3−√x) +. +(17) +We denote the Wronskian by +W += +ψ1∂xψ2 −ψ2∂xψ1 = − +6πi +x(1−x)(9−x). +(18) +6 + +Finally, we set +τ = ψ2 +ψ1 +, +q = e2πiτ. +(19) +We have +dτ += +W +ψ2 +1 +dx +(20) +and +x += +9 η(τ)4η(6τ)8 +η(3τ)4η(2τ)8, +(21) +where η denotes Dedekind’s eta-function. The first few terms read +x += +9q−36q2 +90q3 +O +� +q4� +. +(22) +4 +Uniform weight and ε-factorised differential equations +In this section we investigate the question of uniform weight for bases of master integrals, +which have ε-factorised differential equations. The two-loop sunrise integral with equal non- +zero masses serves as an example. +4.1 +Bases of master integrals +We consider three bases ⃗I, ⃗J and ⃗K for the family of the sunrise integral. The first one, ⃗I, is a +basis without any additional properties and given by +⃗I += + + +I110 +I111 +I211 + +. +(23) +The latter two, ⃗J and ⃗K, put the differential equation into an ε-form: +d⃗J = εB⃗J, +d⃗K = εC⃗K, +(24) +where the (3×3)-matrices B and C are independent of the dimensional regularisation parameter +ε. The basis ⃗J, appearing in [11,20–22], is defined by +J1 += +ε2 I110, +J2 += +ε2 π +ψ1 +I111, +J3 += +ψ2 +1 +2πiεW +d +dxJ2 + 1 +24 +� +3x2 −10x−9 +��ψ1 +π +�2 +J2. +(25) +7 + +In terms of I111 and I211 the master integral J3 is given by +J3 += +� +− ε2 +24 +� +x2 −30x+45 +� ψ1 +π − ε +4 +� +1+√x +�� +3−√x +� ψ1 +π + ε +16 +� +1+√x +�3� +3−√x +� φ1 +π +� +I111 ++ε +4 (1−x)(9−x) ψ1 +π I211. +(26) +Note that the definition of the master integrals ⃗J involves only ψ1 and φ1 (through d +dxψ1), but not +ψ2 nor φ2. +The basis ⃗K, appearing in [9], is defined by +K1 += +ε2 I110, +(27) +K2 += +−ε(1+2ε) +4π +� +1+√x +�� +3−√x +�� +ψ2 − 1 +4 +� +1+√x +�2 φ2 +� +I111 + ε +4π (1−x)(9−x)ψ2I211, +K3 += ++ε(1+2ε) +4π +� +1+√x +�� +3−√x +�� +ψ1 − 1 +4 +� +1+√x +�2 φ1 +� +I111 − ε +4π (1−x)(9−x)ψ1I211. +In the definition of the master integrals ⃗K all periods ψ1,ψ2 and all quasi-periods φ1,φ2 appear. +The master integrals K2 and K3 are related by ψ2 ↔ ψ1, φ2 ↔ φ1 and an overall minus sign. +4.2 +The differential equations +The differential equation in the basis⃗I reads +d⃗I += +A⃗I, +(28) +with +A += + + +0 +0 +0 +0 +−(1+2ε) +3 +0 +−1 +3 (1+2ε)(1+3ε) +1+3ε + + dx +x ++ + + +0 +0 +0 +0 +0 +0 +ε2 +4 +1 +4 (1+2ε)(1+3ε) +−(1+2ε) + + dx +x−1 ++ + + +0 +0 +0 +0 +0 +0 +−ε2 +4 +1 +12 (1+2ε)(1+3ε) +−(1+2ε) + + dx +x−9. +(29) +In this basis, the entries are rational dlog-forms. However, the differential equation is not in +ε-form. +The differential equation in the basis ⃗J reads +d⃗J += +εB⃗J, +(30) +8 + +with +B += + + +0 +0 +0 +0 +ω2 +ω0 +ω3 +ω4 +ω2 + + +(31) +and +ω0 = 2πi dτ += 2πiW +ψ2 +1 +dx, +ω2 = − f2(τ) (2πi)dτ = dx +2x − dx +x−1 − dx +x−9, +ω3 = f3(τ) (2πi)dτ += − 1 +2 +ψ1 +π dx, +ω4 = f4(τ) (2πi)dτ += +(x+3)4 +48x(x−1)(x−9) +�ψ1 +π +�2 +dx. +(32) +f2, f3 and f4 are modular forms of Γ1(6). The minus sign in front of f2 is convention. In terms +of the variable x they are given by +f2 += +1 +24 +� +3x2 −10x−9 +��ψ1 +π +�2 +, +f3 += +− 1 +24x(x−1)(x−9) +�ψ1 +π +�3 +, +f4 += +1 +576 (3+x)4�ψ1 +π +�4 +. +(33) +Their q-expansions are given in appendix A. +The differential equation in the basis ⃗K reads +d⃗K += +εC⃗K, +(34) +with +C += + + +0 +0 +0 +C2,1 +C2,2 +C2,3 +C3,1 +C3,2 +C3,3 + + +(35) +and +C2,1 += +−1 +2 +ψ2 +π dx, +(36) +C2,2 += +iπ +6 +� +(1+x) ψ1 +π +ψ2 +π + +� +3x2 −10x−9 +� ψ2 +π +∂xψ1 +π ++2x(x−1)(x−9) ∂xψ1 +π +∂xψ2 +π +� +dx, +C2,3 += +iπ +6 +� +(1+x) +�ψ2 +π +�2 ++ +� +3x2 −10x−9 +� ψ2 +π +∂xψ2 +π ++2x(x−1)(x−9) +�∂xψ2 +π +�2� +dx, +9 + +C3,1 += +1 +2 +ψ1 +π dx, +C3,2 += +−iπ +6 +� +(1+x) +�ψ1 +π +�2 ++ +� +3x2 −10x−9 +� ψ1 +π +∂xψ1 +π ++2x(x−1)(x−9) +�∂xψ1 +π +�2� +dx, +C3,3 += +−iπ +6 +� +(1+x) ψ1 +π +ψ2 +π + +� +3x2 −10x−9 +� ψ1 +π +∂xψ2 +π ++2x(x−1)(x−9) ∂xψ1 +π +∂xψ2 +π +� +dx. +4.3 +Periods on the maximal cut +In this section we investigate the period matrices on the maximal cut of the sunrise integral. On +the maximal cut of the sunrise integral only the last two master integrals are relevant (either I2,I3 +or J2,J3 or K2,K3). The defining property for basis ⃗K is that the period matrix on the maximal +cut is diagonal and constant. +We denote by +ϕX +i , +X ∈ {I,J,K}, i ∈ {1,2,3}, +(37) +the integrand of the master integral Xi in the loop-by-loop Baikov representation [23]. In the +loop-by-loop Baikov representations we have four integration variables z1 − z4, where z1 − z3 +correspond to the three propagators and z4 to an irreducible scalar product. Let C MaxCut be +the integration domain selecting the maximal cut, i.e. a small counter-clockwise circle around +z1 = 0, a small counter-clockwise circle around z2 = 0 and a small counter-clockwise circle +around z3 = 0. We set z4 = u in accordance with the notation used in eq. (7). We denote by γ1 +and γ2 the two cycles of the elliptic curve. They define the integration domain in the variable u. +We define +C2 = C MaxCut ∪γ1, +C3 = C MaxCut ∪γ2. +(38) +We consider the period matrix +PX += +� � +ϕX +2 |C2 +� +� +ϕX +2 |C3 +� +� +ϕX +3 |C2 +� +� +ϕX +3 |C3 +� +� +. +(39) +In the i-th row of this matrix we then look at the leading term in the expansion in the dimensionsal +regularisation parameter ε for this row. We denote the order of the leading term of row i by +jmin(i). This defines a matrix PX,leading with entries +PX,leading +i j += +coeff +�� +ϕX +i |Cj +� +,ε jmin(i)� +·ε jmin(i). +(40) +One finds +PI,leading += +−8iπ +� +ψ1 +ψ2 +ψ1− 1 +4(1+√x)2φ1 +(1−√x)(3+√x) +ψ2− 1 +4(1+√x)2φ2 +(1−√x)(3+√x) +� +, +10 + +PJ,leading += +2i +� +(2πiε)2 +(2πiε)2τ +0 +−(2πiε) +� +, +PK,leading += +4πε +� 1 +0 +0 +1 +� +. +(41) +As advertised, we see that PK,leading is proportional to the unit matrix. Note that PJ,leading can be +written as +PJ,leading += +2i +� +(2πiε)2 +0 +0 +−(2πiε) +�� +1 +τ +0 +1 +� +. +(42) +This is the decomposition of the period matrix PJ,leading into a semi-simple matrix and an unipo- +tent matrix [24]. +4.4 +Solutions +In the basis ⃗J we may give a solution for the master integrals in terms of iterated integrals of +modular forms. +Let f1(τ), f2(τ), ..., fn(τ) be a set of modular forms. We define the n-fold iterated integral of +these modular forms by +I (f1, f2,..., fn;τ,τ0) += +(2πi)n +τ +� +τ0 +dτ1 +τ1 +� +τ0 +dτ2··· +τn−1 +� +τ0 +dτn f1 (τ1) f2 (τ2)... fn(τn). +(43) +With q = exp(2πiτ) we may equally well write +I (f1, f2,..., fn;τ,τ0) = +q +� +q0 +dq1 +q1 +q1 +� +q0 +dq2 +q2 +... +qn−1 +� +q0 +dqn +qn +f1 (τ1) f2(τ2)... fn(τn), +τ j = 1 +2πi lnqj. +(44) +It will be convenient to introduce a short-hand notation for repeated letters. We use the notation +{ fi}j += +fi, fi,..., fi +� +�� +� +j +(45) +to denote a sequence of j letters fi and more generally +{fi1, fi2,..., fin}j += +fi1, fi2,..., fin,......, fi1, fi2,..., fin +� +�� +� +j copies of fi1, fi2,..., fin +(46) +to denote a sequence of ( j ·n) letters, consisting of j copies of fi1, fi2,..., fin. For example +{ f1, f2}3 += +f1, f2, f1, f2, f1, f2. +(47) +11 + +Our standard choice for the base point τ0 will be τ0 = i∞, corresponding to q0 = 0. This is +unproblematic for modular forms which vanish at the cusp τ = i∞. In this case we have for a +single integration +f = +∞ +∑ +j=1 +ajqj +⇒ +q +� +0 +dq1 +q1 +f = +∞ +∑ +j=1 +aj +j qj. +(48) +For modular forms which attain a finite value at the cusp τ = i∞ we employ the standard “trailing +zero” or “tangential base point” regularisation [10,25,26]: We first take q0 to have a small non- +zero value. The integration will produce terms with ln(q0). Let Rln(q0) be the operator, which +removes all ln(q0)-terms. After these terms have been removed, we may take the limit q0 → 0. +With a slight abuse of notation we set +I ( f1, f2,..., fn;q) = lim +q0→0Rln(q0) + + +q +� +q0 +dq1 +q1 +q1 +� +q0 +dq2 +q2 +... +qn−1 +� +q0 +dqn +qn +f1 (τ1) f2(τ2)... fn (τn) + +. +(49) +We define the boundary constants Bk for the sunrise integral J2 by +lim +q→0Rln(q)J2 += +e +2 +∞ +∑ +n=2 +(−1)n +n +ζnεn ∞ +∑ +k=2 +εkBk. +(50) +The left-hand side corresponds to setting all iterated integrals to zero, including the ones which +are proportional to powers of ln(q). The boundary values Bk are collected in appendix B. Let us +mention that the boundary values Bk are of weight k. The right-hand side of eq. (50) is therefore +of uniform weight. +We may express the master integrals in the basis ⃗J to all orders in the dimensional regulari- +sation parameter in terms of iterated integrals of modular forms. We have +J1 = e +2 +∞ +∑ +n=2 +(−1)n +n +ζnεn +, +J2 = e +−εI(f2;q)+2 +∞ +∑ +n=2 +(−1)n +n +ζnεn + + + +� +∞ +∑ +j=0 +� +ε2jI +� +{1, f4}j ;q +� +− 1 +2ε2j+1I +� +{1, f4}j ,1;q +��� +∞ +∑ +k=2 +εkBk ++ +∞ +∑ +j=0 +ε j+2 +⌊ j +2⌋ +∑ +k=0 +I +� +{1, f4}k ,1, f3,{f2}j−2k ;q +� + + +, +J3 = e +−εI(f2;q)+2 +∞ +∑ +n=2 +(−1)n +n +ζnεn + + + +� +∞ +∑ +j=0 +� +ε2j+1I +� +{ f4,1}j , f4;q +� +− 1 +2ε2jI +� +{ f4,1}j ;q +��� +∞ +∑ +k=2 +εkBk +12 + ++ +∞ +∑ +j=0 +ε j+1 +⌊ j +2⌋ +∑ +k=0 +I +� +{f4,1}k , f3,{ f2}j−2k ;q +� + + +. +(51) +The expression for J2 appeared already in [10], the expression for J3 follows from (see eq. (25)) +J3 += +1 +ε +1 +2πi +d +dτJ2 + f2J2. +(52) +For the first few terms of ε-expansion we have +J1 += +1+ζ2ε2 − 2 +3ζ3ε3 + 7 +10ζ2 +2ε4 +O +� +ε5� +, +J2 += +[B2 +I (1, f3;q)]ε2 ++ +� +B3 − 1 +2B2I (1;q)−B2I ( f2;q)−I (1, f2, f3;q)−I ( f2,1, f3;q) +� +ε3 ++ +� +B4 +ζ2B2 − 1 +2B3I (1;q)−B3I ( f2;q)+ 1 +2B2I (1, f2;q)+ 1 +2B2I ( f2,1;q) ++B2I (1, f4;q)+B2I (f2, f2;q)+ζ2I (1, f3;q)+I (1, f2, f2, f3;q)+I ( f2, f2,1, f3;q) ++I (1, f4,1, f3;q)+I (f2,1, f2, f3;q) +� +ε4 +O +� +ε5� +, +J3 += +εI ( f3;q)+ +� +−1 +2B2 −I ( f2, f3;q) +� +ε2 + +� +−1 +2B3 + 1 +2B2I ( f2;q)+B2I (f4;q) ++ζ2I ( f3;q)+I ( f2, f2, f3;q)+I (f4,1, f3;q) +� +ε3 + +� +−1 +2B4 − 1 +2ζ2B2 + 1 +2B3I (f2;q) ++B3I ( f4;q)− 2 +3ζ3I ( f3;q)−B2I (f4, f2;q)−B2I ( f2, f4;q)− 1 +2B2I (f2, f2;q) +−1 +2B2I (f4,1;q)−ζ2I ( f2, f3;q)−I ( f2, f2, f2, f3;q)−I ( f4, f2,1, f3;q) +−I (f2, f4,1, f3;q)−I (f4,1, f2, f3;q) +� +ε4 +O +� +ε5� +. +(53) +Let us also summarise the boundary values: From eq. (50) and eq. (51) we obtain +lim +q→0Rln(q)J1 += +e +2 +∞ +∑ +n=2 +(−1)n +n +ζnεn +, +lim +q→0Rln(q)J2 += +e +2 +∞ +∑ +n=2 +(−1)n +n +ζnεn ∞ +∑ +k=2 +εkBk, +lim +q→0Rln(q)J3 += +−1 +2e +2 +∞ +∑ +n=2 +(−1)n +n +ζnεn ∞ +∑ +k=2 +εkBk. +(54) +In all three cases the right-hand sides are of uniform weight. +13 + +Given a solution in the basis ⃗J, we easily obtain a solution in the basis ⃗K. The two bases are +related by +⃗K += U⃗J, +(55) +with +U += + + +1 +0 +0 +0 +−(1+2ε) +2πiε −g2 ·τ +τ +0 +g2 +−1 + + +(56) +and +g2 += +1 +24 +�� +3x2 −10x−9 +� ψ1 +π +4x(1−x)(9−x) ∂xψ1 +π +� ψ1 +π . +(57) +In the modular variable τ the quantity g2 is given by +g2 += +f2 +2 π +ψ1 +1 +2πi +d +dτ +ψ1 +π += +4 +� +3b2 +1 −3b1b2 −6b2 +2 −e2 +� +. +(58) +The modular forms b1 and b2 and the quasi-modular form e2 are defined in appendix A. The +quantity g2 is a quasi-modular form of modular weight 2 and depth 1. For γ ∈ Γ1(6) the quantity +g2 transforms as +(g2|2γ)(τ) += +g2(τ)+ 2 +2πi +c +cτ+d , +γ = +� +a +b +c +d +� +, +(59) +where the operator |kγ is defined by +( f|kγ)(τ) += +(cτ+d)−k · f(γ(τ)), +γ(τ) = aτ+b +cτ+d . +(60) +For the first few terms of ε-expansion we have +K2 += +1 +2πi [−B2 +I ( f3,1;q)]ε+ 1 +2πi [−B3 −2B2 +B2I ( f2;q)−I ( f2, f3,1;q)−2I (1, f3;q) +−2g2I (1,1, f3;q)−g2I (1, f3,1;q)−g2B2I (1;q)]ε2 +O +� +ε3� +, +K3 += +−I ( f3;q)ε+ +�1 +2B2 +I ( f2, f3;q)+g2B2 +g2I (1, f3;q) +� +ε2 +O +� +ε3� +. +(61) +Let us look at the boundary values of K2 +lim +q→0Rln(q)K2 += +− B2 +2πiε− (B3 +2B2) +2πi +ε2 +O +� +ε3� +. +(62) +The term +−2B2 +2πi ε2 +(63) +is of weight minus one and spoils the uniform weight property. Hence we conclude that the basis +⃗K is not of uniform weight if we require that the notion of uniform weight is compatible with +restrictions in the kinematic space. +14 + +5 +Purity and simple poles +In this section we address the second main question of this paper: The relation between uniform +weight and simple poles in the elliptic case. +5.1 +Definition of pure functions in the literature +We recapitulate the definitions of unipotent and pure function as given in ref. [18]: +Definition 1. A function is called unipotent, if it satisfies a differential equation without a homo- +geneous term. +To give an example, the functions ln(x) and Li2(x) are unipotent +d +dx ln(x) = 1 +x, +d +dxLi2 (x) = −1 +x ln(1−x), +(64) +while ex is not +d +dxex += +ex. +(65) +Definition 2. Unipotent functions, whose total differential involves only pure functions and one- +forms with at most simple poles are called pure. +The standard example are multiple polylogarithms, whose total differential is given by +dG(z1,...,zr;y) += +r +∑ +j=1 +G(z1,..., ˆz j,...,zr;y) +� +d ln +� +z j−1 −z j +� +−d ln +� +z j+1 −z j +�� +, +(66) +where we set z0 = y and zr+1 = 0. A hat indicates that the corresponding variable is omitted. +In addition one uses the convention that for z j+1 = z j the one-form d ln(z j+1 −z j) equals zero. +Clearly, the one forms +d ln +� +z j+1 −z j +� += +dz j+1 −dz j +z j+1 −z j +(67) +have only simple poles. +5.2 +Iterated integrals of modular forms +Let us now look at iterated integrals of modular forms, as defined in eq. (43). It is clear that +these iterated integrals are unipotent functions, as differentiation removes one integration. We +investigate the order of the poles of the total differential. +We denote by +H += +{ τ ∈ C | Im(τ) > 0 } +(68) +15 + +the complex upper half-plane and by +H += +H∪{i∞}∪Q +(69) +the extended complex upper half-plane. Under the map q = exp(2πiτ) the complex upper half- +plane H is mapped to the punctured open disk +D += +{ q ∈ C | 0 < |q| < 1 } +(70) +and H is mapped to +D += +D∪{0}∪ +� +e2πir | r ∈ Q +� +. +(71) +Let fk(τ) be a modular form of weight k for a congruence subgroup Γ and +ωmodular +k += +2πi fk (τ)dτ. +(72) +For simplicity we assume that +� 1 +1 +0 +1 +� +∈ Γ. In this case fk has the q-expansion [27] +fk += +∞ +∑ +n=0 +anqn. +(73) +(In the general case fk will have an expansion in q +1 +N′ , where N′ is the smallest positive inte- +ger such that fk(τ + N′) = fk(τ). The general case is only from a notational perspective more +elaborate.) In addition we will always assume that modular forms are normalised such that the +coefficients of their q-expansion are algebraic numbers. We view ωmodular +k +as a differential one- +form on D. In the variable q we have +ωmodular +k += +∞ +∑ +n=0 +anqn−1dq. +(74) +This shows immediately that ωmodular +k +is holomorphic on D and has a simple pole at q = 0 if +a0 ̸= 0. Thus, in a neighbourhood of q = 0 the differential one-form ωmodular +k +has at most simple +poles. +Let us now discuss if this extends globally to D. The answer will be no. We have to look at +the other cusps. We investigate the behaviour at +q0 += +e2πi(− d +c), +c,d ∈ Z, c ̸= 0. +(75) +We may derive the behaviour of ωmodular +k +at q0 from the modular properties of fk. We consider +the modular transformation +γ = +� a +b +c +d +� +∈ SL2(Z), +γ−1 = +� +d +−b +−c +a +� +(76) +16 + +and set +τ′ = γ(τ) = aτ+b +cτ+d , +q′ = e2πiτ′. +(77) +This maps τ = −d +c to τ′ = i∞ and q0 to q′ +0 = 0. For the automorphic factor we have +cτ+d += +c +2πi +(q−q0) +q0 ++O +� +(q−q0)2� +. +(78) +( fk|kγ−1)(τ′) has again a q′-expansion as in eq. (73) +� +fk|kγ−1�� +τ′� += +∞ +∑ +n=0 +a′ +n +� +q′�n . +(79) +If fk is a modular form for the congruence subgroup Γ and γ ∈ Γ we have a′ +n = an, otherwise +the coefficients need not be the same. Usually we are interested in the cusps not equivalent to +τ = i∞, this implies γ ∈ SL2 (Z)\Γ. For a′ +0 ̸= 0 we have +ωmodular +k += +a′ +0 +� c +2πi +�−k +qk−1 +0 +dq +(q−q0)k +O +� +(q−q0)−k+1� +. +(80) +Thus we see that whenever fk is non-vanishing at the cusp τ0 = −d +c, the differential one-form +ωmodular +k +has a pole of order k in the variable q at q = q0. Globally, ωmodular +k +has poles up to order +k on D. +5.3 +Elliptic polylogarithms +The discussion of the previous sub-section is not restricted to iterated integrals of modular forms +and carries over to elliptic polylogarithms �Γ. +Let g(k)(z,τ) denote the coefficients of the Kronecker function and set +ωKronecker +k +(z,τ) += +(2πi)2−k +� +g(k−1) (z,τ)dz+(k −1)g(k) (z,τ) dτ +2πi +� +. +(81) +We may view ωKronecker +k +as a differential one-form on the two-dimensional moduli space M1,2. +Coordinates on this moduli space are (z,τ). The elliptic polylogarithms �Γ are iterated integrals +of ωKronecker +k +(z−cj,τ) along z at constant τ. It is known that the functions g(k)(z,τ) have at most +simple poles in z and when restricted to τ = const the elliptic polylogarithms �Γ are pure functions +in the sense of definition 2. However in the applications towards Feynman integrals it is usually +the case that the assumption τ = const is not justified. A variation of the kinematic variables of +the Feynman integral will imply a variation of τ and we have to consider the τ-dependence as +well. For the argument we want to make it is sufficient to restrict to z = a + bτ with a,b ∈ Q +and k ≥ 2. In this case the differential one-forms ωKronecker +k +reduce to the form of ωmodular +k +[24] +and the argument from the previous sub-section applies: In this case the differential one-forms +ωKronecker +k +may have poles up to order k in the variable q (or τ). +17 + +5.4 +Modular transformations +We have seen that locally in the coordinate chart D ∪ {0} the basis ⃗J satisfies the criteria of +definition 2. This coordinate chart includes the point x = 0. Let us now investigate the global +picture. For the sunrise integral we have four singular points x ∈ {0,1,9,∞} and we may cover +the kinematic space with four charts, such that each chart includes exactly one singular point. +In each chart we may construct a basis, which satisfies the criteria of definition 2 locally. In +different charts we will have different coordinates τ and τ′, but also different bases of master +integrals ⃗J and ⃗J′. The coordinates τ and τ′ will be related by a modular transformation. The +modular transformation induces also the transformation between ⃗J and ⃗J′. +Let us discuss the behaviour near the cusp τ0 = −d +c. The modular transformation γ defined +in eq. (76) maps τ0 = −d +c to τ′ +0 = i∞. Let fk be a modular form for a congruence subgroup Γ. +Then by definition fk(τ) is holomorphic on H and ( fk|kγ−1)(τ′) has a q′-expansion as in eq. (79) +for any γ ∈ SL2(Z). This suggest to change in a neighbourhood of τ = −d +c coordinates from q to +q′. The differential one-form +2πi +� +fk|kγ−1�� +τ′� +dτ′ +(82) +has then a simple pole at q′ = 0 (corresponding to τ = −d +c). However, ωmodular +k +as defined in +eq. (72) does not transform under this coordinate change into eq. (82). Instead we find +ωmodular +k += +� +−cτ′ +a +�k−2 ·2πi +� +fk|kγ−1�� +τ′� +dτ′. +(83) +(−cτ′ + a) is the automorphic factor for γ−1. For k ̸= 2 this factor spoils that iterated integrals +of modular forms transform under modular transformations into iterated integrals of modular +forms. However, elliptic Feynman integrals transform nicely: Let us consider for +γ(τ) += +aτ+b +cτ+d , +γ ∈ SL2(Z) +(84) +the combined transformation [21] +⃗J′ += + + +1 +0 +0 +0 +1 +cτ+d +0 +0 +− c +2πiε +cτ+d + +⃗J, +τ′ += +aτ+b +cτ+d . +(85) +One obtains +d⃗J′ += +εB′⃗J′ +(86) +with +B′ += +2πi + + +0 +0 +0 +0 +−( f2|2γ−1)(τ′) +1 +( f3|3γ−1)(τ′) +( f4|4γ−1)(τ′) +−( f2|2γ−1)(τ′) + +dτ′. +(87) +18 + +As Γ(6) is a subgroup of Γ1(6) we have Mk(Γ1(6)) ⊂Mk(Γ(6)) and as Γ(6) is a normal subgroup +of SL2(Z) it follows that +fk|kγ−1 +∈ Mk(Γ(6)). +(88) +We see that the entries of B′ are again differential one-forms of the form as in eq. (72). We may +express ⃗J′ again in terms of iterated integrals of modular forms, this time in the variable q′. It can +be shown that the boundary constants are again of uniform weight. Hence it follows that in the +coordinate chart with coordinate q′ (or τ′) the basis ⃗J′ satisfies the criteria of definition 2 locally. +6 +Conclusions +For Feynman integrals which evaluate to multiple polylogarithms we have a clear understanding +of purity: These are Feynman integrals, whose term of order j in the ε-expansion is pure of +transcendental weight j. We are interested in extending this concept to Feynman integrals beyond +the ones which evaluate to multiple polylogarithms. +This is non-trivial and in this paper we discussed some subtleties: We showed that an ε- +factorised differential equation alone does not lead necessarily lead to a solution which is pure. +The boundary values have to be pure as well. This applies in particular to a basis constructed by +the requirement that the period matrix on the maximal cut is proportional to the unit matrix. The +argument we presented is agnostic to the exact definition of purity beyond the case of multiple +polylogarithms, we only assumed that the definition of transcendental weight in the general case +is compatible with the restriction of the kinematic space to a sub-space. +In the second part of the paper we adopted a particular definition of purity from the literature. +We showed that this definition works only locally – but not globally – for a particular basis of +the two-loop equal mass sunrise integral. Of course, it might well be that this particular basis is +not the optimal one, but another possibility is that the definition of purity needs a more refined +definition. The modular transformation properties, which we discussed in section 5.4, point +towards a possible modification. +We believe that the detailed analysis we carried out in this paper will be helpful for a defi- +nition of purity which not only includes the elliptic case, but also Feynman integrals related to +Calabi-Yau geometries. +Acknowledgements +S.W. thanks the Niels Bohr Institute for hospitality and H.F. thanks the Mainz Institute of Theo- +retical Physics for hospitality. +H.F. is partially supported by a Carlsberg Foundation Reintegration Fellowship, and has re- +ceived funding from the European Union’s Horizon 2020 research and innovation program under +the Marie Sklodowska-Curie grant agreement No. 847523 “INTERACTIONS”. +19 + +A +Modular forms +In this appendix we give the q-expansions of the modular forms f2, f3 and f4, appearing in +eq. (32) and the q-expansions of ψ1 (which is a modular form of modular weight 1). In addition, +we define the Eisenstein series e2, which appears in eq. (58). +We start by introducing a basis {b1,b2} for the modular forms of modular weight 1 for the +Eisenstein subspace E1(Γ1(6)): +b1 = E1(τ;χ1,χ−3), +b2 = E1(2τ;χ1,χ−3), +(89) +where χ1 and χ−3 denote primitive Dirichlet characters with conductors 1 and 3, respectively. In +terms of the coefficients g(k)(z,τ) of the Kronecker function we have +b1 = +√ +3 +6π g(1)(1 +3,τ), +b2 = − +√ +3 +12π +� +g(1)(1 +3,τ)−g(1)(1 +6,τ) +� +. +(90) +Then +f2 += +−6 +� +b2 +1 +6b1b2 −4b2 +2 +� +, +f3 += +36 +√ +3 +� +b3 +1 −b2 +1b2 −4b1b2 +2 +4b3 +2 +� +, +f4 += +324b4 +1. +(91) +In terms of the coefficients g(k)(z,τ) of the Kronecker function we have +f2 += +1 +2π2 +� +3g(2)(1 +2,τ)−g(2)(1 +3,τ)+g(2)(1 +6,τ) +� +, +f3 += +1 +4π3 +� +15g(3)(1 +3,τ)−12g(3)(1 +6,τ) +� +, +f4 += +1 +4π4 +� +−18g(4)(0,τ)−27g(4)(1 +3,τ) +� +. +(92) +The q-expansions are +f2 += +−1 +2 −8q−4q2 −44q3 +4q4 −48q5 −40q6 +O +� +q7� +, +f3 += +−3 +√ +3 +� +q−5q2 +9q3 −11q4 +24q5 −45q6� ++O +� +q7� +, +f4 += +1 +4 +6q+54q2 +222q3 +438q4 +756q5 +1998q6 +O +� +q7� +. +(93) +In addition we have +ψ1 +π += +2 +√ +3(b1 +b2) += +2 +3 +√ +3 +� +1+3q+3q2 +3q3 +3q4 +3q6� ++O +� +q7� +. +(94) +20 + +We define the Eisenstein series e2 by +e2 (τ) += +1 +2(2πi)2 +∑ +′ +(n1,n2)∈Z2\(0,0) +1 +(n1 +n2τ)2. +(95) +The prime at the summation sign denotes the Eisenstein summation prescription defined by +∑ +′ +(n1,n2)∈Z2 +f (z+n1 +n2τ) += +lim +N2→∞ +N2 +∑ +n2=−N2 +� +lim +N1→∞ +N1 +∑ +n1=−N1 +f (z+n1 +n2τ) +� +. +(96) +The q-expansion of e2 starts with +e2 (τ) += +− 1 +24 +q+3q2 +4q3 +7q4 +6q5 +12q6 +O +� +q7� +. +(97) +The Eisenstein series e2 is a quasi-modular form. +B +Boundary values +In this appendix we give the boundary values Bk. These are easily obtained from [10, 28]. We +have +∞ +∑ +k=0 +εkBk += +3 +43−ε +� +h− 2πε +3 +Γ(1+2ε) +Γ(1+ε)2 +� +, +(98) +where +h = 1 +i +� +(−r3)−ε +2F1 (−2ε,−ε;1−ε;r3)− +� +−r−1 +3 +�−ε +2F1 +� +−2ε,−ε;1−ε;r−1 +3 +�� +(99) +and r3 = exp(2πi/3). The hypergeometric function can be expanded systematically in ε with the +methods of [29–32]. The first few terms are given by +2F1 (−2ε,−ε;1−ε;x) = 1+2ε2Li2 (x)+ε3 [2Li3(x)−4Li21 (x,1)] ++ε4[2Li4 (x)−4Li31 (x,1)+8Li211 (x,1,1)]+O +� +ε5� +. +(100) +The first few boundary values are given by +B0 += +0, +B1 += +0, +B2 += +3 +2i +� +Li2 (r3)−Li2 +� +r−1 +3 +�� +, +B3 += +3 +2i +� +−2Li21 (r3,1)−Li3(r3)+2Li21 +� +r−1 +3 ,1 +� ++Li3 +� +r−1 +3 +�� +21 + +−ln(3)B2, +B4 += +3 +2i +� +4Li211 (r3,1,1)−2Li31 (r3,1)+Li4 (r3)−4Li211 +� +r−1 +3 ,1,1 +� ++2Li31 +� +r−1 +3 ,1 +� +−Li4 +� +r−1 +3 +�� +−ln(3)B3 − 1 +2 ln2(3)B2 + 1 +3ζ2B2. +(101) +These can be reduced to polylogarithms of depth 1 as follows [33,34]: +B2 += +3 Im Li2 (r3), +B3 += +24 +5 Im Li3 +� i√ +3 +� +− 17 +90π3 − 1 +10π(ln(3))2 , +B4 += +−63 +10Im Li4 (r3)+ 48 +5 Im Li4 +� i√ +3 +� ++ 17 +90π3 ln(3)+ 1 +30π(ln(3))3. +(102) +References +[1] J. M. Henn, Phys. Rev. Lett. 110, 251601 (2013), arXiv:1304.1806. +[2] F. Cachazo, (2008), arXiv:0803.1988. +[3] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, JHEP 06, 125 (2012), arXiv:1012.6032. +[4] N. Arkani-Hamed, Y. Bai, and T. Lam, JHEP 11, 039 (2017), arXiv:1703.04541. +[5] A. Primo and L. Tancredi, Nucl. Phys. B916, 94 (2017), arXiv:1610.08397. +[6] A. Primo and L. Tancredi, Nucl. Phys. B921, 316 (2017), arXiv:1704.05465. +[7] J. L. Bourjaily, E. Herrmann, and J. Trnka, JHEP 06, 059 (2017), arXiv:1704.05460. +[8] J. L. Bourjaily, N. Kalyanapuram, C. Langer, and K. Patatoukos, +Phys. Rev. D 104, 125009 (2021), +arXiv:2102.02210. +[9] H. Frellesvig, JHEP 03, 079 (2022), arXiv:2110.07968. +[10] L. Adams and S. Weinzierl, Commun. Num. Theor. Phys. 12, 193 (2018), arXiv:1704.08895. +[11] L. Adams and S. Weinzierl, Phys. Lett. B781, 270 (2018), arXiv:1802.05020. +[12] C. Bogner, S. Müller-Stach, and S. Weinzierl, Nucl. Phys. B 954, 114991 (2020), arXiv:1907.01251. +[13] H. Müller and S. Weinzierl, JHEP 07, 101 (2022), arXiv:2205.04818. +[14] S. Pögel, X. Wang, and S. Weinzierl, JHEP 09, 062 (2022), arXiv:2207.12893. +[15] S. Pögel, X. Wang, and S. Weinzierl, (2022), arXiv:2211.04292. +[16] S. Pögel, X. Wang, and S. Weinzierl, (2022), arXiv:2212.08908. +[17] C. Duhr, A. Klemm, C. Nega, and L. Tancredi, (2022), arXiv:2212.09550. +[18] J. Broedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, JHEP 01, 023 (2019), arXiv:1809.10698. +[19] J. Broedel, C. Duhr, F. Dulat, and L. Tancredi, JHEP 05, 093 (2018), arXiv:1712.07089. +[20] I. Hönemann, K. Tempest, and S. Weinzierl, Phys. Rev. D98, 113008 (2018), arXiv:1811.09308. +[21] S. Weinzierl, Nucl. Phys. B 964, 115309 (2021), arXiv:2011.07311. +[22] S. Weinzierl, Feynman Integrals (Springer, 2022), arXiv:2201.03593. +22 + +[23] H. Frellesvig and C. G. Papadopoulos, JHEP 04, 083 (2017), arXiv:1701.07356. +[24] J. Broedel, C. Duhr, F. Dulat, B. Penante, and L. Tancredi, JHEP 08, 014 (2018), arXiv:1803.10256. +[25] F. Brown, (2014), arXiv:1407.5167. +[26] M. Walden and S. Weinzierl, Comput. Phys. Commun. 265, 108020 (2021), arXiv:2010.05271. +[27] T. Miyake, Modular Forms (Springer, 1989). +[28] L. Adams, C. Bogner, and S. Weinzierl, J. Math. Phys. 57, 032304 (2016), arXiv:1512.05630. +[29] S. Moch, P. Uwer, and S. Weinzierl, J. Math. Phys. 43, 3363 (2002), hep-ph/0110083. +[30] S. Weinzierl, Comput. Phys. Commun. 145, 357 (2002), math-ph/0201011. +[31] S. Moch and P. Uwer, Comput. Phys. Commun. 174, 759 (2006), math-ph/0508008. +[32] T. Huber and D. Maitre, Comput. Phys. Commun. 175, 122 (2006), hep-ph/0507094. +[33] H. Frellesvig, D. Tommasini, and C. Wever, JHEP 03, 189 (2016), arXiv:1601.02649. +[34] H. Frellesvig, K. Kudashkin, and C. Wever, JHEP 05, 038 (2020), arXiv:2002.07776. +23 + diff --git a/VdE0T4oBgHgl3EQfVgAw/content/tmp_files/load_file.txt b/VdE0T4oBgHgl3EQfVgAw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a29af46c63f8a8ebc7703ce1542d9587ffc77c96 --- /dev/null +++ b/VdE0T4oBgHgl3EQfVgAw/content/tmp_files/load_file.txt @@ -0,0 +1,658 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf,len=657 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='02264v1 [hep-th] 5 Jan 2023 MITP/23-001 On ε-factorised bases and pure Feynman integrals Hjalte Frellesvig a and Stefan Weinzierl b a Niels Bohr International Academy, University of Copenhagen, Blegdamsvej 17, 2100 København, Denmark b PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität Mainz, D - 55099 Mainz, Germany Abstract We investigate ε-factorised differential equations and purity for Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We are in particular interested in Feynman integrals beyond the ones which evaluate to multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We show that a ε-factorised differential equation does not necessarily lead to pure Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We also point out that a proposed definition of purity works locally, but not globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 1 Introduction The concepts of ε-factorised differential equations [1], purity and uniform transcendental weight, simple poles and constant leading singularities [2–4] play a crucial role in modern techniques for computing Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' These concepts are well understood for Feynman integrals which evaluate to multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However, as soon as we leave this class of function not everything is as clear as we want it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This is already the case for the simplest Feynman integrals beyond the class of multiple polylogarithms, the ones which are associated to an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' It is therefore timely and appropriate to clarify several issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The points which we discuss can be exemplified by the simplest elliptic Feynman integral, the two-loop sunrise integral with equal non-zero masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We start with ε-factorised differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' A ε-factorised differential equation together with boundary values at a given point allows for a systematic solution in terms of iterated inte- grals to any order in the dimensional regularisation parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' But do these iterated integrals have additional nice properties like a definition of transcendental weight or integrands with sim- ple poles only?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In this paper we show that the general answer is no, but there might be bases of master integrals which have more of the nice properties than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This occurs already for the sunrise integral: We know two bases of master integrals, which put the the associated differential equation into an ε-factorised form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The construction of either basis generalises to more complicated integrals, so it is worth examining the two bases in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The first basis is constructed along the lines of an analysis of the maximal cut [5, 6] and/or along the lines of prescriptive unitarity [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Concretely this basis is constructed by the re- quirement that the period matrix on the maximal cut is proportional to the unit matrix [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the sunrise integral we present a cleaned-up basis along these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Throughout this paper we denote this basis by ⃗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The second basis is constructed from Picard-Fuchs operators and leads to a differential equa- tion with modular forms [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the sunrise integral we consider the basis given in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This approach generalises nicely to more complicated Feynman integrals [12–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Throughout this paper we denote this basis by ⃗J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In this paper we work out the relation between the two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The first question we address is the following: Do these bases define master integrals of uniform weight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In principle, this requires a definition of transcendental weight for elliptic Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let us first be agnostic to a full and complete definition of transcendental weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We only make the minimal assumption that the definition of transcendental weight in the elliptic case should be compatible with the restriction of the kinematic space to a sub-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' With this assumption we may restrict to a point in kinematic space where the elliptic curve degenerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The master integrals reduce to multiple polylogarithms, for which the definition of transcendental weight is unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Choosing this point as the boundary point for the integration of the differential equation forces the boundary constants (given by special values of multiple polylogarithms) to be of uniform weight (in the classical sense for multiple polylogarithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In this way we may detect master integrals of non-uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' It turns out that basis ⃗K (constructed by the requirement that the period matrix on the maximal cut is proportional to the unit matrix) has boundary constants of non-uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Hence it is 2 not a basis of uniform weight if we require that the notion of uniform weight is compatible with restrictions in the kinematic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The second question which we address in this paper is the relation between functions of uniform weight and logarithmic singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Functions of uniform weight are also called pure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In order to answer this question we have to adopt a definition of transcendental weight for elliptic Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' A generalisation of weight, which can be applied to the elliptic case, has been defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [18]: Functions which satisfy a differential equation without any homogeneous term are called unipotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Unipotent functions, whose total differential involves only pure functions and one-forms with at most simple poles are called pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Adopting this definition, we investigate if basis ⃗J (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' the modular form basis) for the sunrise integral is of uniform weight in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We find that this is the case locally, but not globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The argument which we present applies not only to the specific example of the equal mass sunrise integral, but to a wide range of elliptic Feynman integrals expressible in terms of the elliptic polylogarithms �Γ [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This paper is organised as follows: In section 2 we start with a toy example, showing that an ε-factorised differential equation alone does not guarantee a solution of uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The boundary values need to be of uniform weight as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The toy example is entirely within the class of multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In section 3 we introduce the standard example of an elliptic Feynman integral: the two-loop sunrise integral with equal non-zero masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We introduce the notation which we will use in later sections of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In section 4 we investigate the first question: Are the known bases, which put the differential equation into an ε-factorised form also of uniform weight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In sub-section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 we introduce three bases⃗I, ⃗J and ⃗K for the sunrise integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The first one⃗I is a pre-canonical basis and serves only in intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The basis ⃗J is the one appearing in [11], while the basis ⃗K is the one appearing in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The associated differential equations are given in sub-section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the bases ⃗J and ⃗K, the differential equations are in ε-factorised form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In sub-section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 we discuss the period matrix on the maximal cut for the bases ⃗J and ⃗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' By construction, the period matrix for the basis ⃗K is proportional to the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In sub-section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='4 we present the solutions for the master integrals for the bases ⃗J and ⃗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We then look at the values at p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' At this point the elliptic curve degenerates and both solutions are given in terms of special values of multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We find that the basis ⃗K is not of uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In section 5 we investigate the second question: What is the relation between purity and simple poles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We start in sub-section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 with recapitulating the definition of purity from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We then show in sub-section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 that this definition does fit the modular form basis locally, but not globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In sub-section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 we demonstrate that our argument extends to Feyn- man integrals expressible in terms of elliptic polylogarithms �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The problem is the behaviour at the finite cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However, modular transformations, which we discuss in sub-section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='4, allow us to cover the kinematic space with coordinate charts such that in each coordinate chart the requirement from the definition of purity holds locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Our conclusions are given in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In appendix A we present the q-expansions of the modular forms and Eisenstein series appearing in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In appendix B we give the boundary constants for the sunrise integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 3 2 A toy example We start with a simple toy example, showing that an ε-factorised differential equation alone does not guarantee a solution of uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The boundary values need to be of uniform weight as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Consider the two functions F1(x) and F2(x) F1 (x) = eεln(x) = 1+εln(x)+ 1 2ε2(ln(x))2 +O � ε3� , F2 (x) = (1+2ε)eεln(x) = 1+ε[2+ln(x)]+ε2 � 2ln(x)+ 1 2 (ln(x))2 � +O � ε3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (1) F1(x) is of uniform weight (where we count algebraic numbers to be of weight zero, ln(x) to be of weight one, and ε to be of weight minus one), while F2(x) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However, both function satisfy the ε-factorised differential equation d dxFi (x) = ε xFi (x), i ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (2) The general solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (2) as a power series in ε reads Fi (x) = C(0) i + � C(1) i +C(0) i ln(x) � ε+ � C(2) i +C(1) i ln(x)+ 1 2C(0) i (ln(x))2 � ε2 +O � ε3� , (3) with boundary values C(j) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For F1(x) the boundary values are C(0) 1 = 1, C(j) 1 = 0 for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (4) For F2(x) the boundary values are C(0) 2 = 1, C(1) 2 = 2, C(j) 2 = 0 for j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (5) For a solution of uniform weight we must have that any non-zero boundary valueC(j) i is of weight j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This is the case for F1(x), but not for F2(x): The boundary value C(1) 2 is of weight zero, for a solution of uniform weight it is supposed to be of weight one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' From this simple example we see that a ε-factorised differential equation alone does not guarantee a solution of uniform weight, we must in addition require that the boundary values C(j) i ε j are of uniform weight as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 3 Feynman integrals and elliptic curves In this section we introduce the standard example of an elliptic Feynman integral: the two-loop sunrise integral with equal non-zero masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This section also serves to set up the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 4 We consider the family of Feynman integrals Iν1ν2ν3 (D,x) = e2εγE � m2�ν123−D � dDk1 iπ D 2 dDk2 iπ D 2 1 � −q2 1 +m2�ν1 � −q2 2 +m2�ν2 � −q2 3 +m2�ν3 , (6) with x = p2/m2, ν123 = ν1 +ν2 +ν3 and q1 = k1, q2 = k2 −k1, q3 = −k2 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Below we will set D = 2−2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The elliptic curve associated to this Feynman integral can be obtained from the maximal cut and is given by a quartic polynomial P(u,v) = 0: E : v2 −u(u+4) � u2 +2(1+x)u+(1−x)2� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (7) We denote the roots of the quartic polynomial by u1 = −4, u2 = − � 1+√x �2 , u3 = − � 1−√x �2 , u4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (8) For 0 < x < 1 the roots are real and ordered as u1 < u2 < u3 < u4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (9) We set U1 = (u3 −u2)(u4 −u1) = 16√x, U2 = (u2 −u1)(u4 −u3) = � 1−√x �3� 3+√x � , U3 = (u3 −u1)(u4 −u2) = � 1+√x �3� 3−√x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (10) We define the modulus and the complementary modulus of the elliptic curve E by k2 = U1 U3 = 16√x (1+√x)3 (3−√x) , ¯k2 = 1−k2 = U2 U3 = (1−√x)3(3+√x) (1+√x)3(3−√x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (11) Our standard choice for the periods and quasi-periods is ψ1 = 4K (k) U 1 2 3 , ψ2 = 4iK �¯k � U 1 2 3 , φ1 = 4[K (k)−E (k)] U 1 2 3 , φ2 = 4iE �¯k � U 1 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (12) The geometric interpretation is as follows: For simplicity we assume that the roots u1-u4 are real and ordered as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The square root v can be taken as a single-valued and continuous function on C\\([u1,u2]∪[u3,u4]) v = √u−u1 √u−u2 √u3 −u√u4 −u, (13) 5 u1 u2 u3 u4 γ1 γ2 Figure 1: Branch cuts and cycles for the computation of the periods of an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' where √x denotes the standard square root with a branch cut along the negative real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the ordering as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (9) v is positive for u ∈]u2,u3[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' It is purely imaginary with positive imaginary part just below the branch cut [u3,u4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let γ1 and γ2 be two cycles which generate the homology group H1(E,Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We choose γ1 and γ2 such that their intersection number is (γ1,γ2) = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Note that the intersection number is anti-symmetric: (γ2,γ1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The periods are alternatively given by ψ1 = � γ1 du v = 2 u3 � u2 du v , ψ2 = � γ2 du v = 2 u3 � u4 du v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (14) In the last expression the square root is evaluated below the cut [u3,u4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Similar formulae can be given for the quasi-periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The derivatives of the periods and quasi-periods are given for i ∈ {1,2} by d dxψi = −1 2ψi d dx (lnU2)+ 1 2φi d dx � ln U2 U1 � , d dxφi = −1 2ψi d dx � ln U2 U3 � + 1 2φi d dx � ln U2 U2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (15) In particular we may use these relations to replace φi by dψi dx or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Explicitly we have 3 � 1+√x �2 φi = 4√x � 2+√x � ψi −4x � 1−√x �� 3+√x � d dxψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (16) Replacing φi by dψi dx is often advantageous to eliminate the square root √x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In the following we will often write ∂x for d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The Legendre relation reads ψ1φ2 −φ1ψ2 = 8πi (1+√x)3 (3−√x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (17) We denote the Wronskian by W = ψ1∂xψ2 −ψ2∂xψ1 = − 6πi x(1−x)(9−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (18) 6 Finally, we set τ = ψ2 ψ1 , q = e2πiτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (19) We have dτ = W ψ2 1 dx (20) and x = 9 η(τ)4η(6τ)8 η(3τ)4η(2τ)8, (21) where η denotes Dedekind’s eta-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The first few terms read x = 9q−36q2 +90q3 +O � q4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (22) 4 Uniform weight and ε-factorised differential equations In this section we investigate the question of uniform weight for bases of master integrals, which have ε-factorised differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The two-loop sunrise integral with equal non- zero masses serves as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 Bases of master integrals We consider three bases ⃗I, ⃗J and ⃗K for the family of the sunrise integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The first one, ⃗I, is a basis without any additional properties and given by ⃗I = \uf8eb \uf8ed I110 I111 I211 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (23) The latter two, ⃗J and ⃗K, put the differential equation into an ε-form: d⃗J = εB⃗J, d⃗K = εC⃗K, (24) where the (3×3)-matrices B and C are independent of the dimensional regularisation parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The basis ⃗J, appearing in [11,20–22], is defined by J1 = ε2 I110, J2 = ε2 π ψ1 I111, J3 = ψ2 1 2πiεW d dxJ2 + 1 24 � 3x2 −10x−9 ��ψ1 π �2 J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (25) 7 In terms of I111 and I211 the master integral J3 is given by J3 = � − ε2 24 � x2 −30x+45 � ψ1 π − ε 4 � 1+√x �� 3−√x � ψ1 π + ε 16 � 1+√x �3� 3−√x � φ1 π � I111 +ε 4 (1−x)(9−x) ψ1 π I211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (26) Note that the definition of the master integrals ⃗J involves only ψ1 and φ1 (through d dxψ1), but not ψ2 nor φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The basis ⃗K, appearing in [9], is defined by K1 = ε2 I110, (27) K2 = −ε(1+2ε) 4π � 1+√x �� 3−√x �� ψ2 − 1 4 � 1+√x �2 φ2 � I111 + ε 4π (1−x)(9−x)ψ2I211, K3 = +ε(1+2ε) 4π � 1+√x �� 3−√x �� ψ1 − 1 4 � 1+√x �2 φ1 � I111 − ε 4π (1−x)(9−x)ψ1I211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In the definition of the master integrals ⃗K all periods ψ1,ψ2 and all quasi-periods φ1,φ2 appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The master integrals K2 and K3 are related by ψ2 ↔ ψ1, φ2 ↔ φ1 and an overall minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 The differential equations The differential equation in the basis⃗I reads d⃗I = A⃗I, (28) with A = \uf8eb \uf8ed 0 0 0 0 −(1+2ε) 3 0 −1 3 (1+2ε)(1+3ε) 1+3ε \uf8f6 \uf8f8 dx x + \uf8eb \uf8ed 0 0 0 0 0 0 ε2 4 1 4 (1+2ε)(1+3ε) −(1+2ε) \uf8f6 \uf8f8 dx x−1 + \uf8eb \uf8ed 0 0 0 0 0 0 −ε2 4 1 12 (1+2ε)(1+3ε) −(1+2ε) \uf8f6 \uf8f8 dx x−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (29) In this basis, the entries are rational dlog-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However, the differential equation is not in ε-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The differential equation in the basis ⃗J reads d⃗J = εB⃗J, (30) 8 with B = \uf8eb \uf8ed 0 0 0 0 ω2 ω0 ω3 ω4 ω2 \uf8f6 \uf8f8 (31) and ω0 = 2πi dτ = 2πiW ψ2 1 dx, ω2 = − f2(τ) (2πi)dτ = dx 2x − dx x−1 − dx x−9, ω3 = f3(τ) (2πi)dτ = − 1 2 ψ1 π dx, ω4 = f4(τ) (2πi)dτ = (x+3)4 48x(x−1)(x−9) �ψ1 π �2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (32) f2, f3 and f4 are modular forms of Γ1(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The minus sign in front of f2 is convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In terms of the variable x they are given by f2 = 1 24 � 3x2 −10x−9 ��ψ1 π �2 , f3 = − 1 24x(x−1)(x−9) �ψ1 π �3 , f4 = 1 576 (3+x)4�ψ1 π �4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (33) Their q-expansions are given in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The differential equation in the basis ⃗K reads d⃗K = εC⃗K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (34) with C = \uf8eb \uf8ed 0 0 0 C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 \uf8f6 \uf8f8 (35) and C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 = −1 2 ψ2 π dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (36) C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 = iπ 6 � (1+x) ψ1 π ψ2 π + � 3x2 −10x−9 � ψ2 π ∂xψ1 π +2x(x−1)(x−9) ∂xψ1 π ∂xψ2 π � dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 = iπ 6 � (1+x) �ψ2 π �2 + � 3x2 −10x−9 � ψ2 π ∂xψ2 π +2x(x−1)(x−9) �∂xψ2 π �2� dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 9 C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 = 1 2 ψ1 π dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 = −iπ 6 � (1+x) �ψ1 π �2 + � 3x2 −10x−9 � ψ1 π ∂xψ1 π +2x(x−1)(x−9) �∂xψ1 π �2� dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' C3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 = −iπ 6 � (1+x) ψ1 π ψ2 π + � 3x2 −10x−9 � ψ1 π ∂xψ2 π +2x(x−1)(x−9) ∂xψ1 π ∂xψ2 π � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 Periods on the maximal cut In this section we investigate the period matrices on the maximal cut of the sunrise integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' On the maximal cut of the sunrise integral only the last two master integrals are relevant (either I2,I3 or J2,J3 or K2,K3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The defining property for basis ⃗K is that the period matrix on the maximal cut is diagonal and constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We denote by ϕX i , X ∈ {I,J,K}, i ∈ {1,2,3}, (37) the integrand of the master integral Xi in the loop-by-loop Baikov representation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In the loop-by-loop Baikov representations we have four integration variables z1 − z4, where z1 − z3 correspond to the three propagators and z4 to an irreducible scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let C MaxCut be the integration domain selecting the maximal cut, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' a small counter-clockwise circle around z1 = 0, a small counter-clockwise circle around z2 = 0 and a small counter-clockwise circle around z3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We set z4 = u in accordance with the notation used in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We denote by γ1 and γ2 the two cycles of the elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' They define the integration domain in the variable u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We define C2 = C MaxCut ∪γ1, C3 = C MaxCut ∪γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (38) We consider the period matrix PX = � � ϕX 2 |C2 � � ϕX 2 |C3 � � ϕX 3 |C2 � � ϕX 3 |C3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (39) In the i-th row of this matrix we then look at the leading term in the expansion in the dimensionsal regularisation parameter ε for this row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We denote the order of the leading term of row i by jmin(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This defines a matrix PX,leading with entries PX,leading i j = coeff �� ϕX i |Cj � ,ε jmin(i)� ε jmin(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (40) One finds PI,leading = −8iπ � ψ1 ψ2 ψ1− 1 4(1+√x)2φ1 (1−√x)(3+√x) ψ2− 1 4(1+√x)2φ2 (1−√x)(3+√x) � , 10 PJ,leading = 2i � (2πiε)2 (2πiε)2τ 0 −(2πiε) � , PK,leading = 4πε � 1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (41) As advertised, we see that PK,leading is proportional to the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Note that PJ,leading can be written as PJ,leading = 2i � (2πiε)2 0 0 −(2πiε) �� 1 τ 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (42) This is the decomposition of the period matrix PJ,leading into a semi-simple matrix and an unipo- tent matrix [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='4 Solutions In the basis ⃗J we may give a solution for the master integrals in terms of iterated integrals of modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let f1(τ), f2(τ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fn(τ) be a set of modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We define the n-fold iterated integral of these modular forms by I (f1, f2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='τ,τ0) = (2πi)n τ � τ0 dτ1 τ1 � τ0 dτ2··· τn−1 � τ0 dτn f1 (τ1) f2 (τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' fn(τn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (43) With q = exp(2πiτ) we may equally well write I (f1, f2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='τ,τ0) = q � q0 dq1 q1 q1 � q0 dq2 q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' qn−1 � q0 dqn qn f1 (τ1) f2(τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' fn(τn), τ j = 1 2πi lnqj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (44) It will be convenient to introduce a short-hand notation for repeated letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We use the notation { fi}j = fi, fi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fi � �� � j (45) to denote a sequence of j letters fi and more generally {fi1, fi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fin}j = fi1, fi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fin,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='., fi1, fi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fin � �� � j copies of fi1, fi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fin (46) to denote a sequence of ( j ·n) letters, consisting of j copies of fi1, fi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For example { f1, f2}3 = f1, f2, f1, f2, f1, f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (47) 11 Our standard choice for the base point τ0 will be τ0 = i∞, corresponding to q0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This is unproblematic for modular forms which vanish at the cusp τ = i∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In this case we have for a single integration f = ∞ ∑ j=1 ajqj ⇒ q � 0 dq1 q1 f = ∞ ∑ j=1 aj j qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (48) For modular forms which attain a finite value at the cusp τ = i∞ we employ the standard “trailing zero” or “tangential base point” regularisation [10,25,26]: We first take q0 to have a small non- zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The integration will produce terms with ln(q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let Rln(q0) be the operator, which removes all ln(q0)-terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' After these terms have been removed, we may take the limit q0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' With a slight abuse of notation we set I ( f1, f2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) = lim q0→0Rln(q0) \uf8ee \uf8f0 q � q0 dq1 q1 q1 � q0 dq2 q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' qn−1 � q0 dqn qn f1 (τ1) f2(τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' fn (τn) \uf8f9 \uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (49) We define the boundary constants Bk for the sunrise integral J2 by lim q→0Rln(q)J2 = e 2 ∞ ∑ n=2 (−1)n n ζnεn ∞ ∑ k=2 εkBk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (50) The left-hand side corresponds to setting all iterated integrals to zero, including the ones which are proportional to powers of ln(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The boundary values Bk are collected in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let us mention that the boundary values Bk are of weight k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (50) is therefore of uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We may express the master integrals in the basis ⃗J to all orders in the dimensional regulari- sation parameter in terms of iterated integrals of modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We have J1 = e 2 ∞ ∑ n=2 (−1)n n ζnεn , J2 = e −εI(f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+2 ∞ ∑ n=2 (−1)n n ζnεn \uf8f1 \uf8f2 \uf8f3 � ∞ ∑ j=0 � ε2jI � {1, f4}j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q � − 1 2ε2j+1I � {1, f4}j ,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q ��� ∞ ∑ k=2 εkBk + ∞ ∑ j=0 ε j+2 ⌊ j 2⌋ ∑ k=0 I � {1, f4}k ,1, f3,{f2}j−2k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q � \uf8fc \uf8fd \uf8fe, J3 = e −εI(f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+2 ∞ ∑ n=2 (−1)n n ζnεn \uf8f1 \uf8f2 \uf8f3 � ∞ ∑ j=0 � ε2j+1I � { f4,1}j , f4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q � − 1 2ε2jI � { f4,1}j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q ��� ∞ ∑ k=2 εkBk 12 + ∞ ∑ j=0 ε j+1 ⌊ j 2⌋ ∑ k=0 I � {f4,1}k , f3,{ f2}j−2k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q � \uf8fc \uf8fd \uf8fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (51) The expression for J2 appeared already in [10], the expression for J3 follows from (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (25)) J3 = 1 ε 1 2πi d dτJ2 + f2J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (52) For the first few terms of ε-expansion we have J1 = 1+ζ2ε2 − 2 3ζ3ε3 + 7 10ζ2 2ε4 +O � ε5� , J2 = [B2 +I (1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)]ε2 + � B3 − 1 2B2I (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−B2I ( f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−I (1, f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−I ( f2,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) � ε3 + � B4 +ζ2B2 − 1 2B3I (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−B3I ( f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+ 1 2B2I (1, f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+ 1 2B2I ( f2,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) +B2I (1, f4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+B2I (f2, f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+ζ2I (1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+I (1, f2, f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+I ( f2, f2,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) +I (1, f4,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+I (f2,1, f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) � ε4 +O � ε5� , J3 = εI ( f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+ � −1 2B2 −I ( f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) � ε2 + � −1 2B3 + 1 2B2I ( f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+B2I (f4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) +ζ2I ( f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+I ( f2, f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+I (f4,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) � ε3 + � −1 2B4 − 1 2ζ2B2 + 1 2B3I (f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) +B3I ( f4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)− 2 3ζ3I ( f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−B2I (f4, f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−B2I ( f2, f4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)− 1 2B2I (f2, f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) −1 2B2I (f4,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−ζ2I ( f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−I ( f2, f2, f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−I ( f4, f2,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) −I (f2, f4,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−I (f4,1, f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) � ε4 +O � ε5� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (53) Let us also summarise the boundary values: From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (50) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (51) we obtain lim q→0Rln(q)J1 = e 2 ∞ ∑ n=2 (−1)n n ζnεn , lim q→0Rln(q)J2 = e 2 ∞ ∑ n=2 (−1)n n ζnεn ∞ ∑ k=2 εkBk, lim q→0Rln(q)J3 = −1 2e 2 ∞ ∑ n=2 (−1)n n ζnεn ∞ ∑ k=2 εkBk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (54) In all three cases the right-hand sides are of uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 13 Given a solution in the basis ⃗J, we easily obtain a solution in the basis ⃗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The two bases are related by ⃗K = U⃗J, (55) with U = \uf8eb \uf8ed 1 0 0 0 −(1+2ε) 2πiε −g2 ·τ τ 0 g2 −1 \uf8f6 \uf8f8 (56) and g2 = 1 24 �� 3x2 −10x−9 � ψ1 π +4x(1−x)(9−x) ∂xψ1 π � ψ1 π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (57) In the modular variable τ the quantity g2 is given by g2 = f2 +2 π ψ1 1 2πi d dτ ψ1 π = 4 � 3b2 1 −3b1b2 −6b2 2 −e2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (58) The modular forms b1 and b2 and the quasi-modular form e2 are defined in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The quantity g2 is a quasi-modular form of modular weight 2 and depth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For γ ∈ Γ1(6) the quantity g2 transforms as (g2|2γ)(τ) = g2(τ)+ 2 2πi c cτ+d , γ = � a b c d � , (59) where the operator |kγ is defined by ( f|kγ)(τ) = (cτ+d)−k · f(γ(τ)), γ(τ) = aτ+b cτ+d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (60) For the first few terms of ε-expansion we have K2 = 1 2πi [−B2 +I ( f3,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)]ε+ 1 2πi [−B3 −2B2 +B2I ( f2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−I ( f2, f3,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−2I (1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) −2g2I (1,1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−g2I (1, f3,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)−g2B2I (1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)]ε2 +O � ε3� , K3 = −I ( f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)ε+ �1 2B2 +I ( f2, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q)+g2B2 +g2I (1, f3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='q) � ε2 +O � ε3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (61) Let us look at the boundary values of K2 lim q→0Rln(q)K2 = − B2 2πiε− (B3 +2B2) 2πi ε2 +O � ε3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (62) The term −2B2 2πi ε2 (63) is of weight minus one and spoils the uniform weight property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Hence we conclude that the basis ⃗K is not of uniform weight if we require that the notion of uniform weight is compatible with restrictions in the kinematic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 14 5 Purity and simple poles In this section we address the second main question of this paper: The relation between uniform weight and simple poles in the elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1 Definition of pure functions in the literature We recapitulate the definitions of unipotent and pure function as given in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [18]: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' A function is called unipotent, if it satisfies a differential equation without a homo- geneous term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' To give an example, the functions ln(x) and Li2(x) are unipotent d dx ln(x) = 1 x, d dxLi2 (x) = −1 x ln(1−x), (64) while ex is not d dxex = ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (65) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Unipotent functions, whose total differential involves only pure functions and one- forms with at most simple poles are called pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The standard example are multiple polylogarithms, whose total differential is given by dG(z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=',zr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='y) = r ∑ j=1 G(z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=', ˆz j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=',zr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='y) � d ln � z j−1 −z j � −d ln � z j+1 −z j �� , (66) where we set z0 = y and zr+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' A hat indicates that the corresponding variable is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In addition one uses the convention that for z j+1 = z j the one-form d ln(z j+1 −z j) equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Clearly, the one forms d ln � z j+1 −z j � = dz j+1 −dz j z j+1 −z j (67) have only simple poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='2 Iterated integrals of modular forms Let us now look at iterated integrals of modular forms, as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' It is clear that these iterated integrals are unipotent functions, as differentiation removes one integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We investigate the order of the poles of the total differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We denote by H = { τ ∈ C | Im(τ) > 0 } (68) 15 the complex upper half-plane and by H = H∪{i∞}∪Q (69) the extended complex upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Under the map q = exp(2πiτ) the complex upper half- plane H is mapped to the punctured open disk D = { q ∈ C | 0 < |q| < 1 } (70) and H is mapped to D = D∪{0}∪ � e2πir | r ∈ Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (71) Let fk(τ) be a modular form of weight k for a congruence subgroup Γ and ωmodular k = 2πi fk (τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (72) For simplicity we assume that � 1 1 0 1 � ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In this case fk has the q-expansion [27] fk = ∞ ∑ n=0 anqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (73) (In the general case fk will have an expansion in q 1 N′ , where N′ is the smallest positive inte- ger such that fk(τ + N′) = fk(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The general case is only from a notational perspective more elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=') In addition we will always assume that modular forms are normalised such that the coefficients of their q-expansion are algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We view ωmodular k as a differential one- form on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In the variable q we have ωmodular k = ∞ ∑ n=0 anqn−1dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (74) This shows immediately that ωmodular k is holomorphic on D and has a simple pole at q = 0 if a0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Thus, in a neighbourhood of q = 0 the differential one-form ωmodular k has at most simple poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let us now discuss if this extends globally to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The answer will be no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We have to look at the other cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We investigate the behaviour at q0 = e2πi(− d c), c,d ∈ Z, c ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (75) We may derive the behaviour of ωmodular k at q0 from the modular properties of fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We consider the modular transformation γ = � a b c d � ∈ SL2(Z), γ−1 = � d −b −c a � (76) 16 and set τ′ = γ(τ) = aτ+b cτ+d , q′ = e2πiτ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (77) This maps τ = −d c to τ′ = i∞ and q0 to q′ 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the automorphic factor we have cτ+d = c 2πi (q−q0) q0 +O � (q−q0)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (78) ( fk|kγ−1)(τ′) has again a q′-expansion as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (73) � fk|kγ−1�� τ′� = ∞ ∑ n=0 a′ n � q′�n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (79) If fk is a modular form for the congruence subgroup Γ and γ ∈ Γ we have a′ n = an, otherwise the coefficients need not be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Usually we are interested in the cusps not equivalent to τ = i∞, this implies γ ∈ SL2 (Z)\\Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For a′ 0 ̸= 0 we have ωmodular k = a′ 0 � c 2πi �−k qk−1 0 dq (q−q0)k +O � (q−q0)−k+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (80) Thus we see that whenever fk is non-vanishing at the cusp τ0 = −d c, the differential one-form ωmodular k has a pole of order k in the variable q at q = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Globally, ωmodular k has poles up to order k on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='3 Elliptic polylogarithms The discussion of the previous sub-section is not restricted to iterated integrals of modular forms and carries over to elliptic polylogarithms �Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let g(k)(z,τ) denote the coefficients of the Kronecker function and set ωKronecker k (z,τ) = (2πi)2−k � g(k−1) (z,τ)dz+(k −1)g(k) (z,τ) dτ 2πi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (81) We may view ωKronecker k as a differential one-form on the two-dimensional moduli space M1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Coordinates on this moduli space are (z,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The elliptic polylogarithms �Γ are iterated integrals of ωKronecker k (z−cj,τ) along z at constant τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' It is known that the functions g(k)(z,τ) have at most simple poles in z and when restricted to τ = const the elliptic polylogarithms �Γ are pure functions in the sense of definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However in the applications towards Feynman integrals it is usually the case that the assumption τ = const is not justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' A variation of the kinematic variables of the Feynman integral will imply a variation of τ and we have to consider the τ-dependence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the argument we want to make it is sufficient to restrict to z = a + bτ with a,b ∈ Q and k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In this case the differential one-forms ωKronecker k reduce to the form of ωmodular k [24] and the argument from the previous sub-section applies: In this case the differential one-forms ωKronecker k may have poles up to order k in the variable q (or τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='4 Modular transformations We have seen that locally in the coordinate chart D ∪ {0} the basis ⃗J satisfies the criteria of definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This coordinate chart includes the point x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let us now investigate the global picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For the sunrise integral we have four singular points x ∈ {0,1,9,∞} and we may cover the kinematic space with four charts, such that each chart includes exactly one singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In each chart we may construct a basis, which satisfies the criteria of definition 2 locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In different charts we will have different coordinates τ and τ′, but also different bases of master integrals ⃗J and ⃗J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The coordinates τ and τ′ will be related by a modular transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The modular transformation induces also the transformation between ⃗J and ⃗J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let us discuss the behaviour near the cusp τ0 = −d c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The modular transformation γ defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (76) maps τ0 = −d c to τ′ 0 = i∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Let fk be a modular form for a congruence subgroup Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Then by definition fk(τ) is holomorphic on H and ( fk|kγ−1)(τ′) has a q′-expansion as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (79) for any γ ∈ SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This suggest to change in a neighbourhood of τ = −d c coordinates from q to q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The differential one-form 2πi � fk|kγ−1�� τ′� dτ′ (82) has then a simple pole at q′ = 0 (corresponding to τ = −d c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However, ωmodular k as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (72) does not transform under this coordinate change into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Instead we find ωmodular k = � −cτ′ +a �k−2 ·2πi � fk|kγ−1�� τ′� dτ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (83) (−cτ′ + a) is the automorphic factor for γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' For k ̸= 2 this factor spoils that iterated integrals of modular forms transform under modular transformations into iterated integrals of modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' However, elliptic Feynman integrals transform nicely: Let us consider for γ(τ) = aτ+b cτ+d , γ ∈ SL2(Z) (84) the combined transformation [21] ⃗J′ = \uf8eb \uf8ed 1 0 0 0 1 cτ+d 0 0 − c 2πiε cτ+d \uf8f6 \uf8f8⃗J, τ′ = aτ+b cτ+d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (85) One obtains d⃗J′ = εB′⃗J′ (86) with B′ = 2πi \uf8eb \uf8ed 0 0 0 0 −( f2|2γ−1)(τ′) 1 ( f3|3γ−1)(τ′) ( f4|4γ−1)(τ′) −( f2|2γ−1)(τ′) \uf8f6 \uf8f8dτ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (87) 18 As Γ(6) is a subgroup of Γ1(6) we have Mk(Γ1(6)) ⊂Mk(Γ(6)) and as Γ(6) is a normal subgroup of SL2(Z) it follows that fk|kγ−1 ∈ Mk(Γ(6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (88) We see that the entries of B′ are again differential one-forms of the form as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We may express ⃗J′ again in terms of iterated integrals of modular forms, this time in the variable q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' It can be shown that the boundary constants are again of uniform weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Hence it follows that in the coordinate chart with coordinate q′ (or τ′) the basis ⃗J′ satisfies the criteria of definition 2 locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 6 Conclusions For Feynman integrals which evaluate to multiple polylogarithms we have a clear understanding of purity: These are Feynman integrals, whose term of order j in the ε-expansion is pure of transcendental weight j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We are interested in extending this concept to Feynman integrals beyond the ones which evaluate to multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This is non-trivial and in this paper we discussed some subtleties: We showed that an ε- factorised differential equation alone does not lead necessarily lead to a solution which is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The boundary values have to be pure as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' This applies in particular to a basis constructed by the requirement that the period matrix on the maximal cut is proportional to the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The argument we presented is agnostic to the exact definition of purity beyond the case of multiple polylogarithms, we only assumed that the definition of transcendental weight in the general case is compatible with the restriction of the kinematic space to a sub-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In the second part of the paper we adopted a particular definition of purity from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We showed that this definition works only locally – but not globally – for a particular basis of the two-loop equal mass sunrise integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Of course, it might well be that this particular basis is not the optimal one, but another possibility is that the definition of purity needs a more refined definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The modular transformation properties, which we discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='4, point towards a possible modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We believe that the detailed analysis we carried out in this paper will be helpful for a defi- nition of purity which not only includes the elliptic case, but also Feynman integrals related to Calabi-Yau geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Acknowledgements S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' thanks the Niels Bohr Institute for hospitality and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' thanks the Mainz Institute of Theo- retical Physics for hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' is partially supported by a Carlsberg Foundation Reintegration Fellowship, and has re- ceived funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 847523 “INTERACTIONS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 19 A Modular forms In this appendix we give the q-expansions of the modular forms f2, f3 and f4, appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (32) and the q-expansions of ψ1 (which is a modular form of modular weight 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In addition, we define the Eisenstein series e2, which appears in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We start by introducing a basis {b1,b2} for the modular forms of modular weight 1 for the Eisenstein subspace E1(Γ1(6)): b1 = E1(τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='χ1,χ−3), b2 = E1(2τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='χ1,χ−3), (89) where χ1 and χ−3 denote primitive Dirichlet characters with conductors 1 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' In terms of the coefficients g(k)(z,τ) of the Kronecker function we have b1 = √ 3 6π g(1)(1 3,τ), b2 = − √ 3 12π � g(1)(1 3,τ)−g(1)(1 6,τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (90) Then f2 = −6 � b2 1 +6b1b2 −4b2 2 � , f3 = 36 √ 3 � b3 1 −b2 1b2 −4b1b2 2 +4b3 2 � , f4 = 324b4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (91) In terms of the coefficients g(k)(z,τ) of the Kronecker function we have f2 = 1 2π2 � 3g(2)(1 2,τ)−g(2)(1 3,τ)+g(2)(1 6,τ) � , f3 = 1 4π3 � 15g(3)(1 3,τ)−12g(3)(1 6,τ) � , f4 = 1 4π4 � −18g(4)(0,τ)−27g(4)(1 3,τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (92) The q-expansions are f2 = −1 2 −8q−4q2 −44q3 +4q4 −48q5 −40q6 +O � q7� , f3 = −3 √ 3 � q−5q2 +9q3 −11q4 +24q5 −45q6� +O � q7� , f4 = 1 4 +6q+54q2 +222q3 +438q4 +756q5 +1998q6 +O � q7� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (93) In addition we have ψ1 π = 2 √ 3(b1 +b2) = 2 3 √ 3 � 1+3q+3q2 +3q3 +3q4 +3q6� +O � q7� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (94) 20 We define the Eisenstein series e2 by e2 (τ) = 1 2(2πi)2 ∑ ′ (n1,n2)∈Z2\\(0,0) 1 (n1 +n2τ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (95) The prime at the summation sign denotes the Eisenstein summation prescription defined by ∑ ′ (n1,n2)∈Z2 f (z+n1 +n2τ) = lim N2→∞ N2 ∑ n2=−N2 � lim N1→∞ N1 ∑ n1=−N1 f (z+n1 +n2τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (96) The q-expansion of e2 starts with e2 (τ) = − 1 24 +q+3q2 +4q3 +7q4 +6q5 +12q6 +O � q7� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (97) The Eisenstein series e2 is a quasi-modular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' B Boundary values In this appendix we give the boundary values Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' These are easily obtained from [10, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' We have ∞ ∑ k=0 εkBk = 3 43−ε � h− 2πε 3 Γ(1+2ε) Γ(1+ε)2 � , (98) where h = 1 i � (−r3)−ε 2F1 (−2ε,−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='r3)− � −r−1 3 �−ε 2F1 � −2ε,−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='r−1 3 �� (99) and r3 = exp(2πi/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The hypergeometric function can be expanded systematically in ε with the methods of [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' The first few terms are given by 2F1 (−2ε,−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='x) = 1+2ε2Li2 (x)+ε3 [2Li3(x)−4Li21 (x,1)] +ε4[2Li4 (x)−4Li31 (x,1)+8Li211 (x,1,1)]+O � ε5� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (100) The first few boundary values are given by B0 = 0, B1 = 0, B2 = 3 2i � Li2 (r3)−Li2 � r−1 3 �� , B3 = 3 2i � −2Li21 (r3,1)−Li3(r3)+2Li21 � r−1 3 ,1 � +Li3 � r−1 3 �� 21 −ln(3)B2, B4 = 3 2i � 4Li211 (r3,1,1)−2Li31 (r3,1)+Li4 (r3)−4Li211 � r−1 3 ,1,1 � +2Li31 � r−1 3 ,1 � −Li4 � r−1 3 �� −ln(3)B3 − 1 2 ln2(3)B2 + 1 3ζ2B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (101) These can be reduced to polylogarithms of depth 1 as follows [33,34]: B2 = 3 Im Li2 (r3), B3 = 24 5 Im Li3 � i√ 3 � − 17 90π3 − 1 10π(ln(3))2 , B4 = −63 10Im Li4 (r3)+ 48 5 Im Li4 � i√ 3 � + 17 90π3 ln(3)+ 1 30π(ln(3))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' (102) References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Henn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 110, 251601 (2013), arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Cachazo, (2008), arXiv:0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Arkani-Hamed, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Bourjaily, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Cachazo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Trnka, JHEP 06, 125 (2012), arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='6032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Arkani-Hamed, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Bai, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Lam, JHEP 11, 039 (2017), arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='04541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Primo and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tancredi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' B916, 94 (2017), arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='08397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Primo and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tancredi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' B921, 316 (2017), arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='05465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Bourjaily, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Herrmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Trnka, JHEP 06, 059 (2017), arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='05460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Bourjaily, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Kalyanapuram, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Langer, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Patatoukos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' D 104, 125009 (2021), arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='02210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Frellesvig, JHEP 03, 079 (2022), arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='07968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Adams and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 12, 193 (2018), arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='08895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Adams and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' B781, 270 (2018), arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='05020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Bogner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Müller-Stach, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' B 954, 114991 (2020), arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='01251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Müller and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, JHEP 07, 101 (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='04818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Pögel, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, JHEP 09, 062 (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='12893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Pögel, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='04292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Pögel, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='08908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Duhr, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Klemm, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Nega, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tancredi, (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='09550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Broedel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Duhr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Dulat, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Penante, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tancredi, JHEP 01, 023 (2019), arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='10698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Broedel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Duhr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Dulat, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tancredi, JHEP 05, 093 (2018), arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='07089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [20] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Hönemann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tempest, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' D98, 113008 (2018), arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='09308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' B 964, 115309 (2021), arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='07311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Feynman Integrals (Springer, 2022), arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='03593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 22 [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Frellesvig and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Papadopoulos, JHEP 04, 083 (2017), arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='07356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Broedel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Duhr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Dulat, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Penante, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tancredi, JHEP 08, 014 (2018), arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='10256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [25] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Brown, (2014), arXiv:1407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='5167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Walden and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 265, 108020 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='05271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Miyake, Modular Forms (Springer, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Adams, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Bogner, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 57, 032304 (2016), arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='05630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Moch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Uwer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 43, 3363 (2002), hep-ph/0110083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Weinzierl, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 145, 357 (2002), math-ph/0201011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Moch and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Uwer, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 174, 759 (2006), math-ph/0508008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Huber and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Maitre, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' 175, 122 (2006), hep-ph/0507094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [33] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Frellesvig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Tommasini, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Wever, JHEP 03, 189 (2016), arXiv:1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='02649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Frellesvig, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Kudashkin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content=' Wever, JHEP 05, 038 (2020), arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} +page_content='07776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdE0T4oBgHgl3EQfVgAw/content/2301.02264v1.pdf'} 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Eldamak +December 2022 +1 +Introduction +According to the World Health Organization (WHO), 30 million people are in need of prosthetic +and orthotic devices [1]. Some people are born with this limb loss, while others lose limbs due to +diseases such as Cancer, diabetes, and work accidents. Additionally, limb amputation is among +the most severe and heavily reported injuries among veterans during war [2, 3]. Example of +female with hand amputation is shown in Figure 1. +Figure 1: Female with Prosthetic limb [4] +The medical applications of integrated circuit technology have recently made significant ad- +vances, thus improving human quality of life. Moreover, the use of microelectronics integration +1 +arXiv:2301.00163v1 [eess.SP] 31 Dec 2022 + +dominates a lot of medical applications, especially portable and wearable battery-operated de- +vices. Bio-signals mostly arise from natural physiological processes, such as cardiac potentials +(ECG -electro-cardiogram), potentials of the ocular tissue (EOG - electro-oculogram), potentials +of the muscular tissues (electro-myogram -EMG), brain potential (electro-encephalogram -EEG), +and respiratory signals , etc. Electro-myogram - EMG is an important factor for muscle disease +diagnosis. Furthermore, it’s the key factor in connecting any amputee to a prosthetic limb. This +can be done through extracting the EMG signal from the body using a readout electronics that +can detect the muscles electrical activity. Consequently, the extracted signal is processed and +used to control the prosthetic limb. Thus, the objective of this report is to provide the reader +with the basic understanding of integrated solutions for controlling prosthetic limbs either arms +or legs. +The top level block diagram of a smart EMG acquisition system is shown in Fig. 2. The +system includes a self-powered readout portable acquisition device for measuring the patient’s +EMG signal in order to send it to a controller that can be used to emulate the right action +to the prosthetic limb similar to the same action in a normal person. It should be noted that +miniaturized EMG acquisition system idea, which continuously monitor muscles activity, can be +extended to different applications such as physical rehabilitation and prosthesis. +Figure 2: Block diagram of a general smart sEMG recorder [5] +2 + +Tissue Interface +ProcessingUnit +Sensor Interface2 +System Architecture +An electronic system can control a prosethetic device by monitoring the EMG signals of the arm, +and use those signals to control the prothetic arm. Moreover, the devices can be battery-free by +being powered solely using energy harvesting from the ambient. +Since these prosthetic devices requires precise fitting to the residual limb, pressure and tem- +perature sensor at the skin-prosthetic interface are added to the system. Pressure sensors are +needed for monitoring the prosthetic limb to avoid the development of regions of high pressure +as the limb moves during walking or grasping objects. Temperature sensor are necessary as high +temperature can accelerate tissue damage [6]. The signals from the sensor at the skin-prosthetic +can be transmitted to the outer surface of the prosthetic socket using Near Field Communication +(NFC) or to a smart phone using Bluetooth Low Energy (BLE) as shown in Figure 3. +Figure 3: Illustration of sensors mounted at the skin-prosthetic interface transmitting data to +the device at the outer surface of the prosthetic leg using NFC and to smart phone using BLE +[6]. +3 +Block Diagram +Three major research directions are available when designing an EMG acquisition system. The +first is to acquire the signal from the surrounding noisy environment using a sensor interface +3 + +Residual +limb +NFC/BLE +modules +Multimodal +battery-free +sensors +Prosthetic leg +Prosthetic +Portable +socket +NFC/BLE +electronicdevice +modulesFigure 4: Block diagram of the proposed smart sEMG recorder including sensors, AFE, and RF +integrated system [5] +circuit that’s designed in CMOS technology. The second involves reducing the form factor and +power consumption of the acquisition system. The third is the signal conversion to the digital +world and the interface with the digital controller. At this point, the extracted EMG signal +is in a digital form and can be processed through FPGA or any other processor to control a +Pprosthetic limb. +A typical block diagram of the proposed EMG acquisition system is shown in Fig. +4. +The system consists of an EMG sensor, analog front end (AFE), and radio frequency (RF) +transmission unit. The AFE is typically composed of an analog amplification, filtration, analog +to digital converter (ADC), and controller to process the digital signal and send it to a prosthetic +limb. The acquisition system design can be integrated on a single chip, then the digital data is +fed to FPGA or a controller. +In addition, because the integrated solution takes a considerable time during design, fabrica- +tion, and testing phases, a discrete solution in parallel with the integrated one can be used as a +proof of concept to validate the proposed methodology. +Figure 5 shows a detailed system block diagram of the proposed smart sEMG acquisition +system. An analog multiplexer is inserted to choose between different EMG electrodes in the +smart sEMG recorder shown in the figure. The design of each of the building blocks involves +4 + +AFE+RF +NSPU +FlexBandFigure 5: Detailed block diagram of the proposed smart sEMG acquisition system [5] +several design challenges requiring some research. The following section includes a list of major +research directions that can be pursued. +4 +Circuit Implementation +In the following subsections, the basic system building blocks are introduced. First, the EMG +sensor specifications are explored. +Second, the low noise amplifier LNA design is presented. +Third, the filter design and bandwidth are provided. Fourth, the signal conversion from analog +to digital is presented through an ADC. Last, digital signal processing through FPGA is explored. +4.1 +Sensor Specifications +EMG sensor placement plays an important role in signal acquisition. According to its orientation +and position, the EMG signal strength varies significantly. This effect is shown in Fig 6. As +seen, by placing the sensor in the middle of muscle fiber, the maximum signal strength can be +easily obtained. Otherwise, the signal degrades significantly when placing the sensor far away +from the middle. +EMG sensor can be represented in different forms. It can be in either needle that is inserted +into the muscle or surface electrode that picks the signal from the skin. An example of surface +5 + +Smart sEMG Recorder +FPGA +16-Channel Recorder +GBDT based NSPU +Flexible +16x +GBDT Core +Feature +Band +LNA +Extractors +00 +MUX +PGA +ADC +16:1 +LNA +Pre- +Processing +Result +Model +Digital Logic +Data Ready| CLK_ADC +Generator +Loader +nRF52 +Ping Pong Buffer +Recorder +TF Card +BLE +CIC Filter +Interface +Buffer A +Buffer B +Interface i +X +:= +TF Card +Mobile Devices +Possible Applications +Offline trainingFigure 6: Effect of EMG sensor position [7] +EMG sensor specifications that have to be met through out the design are as follow shown in +Fig. 7. +4.2 +Low Noise Amplifier Design +It’s the first and the major block in the EMG chain that comes after the sensor. The measurement +sensitivity and accuracy is determined in this stage. This complicates the design and requires a +large amount of adaptability to accommodate the input signal. The previous stage, which is the +EMG sensor, adds large parasitic capacitance at the input of this stage, and thus reduces gain, +bandwidth, noise performance and the sensitivity of the amplifier. +Sources of noise and interference like flicker noise, electrodes offset, and 60 Hz power line +noise can affect the whole acquisition procedure. The bandwidth of the EMG signal is up to +6 + +Raw EMG output +Innervation Zone +Correct Placement +Midline of the muscle belly +between an innervation zone +and a myotendon junction +Midline Offset +Myotendon JunctionFigure 7: sensor specifications [8] +500 Hz with amplitude that ranges from 0.1 to 5 mV and the high-frequency noise can be easily +removed using a low pass filter. However, low-frequency noise and DC offset fall within the +EMG bandwidth and hence require different rejection techniques. Chopping technique is one +of the best candidates to modulate the offset and flicker noise to a higher spectrum which in +turn enable the acquisition system to effectively suppress the interference from ambient and 1/f +noise. Different architectures with different requirements in terms of input signal levels, BW and +amplitudes are proposed in literature [9, 10]. Figure 8 shows the block diagram of implemented +analog front-end for acquiring of EEG, ECG, and EMG signals [9]. The shown diagram consists +of a chopper instrumentation amplifer in addition to capacitive coupling, filter stage to remove +the chopping spikes, a digitally controlled variable gain amplifier. +4.3 +Filter Design +A Gm-C filter cab be used in the design. A standard architecture is shown in Figure 9. Offset +from the electrodes can be canceled using current-mode DAC [11]. Power Consumption of this +7 + +DataLITE Wireless EMG Amplifier +Wired EMG Amplifier +Product Ref +LE230FW +SX230FW +42 × 24 × 14 mm +38 x 20 +Dimensions +Two 4 mm snap connectors on 100 mm wires +Two 4 mm snap connectors on 100 mm wires +Mass +17 g (excluding cable and plug) +8g (excluding cable and plug) +Bandwidth +10 - 250,470, 950, 5000Hz +20 - 460Hz +5Hz - 480Hz +Additional Bandwidths +N/A +5Hz - 1000Hz +Contact Diameter +Dependant on electrode size +Contact Center Spacing +Variable +Electrodes +Disposable +CMRR @ 60 Hz (dB) +> 96 dB (typically 110 dB) +Full Scale ++/- 6 mV Peak to Peak ++/- 3 mV Peak to Peak +Gain ++/- 60 microvolts to +/- 6 millivolts +Standard unit x1000 (100 als0 available) +Input Impedance +>100 Mohms +Accuracy ++/- 1.0% ++/- 2% full scale +Noise +<5μv +Supply Voltage +N/A ++3.50 to +5.5 Vdc +Battery Life +Up to 8 hours +N/A +Battery Type +Rechargeable Li-lon Polymer +N/A +Wireless Transmission +Tolerant for 100 mS +N/A +Data Loss +1.25m cable +Range from Interface +Wireless range up to 30m +(custom lengths available on request) +Compatible Interfaces +DataLITE PIONEER, ADVANCE, EXPLORE +DataLOG, DataLINK, Amplifier or 3rd partyFigure 8: Architecture of the bio-potential readout front-end for the acquisition of EEG, ECG, +and EMG signals [9] +topology can also be reduced by low-voltage supply operation [10]. +Figure 9: Transistor level implementation of Gm-C filter. DDA: Differential Difference Amplifier +[11]. +4.4 +ADC Design +Non-uniform sampling can minimize the power consumption of ADC while digitizing activity- +dependent biological signals. For example, a continous-time (CT) charge-based ADC that ac- +8 + +DC Level +Select BW Select Gain +Programmable +1pF +Gain Stage +BW Select +Cext2 +OTA2 +■out +Buffer +vin+ +IA +C12 +OTA +vin- +BW Select +Cs +C12= 20pF +AC +23 +21 +Coupled +Chopping +C22 +Chopped +Spike +C11 +IA +Filter +BIOPOTENTIAL +Bias Current +I= +clk +CLK Generator +ASIC +GeneratorVDDA +Velectrode +VBIAS +on +Vt +IIN +TIN +VBODY +rst +Voutp +Cint +H +Vref +TcN +VBN +TcN +IFBP +IFBN +DDA +TL +LT +Hquires samples when the input crosses a specific threshold is shown in Figure 10. The ADC +works by storing the analog equivalent of the last digitized input as a voltage across the across +the capacitor 𝐶𝑏. Once the input signal crosses this voltage, a pulse with length 𝑇𝑃 is generated +to charge or discharge the capacitor 𝐶𝑏 by 𝑉𝐿𝑆𝐵 using one of the current sources connected to +the supply and ground [12]. Non-uniform sampling adapts to the instantaneous bandwidth of +the signal, consequently the dynamic power consumption scales with the activity of the input +signal. The FOM of the CT charge-based ADC can be improved by reducing the power supply +further [10]. +Figure 10: Top level architecture of Continous-time (CT) charge based ADC with non-uniform +sampling rate [12]. +4.5 +FPGA Processing +Machine learning algorithms such as Support Vector Machine (SVM) have allowed for on-chip +feature extraction and classification of biomedical signals [13]. Machine learning can also be +deployed in the domain of prosthetic devices for precise control. +Figure 11 depicts the con- +troller of prosthetic device which can be implemented using Field Programmable Gate Array +(FPGA). Figure 12 depicts the experimental setup for analyzing the data from high density +EMG acquisition system using Xilinix Zedboard [14]. +9 + +Biasing +Pulse ++OPulseUp +Generator +Comparator +Vin O +UP +Vb +DOWN +OUT ++00UT<7:0> +Cb +RESET +ORESET +Pulse ++OPulseDown +Generator +Conf guration +Register +aLkT +OCLK +DO +SI +ISO +VFigure 11: Top level architecture of controller of prosthetic hand including feature extraction +and classification [14]. +Figure 12: Experimental Setup of EMG acquisition and processing using Xilinix ZedBoard [14]. +4.6 +Energy Harvesting +The electrical power harvested from the environment (specially, thermal energy) can power +the ultra-low-power EMG Sensor. +We have previously developed energy harvesting systems +from various sources and high-efficiency DC-DC converters [15, 16]. For example, the system +architecture of power management IC for solar energy harvesting applications , designed by the +author, and chip micrograph are shown in Figure 13. +10 + +class +192 HD EMG +decision +channels +Data +Feature +Classification +Acquisition +Extraction +Embedded Prosthesis Controller +prosthesis +movementI92 ch. HD EMG +ZedBoard for EMG +electrode array +signal processing +Michelangelo +HD EMG +PC for +hand prosthesis +DAQ +comm.Figure 13: System architecture of power management IC for solar energy harvesting applications, +designed by one of the team members, and chip micrograph [15] +5 +Conclusion +This paper provided a survey about EMG acquisition systems for prostehtics and orthotic de- +vices. +References +[1] Shirley Ryan AbilityLab. Facts about limb loss. [Online]. Available: https://www.sralab. +org/research/labs/bionic-medicine/news/facts-about-limb-loss +[2] UK +Ministry +of +Defence. +Uk +service +personnel +amputations: +fi- +nancial +year +2019/2020. +[Online]. +Available: +https://www.gov.uk/ +government/statistics/uk-service-personnel-amputations-financial-year-20192020/ +afghanistan-and-iraq-amputation-statistics-1-april-2015-to-31-march-2020 +[3] L. G. Stansbury, S. J. Lalliss, J. G. Branstetter, M. R. Bagg, and J. B. Holcomb, +“Amputations in U.S. military personnel in the current conflicts in Afghanistan and Iraq,” +Journal of Orthopaedic Trauma, vol. 22, no. 1, pp. 43–46, 2008. [Online]. Available: +https://journals.lww.com/00005131-200801000-00009 +[4] iStock +. +Image +of +a +female +with +a +prosthetic +limb. +[Online]. +Available: +https: +//www.istockphoto.com/ +[5] W. Song, Q. Han, Z. Lin, N. Yan, D. Luo, Y. Liao, M. Zhang, Z. Wang, X. Xie, A. Wang +11 + +3 mirr +VON +VLOAD +2.2mm +Startup +VcBUF +smtehe +Switch Matrix +Gapetsi +S2 +Switch +DXADE +Current +VBA +VINDN市 +Matrix +Ref. +Startup +and +Drivers +Dnivers +Mp1 +Configuration Block +Modell +MIPP +lectior +VIN +VLOAD +S +DA +Pulse Generation Block +Test Block +Φ1 +Φ2 +En +A +PTrig +Dvlee +VINDN +VBA +AE +Test Block +险电 +Voltase: +Curont +Boost2et al., “Design of a flexible wearable smart sEMG recorder integrated gradient boosting +decision tree based hand gesture recognition,” IEEE transactions on biomedical circuits +and systems, vol. 13, no. 6, pp. 1563–1574, 2019. +[6] J. W. Kwak, M. Han, Z. Xie, H. U. Chung, J. Y. Lee, R. Avila, J. Yohay, X. Chen, +C. Liang, M. Patel, I. Jung, J. Kim, M. Namkoong, K. Kwon, X. Guo, C. Ogle, +D. Grande, D. Ryu, D. H. Kim, S. Madhvapathy, C. Liu, D. S. Yang, Y. Park, +R. Caldwell, A. Banks, S. Xu, Y. Huang, S. Fatone, and J. A. Rogers, “Wireless sensors for +continuous, multimodal measurements at the skin interface with lower limb prostheses,” +Science Translational Medicine, vol. 12, no. 574, p. eabc4327, 2020. [Online]. Available: +https://www.science.org/doi/abs/10.1126/scitranslmed.abc4327 +[7] MyoWare, +“3-lead +muscle +/ +electromyography +sensor +for +microcontroller +applications.” +[Online]. +Available: +https://www.mouser.com/datasheet/2/813/ +MyowareUserManualAT-04-001-1223951.pdf +[8] B. Ltd, “Surface emg amplifier.” [Online]. Available: +https://www.biometricsltd.com/ +surface-emg-sensor.htm#popupSpecAmplifier +[9] R. F. Yazicioglu, “A 60𝜇w 60 nV/ +√ +𝐻𝑧 readout front-end for portablebiopotential acqui- +sition systems,” in IEEE International Solid-State Circuits Conference Digest of Technical +Papers, Feb. 2006, 2006. +[10] S. Orguc, H. S. Khurana, H.-S. Lee, and A. P. Chandrakasan, “0.3 v ultra-low power sensor +interface for emg,” in ESSCIRC 2017-43rd IEEE European Solid State Circuits Conference. +IEEE, 2017, pp. 219–222. +[11] D. Wendler, D. D. Dorigo, M. Amayreh, A. Bleitner, M. Marx, and Y. Manoli, “A +0.00378mm2 scalable neural recording front-end for fully immersible neural probes based on +a two-step incremental delta-sigma converter with extended counting and hardware reuse,” +in 2021 IEEE International Solid- State Circuits Conference (ISSCC), vol. 64, 2021, pp. +398–400. +[12] M. Maslik, Y. Liu, T. S. Lande, and T. G. Constandinou, “Continuous-time acquisition of +biosignals using a charge-based ADC topology,” IEEE Transactions on Biomedical Circuits +and Systems, vol. 12, no. 3, pp. 471–482, 2018. +12 + +[13] J. Yoo, L. Yan, D. El-Damak, M. A. B. Altaf, A. H. Shoeb, and A. P. Chandrakasan, +“An 8-channel scalable EEG acquisition soc with patient-specific seizure classification and +recording processor,” IEEE Journal of Solid-State Circuits, vol. 48, no. 1, pp. 214–228, +2013. +[14] A. Boschmann, G. Thombansen, L. Witschen, A. Wiens, and M. Platzner, “A zynq-based +dynamically reconfigurable high density myoelectric prosthesis controller,” in Design, +Automation & Test in Europe Conference & Exhibition (DATE), 2017. +IEEE, 2017, pp. +1002–1007. [Online]. Available: http://ieeexplore.ieee.org/document/7927137/ +[15] D. El-Damak and A. P. Chandrakasan, “A 10 nw-1 𝜇w power management ic with integrated +battery management and self-startup for energy harvesting applications,” IEEE Journal of +Solid-State Circuits, vol. 51, no. 4, pp. 943–954, 2016. +[16] P. Garcha, D. El-Damak, N. Desai, J. Troncoso, E. Mazotti, J. Mullenix, S. Tang, D. Tromb- +ley, D. Buss, J. Lang, and A. Chandrakasan, “A 25 mV-startup cold start system with on- +chip magnetics for thermal energy harvesting,” in ESSCIRC 2017 - 43rd IEEE European +Solid State Circuits Conference, 2017, pp. 127–130. +13 + diff --git a/XtAyT4oBgHgl3EQfWfc6/content/tmp_files/load_file.txt b/XtAyT4oBgHgl3EQfWfc6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbd6eccf2a6a2836e49f9415b1e36703ddded464 --- /dev/null +++ b/XtAyT4oBgHgl3EQfWfc6/content/tmp_files/load_file.txt @@ -0,0 +1,395 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf,len=394 +page_content='A Survey about Acquisition System Design for Myoelectric Prosthesis Dina Reda Eldamak December 2022 1 Introduction According to the World Health Organization (WHO), 30 million people are in need of prosthetic and orthotic devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Some people are born with this limb loss, while others lose limbs due to diseases such as Cancer, diabetes, and work accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Additionally, limb amputation is among the most severe and heavily reported injuries among veterans during war [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Example of female with hand amputation is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 1: Female with Prosthetic limb [4] The medical applications of integrated circuit technology have recently made significant ad- vances, thus improving human quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Moreover, the use of microelectronics integration 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='00163v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='SP] 31 Dec 2022 dominates a lot of medical applications, especially portable and wearable battery-operated de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Bio-signals mostly arise from natural physiological processes, such as cardiac potentials (ECG -electro-cardiogram), potentials of the ocular tissue (EOG - electro-oculogram), potentials of the muscular tissues (electro-myogram -EMG), brain potential (electro-encephalogram -EEG), and respiratory signals , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Electro-myogram - EMG is an important factor for muscle disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Furthermore, it’s the key factor in connecting any amputee to a prosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' This can be done through extracting the EMG signal from the body using a readout electronics that can detect the muscles electrical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Consequently, the extracted signal is processed and used to control the prosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Thus, the objective of this report is to provide the reader with the basic understanding of integrated solutions for controlling prosthetic limbs either arms or legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The top level block diagram of a smart EMG acquisition system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The system includes a self-powered readout portable acquisition device for measuring the patient’s EMG signal in order to send it to a controller that can be used to emulate the right action to the prosthetic limb similar to the same action in a normal person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' It should be noted that miniaturized EMG acquisition system idea, which continuously monitor muscles activity, can be extended to different applications such as physical rehabilitation and prosthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 2: Block diagram of a general smart sEMG recorder [5] 2 Tissue Interface ProcessingUnit Sensor Interface2 System Architecture An electronic system can control a prosethetic device by monitoring the EMG signals of the arm, and use those signals to control the prothetic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Moreover, the devices can be battery-free by being powered solely using energy harvesting from the ambient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Since these prosthetic devices requires precise fitting to the residual limb, pressure and tem- perature sensor at the skin-prosthetic interface are added to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Pressure sensors are needed for monitoring the prosthetic limb to avoid the development of regions of high pressure as the limb moves during walking or grasping objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Temperature sensor are necessary as high temperature can accelerate tissue damage [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The signals from the sensor at the skin-prosthetic can be transmitted to the outer surface of the prosthetic socket using Near Field Communication (NFC) or to a smart phone using Bluetooth Low Energy (BLE) as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 3: Illustration of sensors mounted at the skin-prosthetic interface transmitting data to the device at the outer surface of the prosthetic leg using NFC and to smart phone using BLE [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 3 Block Diagram Three major research directions are available when designing an EMG acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The first is to acquire the signal from the surrounding noisy environment using a sensor interface 3 Residual limb NFC/BLE modules Multimodal battery-free sensors Prosthetic leg Prosthetic Portable socket NFC/BLE electronicdevice modulesFigure 4: Block diagram of the proposed smart sEMG recorder including sensors, AFE, and RF integrated system [5] circuit that’s designed in CMOS technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The second involves reducing the form factor and power consumption of the acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The third is the signal conversion to the digital world and the interface with the digital controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' At this point, the extracted EMG signal is in a digital form and can be processed through FPGA or any other processor to control a Pprosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' A typical block diagram of the proposed EMG acquisition system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The system consists of an EMG sensor, analog front end (AFE), and radio frequency (RF) transmission unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The AFE is typically composed of an analog amplification, filtration, analog to digital converter (ADC), and controller to process the digital signal and send it to a prosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The acquisition system design can be integrated on a single chip, then the digital data is fed to FPGA or a controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' In addition, because the integrated solution takes a considerable time during design, fabrica- tion, and testing phases, a discrete solution in parallel with the integrated one can be used as a proof of concept to validate the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 5 shows a detailed system block diagram of the proposed smart sEMG acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' An analog multiplexer is inserted to choose between different EMG electrodes in the smart sEMG recorder shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The design of each of the building blocks involves 4 AFE+RF NSPU FlexBandFigure 5: Detailed block diagram of the proposed smart sEMG acquisition system [5] several design challenges requiring some research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The following section includes a list of major research directions that can be pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4 Circuit Implementation In the following subsections, the basic system building blocks are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' First, the EMG sensor specifications are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Second, the low noise amplifier LNA design is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Third, the filter design and bandwidth are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Fourth, the signal conversion from analog to digital is presented through an ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Last, digital signal processing through FPGA is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='1 Sensor Specifications EMG sensor placement plays an important role in signal acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' According to its orientation and position, the EMG signal strength varies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' This effect is shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' As seen, by placing the sensor in the middle of muscle fiber, the maximum signal strength can be easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Otherwise, the signal degrades significantly when placing the sensor far away from the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' EMG sensor can be represented in different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' It can be in either needle that is inserted into the muscle or surface electrode that picks the signal from the skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' An example of surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Smart sEMG Recorder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='FPGA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='16-Channel Recorder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='GBDT based NSPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Flexible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='16x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='GBDT Core 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='TF Card ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='BLE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='CIC Filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Buffer A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Buffer B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Interface i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=':= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='TF Card ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Mobile Devices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Possible Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Offline trainingFigure 6: Effect of EMG sensor position [7] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='EMG sensor specifications that have to be met through out the design are as follow shown in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='2 Low Noise Amplifier Design It’s the first and the major block in the EMG chain that comes after the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The measurement sensitivity and accuracy is determined in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' This complicates the design and requires a large amount of adaptability to accommodate the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The previous stage, which is the EMG sensor, adds large parasitic capacitance at the input of this stage, and thus reduces gain, bandwidth, noise performance and the sensitivity of the amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Sources of noise and interference like flicker noise, electrodes offset, and 60 Hz power line noise can affect the whole acquisition procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The bandwidth of the EMG signal is up to 6 Raw EMG output Innervation Zone Correct Placement Midline of the muscle belly between an innervation zone and a myotendon junction Midline Offset Myotendon JunctionFigure 7: sensor specifications [8] 500 Hz with amplitude that ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='1 to 5 mV and the high-frequency noise can be easily removed using a low pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' However, low-frequency noise and DC offset fall within the EMG bandwidth and hence require different rejection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chopping technique is one of the best candidates to modulate the offset and flicker noise to a higher spectrum which in turn enable the acquisition system to effectively suppress the interference from ambient and 1/f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Different architectures with different requirements in terms of input signal levels, BW and amplitudes are proposed in literature [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 8 shows the block diagram of implemented analog front-end for acquiring of EEG, ECG, and EMG signals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The shown diagram consists of a chopper instrumentation amplifer in addition to capacitive coupling, filter stage to remove the chopping spikes, a digitally controlled variable gain amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='3 Filter Design A Gm-C filter cab be used in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' A standard architecture is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Offset from the electrodes can be canceled using current-mode DAC [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Power Consumption of this 7 DataLITE Wireless EMG Amplifier Wired EMG Amplifier Product Ref LE230FW SX230FW 42 × 24 × 14 mm 38 x 20 Dimensions Two 4 mm snap connectors on 100 mm wires Two 4 mm snap connectors on 100 mm wires Mass 17 g (excluding cable and plug) 8g (excluding cable and plug) Bandwidth 10 - 250,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='470,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 950,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 5000Hz 20 - 460Hz 5Hz - 480Hz Additional Bandwidths N/A 5Hz - 1000Hz Contact Diameter Dependant on electrode size Contact Center Spacing Variable Electrodes Disposable CMRR @ 60 Hz (dB) > 96 dB (typically 110 dB) Full Scale +/- 6 mV Peak to Peak +/- 3 mV Peak to Peak Gain +/- 60 microvolts to +/- 6 millivolts Standard unit x1000 (100 als0 available) Input Impedance >100 Mohms Accuracy +/- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='0% +/- 2% full scale Noise <5μv Supply Voltage N/A +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='50 to +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='5 Vdc Battery Life Up to 8 hours N/A Battery Type Rechargeable Li-lon Polymer N/A Wireless Transmission Tolerant for 100 mS N/A Data Loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='25m cable Range from Interface Wireless range up to 30m (custom lengths available on request) Compatible Interfaces DataLITE PIONEER, ADVANCE, EXPLORE DataLOG, DataLINK, Amplifier or 3rd partyFigure 8: Architecture of the bio-potential readout front-end for the acquisition of EEG, ECG, and EMG signals [9] topology can also be reduced by low-voltage supply operation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 9: Transistor level implementation of Gm-C filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' DDA: Differential Difference Amplifier [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='4 ADC Design Non-uniform sampling can minimize the power consumption of ADC while digitizing activity- dependent biological signals.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='IFBN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='DDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='TL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='LT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Hquires samples when the input crosses a specific threshold is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The ADC works by storing the analog equivalent of the last digitized input as a voltage across the across the capacitor 𝐶𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Once the input signal crosses this voltage, a pulse with length 𝑇𝑃 is generated to charge or discharge the capacitor 𝐶𝑏 by 𝑉𝐿𝑆𝐵 using one of the current sources connected to the supply and ground [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Non-uniform sampling adapts to the instantaneous bandwidth of the signal, consequently the dynamic power consumption scales with the activity of the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' The FOM of the CT charge-based ADC can be improved by reducing the power supply further [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 10: Top level architecture of Continous-time (CT) charge based ADC with non-uniform sampling rate [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='5 FPGA Processing Machine learning algorithms such as Support Vector Machine (SVM) have allowed for on-chip feature extraction and classification of biomedical signals [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Machine learning can also be deployed in the domain of prosthetic devices for precise control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 11 depicts the con- troller of prosthetic device which can be implemented using Field Programmable Gate Array (FPGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 12 depicts the experimental setup for analyzing the data from high density EMG acquisition system using Xilinix Zedboard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 9 Biasing Pulse +OPulseUp Generator Comparator Vin O UP Vb DOWN OUT +00UT<7:0> Cb RESET ORESET Pulse +OPulseDown Generator Conf guration Register aLkT OCLK DO SI ISO VFigure 11: Top level architecture of controller of prosthetic hand including feature extraction and classification [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Figure 12: Experimental Setup of EMG acquisition and processing using Xilinix ZedBoard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='6 Energy Harvesting The electrical power harvested from the environment (specially, thermal energy) can power the ultra-low-power EMG Sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' We have previously developed energy harvesting systems from various sources and high-efficiency DC-DC converters [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' For example, the system architecture of power management IC for solar energy harvesting applications , designed by the author, and chip micrograph are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 10 class 192 HD EMG decision channels Data Feature Classification Acquisition Extraction Embedded Prosthesis Controller prosthesis movementI92 ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' HD EMG ZedBoard for EMG electrode array signal processing Michelangelo HD EMG PC for hand prosthesis DAQ comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='Figure 13: System architecture of power management IC for solar energy harvesting applications, designed by one of the team members, and chip micrograph [15] 5 Conclusion This paper provided a survey about EMG acquisition systems for prostehtics and orthotic de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' References [1] Shirley Ryan AbilityLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Facts about limb loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='sralab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' org/research/labs/bionic-medicine/news/facts-about-limb-loss [2] UK Ministry of Defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Uk service personnel amputations: fi- nancial year 2019/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='uk/ government/statistics/uk-service-personnel-amputations-financial-year-20192020/ afghanistan-and-iraq-amputation-statistics-1-april-2015-to-31-march-2020 [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Stansbury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Lalliss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Branstetter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Bagg, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Holcomb, “Amputations in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' military personnel in the current conflicts in Afghanistan and Iraq,” Journal of Orthopaedic Trauma, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 43–46, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https://journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='lww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='com/00005131-200801000-00009 [4] iStock .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Image of a female with a prosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='istockphoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='com/ [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Song, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Lin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Liao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Xie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Wang 11 3 mirr VON VLOAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='2mm Startup VcBUF smtehe Switch Matrix Gapetsi S2 Switch DXADE Current VBA VINDN市 Matrix Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Startup and Drivers Dnivers Mp1 Configuration Block Modell MIPP lectior VIN VLOAD S DA Pulse Generation Block Test Block Φ1 Φ2 En A PTrig Dvlee VINDN VBA AE Test Block 险电 Voltase: Curont Boost2et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=', “Design of a flexible wearable smart sEMG recorder integrated gradient boosting decision tree based hand gesture recognition,” IEEE transactions on biomedical circuits and systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 1563–1574, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Kwak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Avila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Yohay, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Liang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Patel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Jung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Namkoong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Kwon, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Ogle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Grande, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Ryu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Madhvapathy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Park, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Caldwell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Banks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Fatone, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Rogers, “Wireless sensors for continuous, multimodal measurements at the skin interface with lower limb prostheses,” Science Translational Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 574, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' eabc4327, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='org/doi/abs/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='1126/scitranslmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='abc4327 [7] MyoWare, “3-lead muscle / electromyography sensor for microcontroller applications.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='mouser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='com/datasheet/2/813/ MyowareUserManualAT-04-001-1223951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='pdf [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Ltd, “Surface emg amplifier.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='biometricsltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='com/ surface-emg-sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='htm#popupSpecAmplifier [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Yazicioglu, “A 60𝜇w 60 nV/ √ 𝐻𝑧 readout front-end for portablebiopotential acqui- sition systems,” in IEEE International Solid-State Circuits Conference Digest of Technical Papers, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 2006, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Orguc, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Khurana, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Lee, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chandrakasan, “0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='3 v ultra-low power sensor interface for emg,” in ESSCIRC 2017-43rd IEEE European Solid State Circuits Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 219–222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Wendler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Dorigo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Amayreh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Bleitner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Marx, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Manoli, “A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='00378mm2 scalable neural recording front-end for fully immersible neural probes based on a two-step incremental delta-sigma converter with extended counting and hardware reuse,” in 2021 IEEE International Solid- State Circuits Conference (ISSCC), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 64, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 398–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Maslik, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Lande, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Constandinou, “Continuous-time acquisition of biosignals using a charge-based ADC topology,” IEEE Transactions on Biomedical Circuits and Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 471–482, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 12 [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Yoo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' El-Damak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Altaf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Shoeb, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chandrakasan, “An 8-channel scalable EEG acquisition soc with patient-specific seizure classification and recording processor,” IEEE Journal of Solid-State Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 214–228, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Boschmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Thombansen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Witschen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Wiens, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Platzner, “A zynq-based dynamically reconfigurable high density myoelectric prosthesis controller,” in Design, Automation & Test in Europe Conference & Exhibition (DATE), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 1002–1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Available: http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content='org/document/7927137/ [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' El-Damak and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chandrakasan, “A 10 nw-1 𝜇w power management ic with integrated battery management and self-startup for energy harvesting applications,” IEEE Journal of Solid-State Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 943–954, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Garcha, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' El-Damak, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Desai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Troncoso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Mazotti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Mullenix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Tang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Tromb- ley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Buss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Lang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' Chandrakasan, “A 25 mV-startup cold start system with on- chip magnetics for thermal energy harvesting,” in ESSCIRC 2017 - 43rd IEEE European Solid State Circuits Conference, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 127–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'} diff --git a/Y9FST4oBgHgl3EQfADiR/content/tmp_files/2301.13697v1.pdf.txt b/Y9FST4oBgHgl3EQfADiR/content/tmp_files/2301.13697v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e62c526de132d52d980fcfdb68a97eac6e2109e7 --- /dev/null +++ b/Y9FST4oBgHgl3EQfADiR/content/tmp_files/2301.13697v1.pdf.txt @@ -0,0 +1,883 @@ +Scalability of non-adiabatic effects in lithium-decorated graphene superconductor +Dominik Szcz¸e´sniak +Department of Theoretical Physics, Faculty of Science and Technology, +Jan D�lugosz University in Cz¸estochowa, 13/15 Armii Krajowej Ave., 42200 Cz¸estochowa, Poland +(Dated: February 1, 2023) +The analysis is conducted to unveil how the non-adiabatic effects scale within the superconducting +phase of lithium-decorated graphene (LiC6). Based on the Eliashberg formalism it is shown that +the non-adiabatic effects notably reduce essential superconducting parameters in LiC6 and arise as +a significant oppressor of the discussed phase. Moreover, nonadiabaticity is found to scale with +the strength of superconductivity, proportionally to the phonon energy scale and inversely with +respect to the electron-phonon coupling. These findings are partially in contrast to other theoretical +studies and show that superconductivity in LiC6 is more peculiar than previously anticipated. In +this context, the guidelines for enhancing superconducting phase in LiC6 and sibling materials are +also proposed. +I. +INTRODUCTION +In the literature, a number of scenarios exist on how +to potentially induce conventional superconductivity in +graphene, promising novel applications of this intrigu- +ing carbon allotrope [1–12]. These strategies are mainly +aimed at the structural modifications of graphene to +enhance number of charge carriers at the Fermi level +and alter intrinsic semimetallic character of this mate- +rial. Among the resulting structures, lithium-decorated +graphene (LiC6) is here of particular interest since it +is suggested to host phonon-mediated superconducting +state generated via process analogous to the intercalation +of graphite [10]. This approach relies on the introduction +of adatoms that break chiral symmetry of graphene and +lifts Fermi level to the van Hove singularity [10, 13–15]. +However, superconductivity in LiC6 not only derives from +the well-established method of synthesis but also appears +to be actually feasible, as partially confirmed within the +experiment [16]. +According to the above, LiC6 may be considered as a +proving ground for the phonon-mediated superconduc- +tivity at low-dimensions. +Indeed, in recent years the +superconducting phase in LiC6 received notable atten- +tion in terms of its fundamental properties. +The re- +lated studies were devoted, but not limited to, the role of +strong electron-phonon coupling [17], character of super- +conducting gap [5], symmetry-breaking effects [13], ways +of enhancing superconducting state [7], or the substrate +ievmpact on superconductivity [9]. Beside listed direc- +tions of research, LiC6 appears also to be a perfect exam- +ple of superconducting material for studying the influence +of non-adiabatic pairing [18]. This aspect is particularly +intriguing since non-adiabatic effects tend to manifest +themselves relatively rarely in conventional superconduc- +tors. The reason for that relates to the non-comparable +electronic and phononic energy scales in most supercon- +ducting materials with the electron-phonon pairing mech- +anism [19–21]. In other words, it can be often assumed +that electrons follow adiabatically ionic oscillations and +that the corresponding superconducting state can be de- +scribed in a self-consistent manner [22, 23]. +However, +this is not the case when superconducting materials ex- +hibit shallow conduction band such as the fullerenes [24], +fullerides [23], bismuthates [25], transition-metal-oxides +[26] or the discussed LiC6 [18] (see [27] for the review of +nonadiabatic superconductors). +So far, the characteristic energy scales are one of +the few signatures of non-adiabatic superconductivity in +LiC6. Other than that recent theoretical studies show +non-adiabatic effects to notably reduce magnitude of de- +pairing interaction in the discussed material, in compar- +ison to the adiabatic regime [18]. These findings are also +supplemented by the considerations suggesting that non- +adiabaticity contributes to the electron-phonon coupling +(λ) and modulates the transition temperature (TC) in +doped graphene [22] or two-dimensional superconductors +in general [28]. Still, little is known about the scalability +of non-adiabatic effects in LiC6. In the first approxima- +tion, it can be only qualitatively argued that their impact +changes according to the Migdal’s ratio (known also as +the expansion ratio) given by m = λωD/EF , where EF +is the Fermi energy and ωD denotes Debye’s frequency +[19–21]. In fact, although m-ratio can provide some in- +formation on the scalability problem it is mostly used to +determine whether or not given material can be described +within the Migdal’s theorem [29] i.e. within adiabatic or +non-adiabatic regime. Hence, the measurable and direct +role of the above parameters and their variations in shap- +ing non-adiabatic superconductivity in LiC6 is somewhat +hindered. In details, it is unknown how changes in the m- +ratio components modify experimentally observable ther- +modynamic properties such as the superconducting gap +or the transition temperature. This is to say, what trends +in thermodynamics can be expected due to the strength +of the non-adiabatic effects. As a result, the relevancy +of the energy scales and the electron-phonon coupling +in the non-adiabatic limit is also not well-estimated yet. +Therefore, addressing these aspects would be of great im- +portance to the better understanding of superconductiv- +ity in LiC6 and potentially other sibling low-dimensional +materials. Moreover, it should also help in assessing im- +pact of the external factors that can be applied to mod- +ify the aforementioned properties and ultimately enhance +arXiv:2301.13697v1 [cond-mat.supr-con] 31 Jan 2023 + +2 +the superconducting state even further. +To provide deeper insight into the scalability of non- +adiabatic effects in LiC6, the present study analyzes be- +havior of the discussed material in the adiabatic and +non-adiabatic limit when the expansion ratio parame- +ters vary. This is done within the Eliashberg formalism +that generalizes conventional Bardeen-Cooper-Schrieffer +(BCS) theory of superconductivity [30, 31] by incorporat- +ing the strong-coupling, retardation and non-adiabatic +effects [19–21, 32, 33]. As a result it allows to consider +both regimes of interest and relate predictions on the piv- +otal thermodynamics to the potential factors responsible +for the m-ratio variations. Here the latter is modeled in +reference to [7], by recalling the fact that the deforma- +tion potential is able to simultaneously influence energy +scales and the electron-phonon coupling in a given su- +perconducting material. This effect is captured via the +percentage change in graphene lattice constant given as +δ = |a − a0| /a0 × 100%, where a0(a) is the unmodified +(modified) lattice constant value. In what follows, sev- +eral levels of δ are considered allowing for tracing changes +in the m-ratio components on the same footing and in +the direct relation to the experimentally observable case +(δ = 0%). Based on that it is possible to unveiled how the +non-adiabatic effects scale in LiC6 and what can be done +to eliminate their potentially negative consequences. +II. +METODOLOGY +The theoretical formalism of choice is provided here +by following the study of Freericks et al. [34], where con- +venient form of the generalized Eliashberg equations for +considering the non-adiabatic superconductivity is pre- +sented. This theoretical approach is based on the per- +turbative theory introduced originally by Pietronero et +al. in [19–21], which incorporates non-adiabatic effects +via vertex corrections to the electron-phonon interac- +tion. However, the theoretical scenario given in [34] in- +cludes specific computational techniques for better ac- +curacy and efficiency, such as the perturbative theory +on the imaginary-axis and the high-frequency resum- +mation schemes. +In this manner, the resulting equa- +tions provide compromise between predictive capabili- +ties and computational requirements. Note that such ap- +proach was already proved successful in describing non- +adiabatic superconductivity not only in LiC6 but also +other phonon-mediated superconductors such as lead [34] +or bismuthates [35]. +In respect to the above, inital approximations are as- +sumed in accordance to [34] and the character of the +superconducting phase in LiC6. +In particular, (i) the +direct dependence on momentum is neglected for the +electron-phonon matrix elements, in correspondence to +the isotropic nature of superconducting gap in LiC6, +(ii) the depairing correlations are modeled only by +the first-order Coulomb pseudopotential terms, due to +the fact that higher-order contributions are negligibly +small for phonon-mediated superconductors, (iii) sim- +ilarly only the lowest-order vertex corrections to the +electron-phonon interaction are considered to describe +the non-adiabatic effects, since the Fermi liquid picture +in LiC6 appears to be conserved. As a results, it is possi- +ble to derive self-consistent Eliashberg equations beyond +Midal’s theorem within perturbation scheme. In details, +their form on the imaginary axis for the order parameter +function (φn = φ (iωn)) and the wave function renormal- +ization factor (Zn = Z (iωn)) is following: +φn = πkBT +M +� +m=−M +Kn,m − µ⋆ +m +� +ω2mZ2m + φ2m +φm − Vφ, +(1) +Zn = 1 + πkBT +ωn +M +� +m=−M +Kn,m +� +ω2mZ2m + φ2m +ωmZm − VZ, (2) +where, kBT is the inverse temperature, with kB denot- +ing the Boltzmann constant. +In what follows, ωn = +πkBT (2n + 1) is the n-th Matsubara frequency with the +cutoff M = 1100 for numerical stability above T = 2 K. +Moreover, Kn,m stands for the electron-phonon pairing +kernel given as: +Kn,m = 2 +� ωD +0 +dω +ω +ω2 + 4π2 (kBT)2 (n − m)2 α2F (ω) , +(3) +with ω being the phonon frequency, α describing the av- +erage electron-phonon coupling and F (ω) denoting the +phonon density of states. Note that the product of the +two latter is known as the electron-phonon spectral func- +tion which provides most important information about +a physical system within the Eliashberg formalism [33]. +Here, several α2F (ω) functions are considered, each of +them corresponding to the different δ-value in order to +analyze behavior of LiC6 when the Migdal’s parameter +and its components very. +For this purpose, the exact +forms of the α2F (ω) functions are assumed after [7] for +δ ∈ ⟨0, 3, 5, 7, 10⟩ %, where the first case corresponds to +the experimentally observed superconducting phase of +LiC6 whereas the remaining functions describe poten- +tial variations from its pristine form. +The remaining +information is given via the Coulomb pseudopotential +µ⋆ +n = µ⋆θ (ωc − |ωn|), with θ standing for the the Heavi- +side function and ωc for the cut-off frequency. To consider +all the δ cases on equal footing the conventional value of +µ⋆ = 0.1 [36] which is close to the magnitude of Coulomb +depairing interaction predicted for LiC6 at δ = 0% [18]. +Finally, Vφ and VZ are the lowest-order vertex correc- + +3 +tion terms of the following form: +Vφ = π3 (kBT)2 +4EF +M +� +m=−M +M +� +m′=−M +Kn,mKn,m′ +× +1 +� +(ω2mZ2m + φ2m) (ω2 +m′Z2 +m′ + φ2 +m′) +× +1 +� +(ω2 +m′′Z2 +m′′ + φ2 +m′′) +× (φmφm′φm′′ + 2φmωm′Zm′ωm′′Zm′′ +− ωmZmωm′Zm′φm′′) , +(4) +and +VZ = π3 (kBT)2 +4EF ωn +M +� +m=−M +M +� +m′=−M +Kn,mKn,m′ +× +1 +� +(ω2mZ2m + φ2m) (ω2 +m′Z2 +m′ + φ2 +m′) +× +1 +� +(ω2 +m′′Z2 +m′′ + φ2 +m′′) +× (ωmZmωm′Zm′ωm′′Zm′′ + 2ωmZmφm′φm′′ +− φmφm′ωm′′Zm′′) . +(5) +Based on the above, when vertex corrections are consid- +ered within the Eqs. (1) and (2) they are refereed here to +as the non-adiabatic Eliashberg equations (N-E), other- +wise, when the corrections are neglected, the formalism +is reduced to the adiabatic Eliashberg equations (A-E). +In what follows, by solving Eqs. (1) and (2) it is pos- +sible to obtain estimates on the most important ther- +modynamic properties of superconducting state in LiC6. +Specifically, the central role in such analysis is played +by the order parameter function that is obtained from +Eqs. (1) and (2) as: ∆n(T) = φn/Zn. Here of special +interest is the maximum value (m = 1) of ∆n(T) which +contains information on the transition temperature and +the superconducting gap half-width. The former is de- +termined based on the relation ∆m=1(TC) = 0, whereas +the latter is given by ∆m=1(T0), with T0 = 2 K being +the lowest temperature assumed for calculations. Since +the aforementioned α2F (ω) function is dependent on the +characteristic energy scales and the electron-phonon cou- +pling constant, the described solutions of the Eliashberg +equations also inherit such dependence, allowing for the +analysis of interest. +III. +THE RESULTS AND DISCUSSION +In Fig. +1, the main numerical results are presented +as obtained by solving Eqs. (1) and (2) iteratively with +respect to the temperature (see [25] and [34] for more +details on the computational methods used here). In de- +tails, Fig. 1 depicts the behavior of ∆m=1(T) function +for T ∈ ⟨T0, TC⟩ at the assumed levels of lattice constant +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +0 +1 +2 +3 +4 +5 +6 + + +∆m=1 (mev) +T (K) +A-E: + δ=0% + δ=3% + δ=5% + δ=7% + δ=10% +N-E: + δ=0% + δ=3% + δ=5% + δ=7% + δ=10% +FIG. 1: The maximum value of order parameter (∆m=1(T)) +as a function of temperature in LiC6. +The results are de- +picted for the selected values of lattice constant deviation in +graphene (δ) as obtained within the adiabatic (closed sym- +bols) and non-adiabatic (open symbols) regime of the Eliash- +berg equations. Solid lines constitute the guides for an eye. +deviation denoted by δ. Note that the 0% case corre- +sponds to the unaltered LiC6 material, hosting the ex- +perimentally observable superconducting phase. On the +other hand, the remaining cases describe situation when +crystal lattice of graphene changes according to the al- +ready mentioned expression: δ = |a − a0| /a0 × 100%, +with a0(a) standing for the unmodified (modified) lattice +constant. Moreover, as allowed by the employed formal- +ism, the discussed thermal behavior of ∆m=1(T) func- +tion is plotted for the adiabatic (open symbols) and non- +adiabatic (closed symbols) regime. In all figures, symbols +relate to the exact numerical results of the Eliashberg +equations and solid lines constitute guides for an eye. +The result depicted in Fig. 1 reveal several general as- +pects of the superconducting state in LiC6. In particular, +it can be observed that for δ > 3% the increase of the +δ value causes notable increase of the ∆m=1(T) in the +entire temperature range for both considered regimes. +This trend is not conserved only when comparing re- +sult obtained at the two lowest levels of δ, as caused +by the δ-driven increase of charge transfer that emp- +ties interlayer states and notably reduces the electron- +phonon coupling constant at δ = 3% with respect to +the 0% case [7]. Nonetheless, the observed effect means +that above some level of δ the superconducting state +in LiC6 is clearly enhanced. +This observation can be +quantified by deducing the transition temperature values +from the obtained results. In particular, TC = 8.78 K +at δ = 0% and TC ∈ ⟨8.48, 34.61⟩ K for δ ∈ ⟨3, 10⟩ % +within the A-E limit, whereas TC = 7.29 K at δ = 0% +and TC ∈ ⟨6.59, 28.46⟩ K for δ ∈ ⟨3, 10⟩ % when consid- +ering the N-E equations. Note that these observations +are in qualitative agreement with the previous studies +conducted within the adiabatic limit by using the Allen- + +4 +Dynes formula in [7]. The difference between these data +sets and results given in [7] is due to the fact that the as- +sumed Eliashberg equations incorporate strong-coupling, +retardation, and non-adiabatic effects which are missing +in the Allen-Dynes formula. At this point, it is also in- +structive to note that the TC value estimated at δ = 0% +is slightly higher in comparison to the predictions made +within the Eliashberg formalism for the experimentally +derived electron-phonon spectral function, as presented +in [18]. This discrepancy is obviously caused by the as- +sumed value of µ⋆, smaller than the one suggested in +[18]. The reason to make such assumption is to allow +for better comparison not only with the BCS-derived re- +sults given in [7] but also other two-dimensional super- +conductors, which are often still hypothetical structures +and their superconducting state is described by µ⋆ ∼ 0.1 +(see e.g. [37, 38]). Note that even if µ⋆ would be as- +sumed here after [18], the main outcomes and findings of +the present analysis would not change. This includes es- +timates on the superconducting gap half-width that can +be made based on the results plotted in Fig. 1. This is to +say, the general behavior of ∆m=1(T0) parameter is the +same as in the case of TC and it will not change qualita- +tively when assuming other µ⋆ value. Specifically, in the +present study ∆m=1(T0) = 1.39 meV at δ = 0% whereas +∆m=1(T0) ∈ ⟨1.30, 5.55⟩ meV for δ ∈ ⟨3, 10⟩ % in the A-E +regime, while the N-E equations yield ∆m=1(T0) = 1.22 +meV at δ = 0% and ∆m=1(T0) ∈ ⟨1.11, 4.84⟩ meV for +δ ∈ ⟨3, 10⟩ %. Note that results on TC and ∆m=1(T0) +can be supplemented by introducing their characteristic +ratio, familiar in the BCS theory and given by [30, 31, 33]: +R = 2∆m=1(T0) +kBTC +. +(6) +The Eq. (6) not only allows for additional insight into +the considered problem but also provides yet another ob- +servable for future comparisons with the experiment. As +it can be expected, the obtained values of R follow the +same trends like the TC and ∆m=1(T0) parameters. The +values of R base on the A-E equations are R = 3.67 at +δ = 0% and R ∈ ⟨3.55, 3.72⟩ for δ ∈ ⟨3, 10⟩ %. On the +other hand, the N-E equations give R = 3.89 at δ = 0% +and R ∈ ⟨3.92, 3.98⟩ for δ ∈ ⟨3, 10⟩ %. Still, both sets +present values higher than the level suggested within the +BCS theory and equal to 3.53. It means that the strong- +coupling and retardation effects play relatively impor- +tant role in shaping the superconducting state in LiC6. +This observation is in agreement with the strength of +the electron-phonon coupling reported in [7] and previ- +ous findings given in [18]. +Beside the above observations, it is also crucial to note +that the reported results suggest simultaneous changes +in the energy scales and the electron-phonon coupling +constant due to the variations of δ parameter. Indeed, +all of these characteristic parameters exhibit increasing +or decreasing trends along with the growing δ value (see +Fig. 2 (A)). For convenience, their cumulative behavior +is depicted in Fig. 2 (B) in terms of already introduced +120 +140 +160 +180 +1000 +1200 +1400 +1600 +1800 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0 +2 +4 +6 +8 +10 +4 +8 +12 +16 +20 +24 +28 + +ωD (meV) EF (meV) +(A) +λ + +m + m=ωD/EF + m=λωD/EF +(B) +Dx (%) + x=1 + x=2 + x=3 +δ (%) +(C) +FIG. 2: The behavior of (A) the Fermi energy (EF ), Debye’s +frequency (ωD) and electron-phonon coupling constant (λ), +(B) the dressed (m = λωD/EF ) and bare (m = ωD/EF ) +Migdal’s ratio as well as (C) the percentage differences be- +tween adiabatic and non-adiabatic estimates for the critical +temperature (D1), superconducting gap half-width (D2) and +their cumulative ratio (D3) at the selected values of the lat- +tice deviation in graphene (δ) in the LiC6 superconductor. +The closed symbols depicts exact results, the solid lines are +the guides for an eye and the color arrows points to the cor- +responding axes. +dressed Migdal’s ratio (m = λωD/EF ) but also its bare + +5 +TABLE I: The parameters of superconducting state in LiC6 for the selected values of the lattice deviation in graphene (δ). +In a consecutive order, the parameters are: the electron-phonon coupling constant (λ), the Debye’s frequency (ωd), the Fermi +energy (EF ), the bare (ωd/EF ) and dressed (λωd/EF ) Migdal’s ratio, the transition temperature (TC), the superconducting +gap half-width (∆m=1(T0)) as well as the thermodynamic ratio for the two last ones (R). Note that, where necessary, the +results are presented for the in terms of the adiabatic (A − E) and non-adiabatic (N − E) regime. Moreover, the percentage +differences between estimates in these two limits for TC (D1), ∆m=1(T0) (D2) and R (D3) are also given. +A-E +N-E +δ +λ +ωd +EF +ωd/EF +λωd/EF +TC +∆m=1(T0) +R +TC +∆m=1(T0) +R +D1 +D2 +D3 +(%) +(meV) +(meV) +(K) +(meV) +(K) +(meV) +(%) +(%) +(%) +0 +0.61 +141.21 +1100 +0.128 +0.078 +8.78 +1.39 +3.67 +7.29 +1.22 +3.89 +18.54 +13.03 +6.37 +3 +0.47 +171.34 +1300 +0.132 +0.062 +8.48 +1.30 +3.55 +6.59 +1.11 +3.92 +25.08 +15.77 +9.91 +5 +0.49 +158.82 +1624 +0.098 +0.048 +12.61 +1.94 +3.58 +9.90 +1.68 +3.93 +24.08 +14.36 +9.32 +7 +0.55 +148.16 +1726 +0.086 +0.047 +18.86 +2.95 +3.63 +14.98 +2.56 +3.97 +22.93 +14.16 +8.95 +10 +0.73 +132.79 +1800 +0.074 +0.054 +34.61 +5.55 +3.72 +28.26 +4.84 +3.98 +20.20 +13.67 +6.75 +counterpart (m = ωD/EF ). In what follows, it is argued +here that the scalability of non-adiabatic effects in LiC6 +can be traced with respect to the pivotal parameters en- +tering Migdal’s ratio. In this context, first it should be +noted that for each considered δ value the non-adiabatic +equations yield lower ∆m=1(T) values than their adia- +batic counterparts (see Fig. 1). This directly relates to +the fact that the transition temperature values as well as +the estimates of the superconducting gap are lower in the +non-adiabatic regime when comparing to the adiabatic +one. As a results, the percentage difference between esti- +mates made in two considered regimes can be introduced +as a measure of non-adiabatic effects impact on super- +conducting phase in LiC6. This new measure is depicted +for all considered thermodynamic parameters in Fig. 2 +(C). In details, the percentage difference between TC val- +ues determined in the adiabatic and non-adiabatic limits +is D1 = 18.54% at δ = 0% and D1 = ⟨25.08, 20.20⟩% +for δ ∈ ⟨3, 10⟩ %. Similarly, the same percentage mea- +sure but for the ∆m=1(T0) parameter is D2 = 13.03% +for δ = 0% and D2 = ⟨15.77, 13.67⟩% when δ ∈ ⟨3, 10⟩ %. +Finally, the cumulative ratio gives the corresponding per- +centage D3 = 6.37% for δ = 0% and D3 = ⟨9.91, 6.75⟩% +when again δ ∈ ⟨3, 10⟩ %. Based on these results, it is +clear that the critical temperature is the most influenced +by the non-adiabatic effects from all three considered pa- +rameters. It also presents the most visible signature of +the charge transfer from interlayer states at δ ∈ 3%. On +the contrary, the cumulative ratio shows the smallest dis- +crepancies. Still all three parameters exhibit the same +qualitative behavior i.e. +the percentage difference be- +tween adiabatic and non-adiabatic results increases up +to δ ∈ 3% and then starts to almost linearly decrease as +the δ takes higher values. +The final observations can be made when comparing +results presented in Fig. 2 (C) with the estimates de- +picted in Figs. 2 (A) and (B). In particular, it can quali- +tatively argued that the behavior of percentage measures +does not fully resemble the Migdal’s ratio dependence on +the δ value, although the latter is considered to be the +first approximation approach to provide information on +the scalability of non-adiabatic effects in a superconduc- +tors. Precisely speaking, only the bare ratio can be con- +sidered somewhat similar in behavior to the percentage +measures, whereas its dressed value presents almost in- +verse character with respect to the parameters given in +Fig. 2 (C). To inspect these discrepancies even further it +is instructive to compare the percentage difference mea- +sures with the characteristic component parameters of +the Migdal’s ratio, as plotted in Figs. 2 (A). The out- +come is that only the Debye’s energy scales qualitatively +the same as the percentage difference measures, while the +electron-phonon coupling gives inverse characteristic and +the electronic energy scale is practically nowhere similar +to the results given in Figs. 2 (C). +IV. +SUMMARY AND CONCLUSIONS +In summary, the presented analysis provides new in- +sight into the superconducting properties of LiC6 in +terms of its non-adiabatic characteristic. It shows how +the non-adiabatic effects scale in LiC6 with respect to the +deviation of lattice constant in graphene, which can be +considered as an exemplary factor that modifies strength +of the superconducting state. In details, the discussed +scalability is expressed here in terms of the percentage +difference between estimates of pivotal thermodynamic +parameters (the transition temperature, superconduct- +ing gap and their ratio) obtained within the adiabatic +and non-adiabatic regime, allowing for further compari- +son with the Migdal’s expansion ratio that characterizes +nonadiabaticity in the first approximation. For conve- +nience, all the obtained numerical results are summarized +in Tab. I. +Based on the above findings it is possible to draw sev- +eral conclusions related, but no limited to, the scalability +of non-adiabatic effects in LiC6. In details: +(i) The introduction of vertex corrections to the +electron-phonon interaction causes notable changes +in the pivotal thermodynamic parameters of the su- +perconducting state in LiC6, in particular their de- +crease in comparison to the adiabatic limit (see Fig. + +6 +2 (C)). This is to say, the superconducting state +appears to have strongly non-adiabatic character, +where non-adiabatic effects act as an important op- +pressor of superconductivity in LiC6. Notably the +superconducting state sustains its non-adiabatic +character even when superconductivity in LiC6 is +strongly enhanced. Still the non-adiabatic effects +are observed to visibly vary with the strength of +superconducting phase. Moreover, it is found that +nonadiabaticity is supplemented by the strong- +coupling and retardation effects, meaning that LiC6 +is a somewhat unorthodox phonon-mediated super- +conductor. +(ii) The considered thermodynamic parameters present +the same qualitative behavior under the influence +of non-adiabatic effects when the strength of super- +conductivity is varied (see Fig. 2 (C)). Nonetheless, +the critical temperature is suggested to be partic- +ularly sensitive to nonadiabaticity, while supercon- +ducting gap is showing much smaller dependence +on the variation of the discussed effects. As a re- +sult, this opens new prospect for increasing tran- +sition temperature value in LiC6 according to the +presented here findings, saying that smaller mag- +nitude of non-adiabatic effects leads to the higher +transition temperature (see Tab. I). Note that this +trend is not conserved when including results for +the unaltered LiC6, due to the decreased charge +transfer from the interlayer states in comparison +to other considered cases. It can be additionally +argued that such trend may be considered general +for other graphene-based superconductor which ex- +hibit similar dependence on nonadiabaticity (see +e.g. recent study on the electron-doped graphene +[39]). +(iii) The deeper inspection of the obtained results +shows that the non-adiabatic effects scale with +the strength of superconductivity, proportionally +to the phonon energy scale and inversely to the +electron-phonon coupling magnitude (see Figs. 2 +(A) and (C)). Note that, while the former observa- +tion agrees with the predictions of both considered +forms of the Migdal’s ratio, the latter is in contrast +to what can be expected based on the dressed pa- +rameter and previous theoretical studies consider- +ing superconductivity in two-dimensional systems +[22, 28]. However, the mentioned trends does not +take into account existing interplay between all +components of the Migdal’s ratio and the fact that +the electron-phonon coupling constant increases al- +most linearly with the electronic energy scale (see +Tab. I). As a results, strong electron-phonon cor- +relations correspond to the relatively wide conduc- +tion band that causes suppression of nonadiabatic- +ity. This argument is confirmed qualitatively by the +observed here cumulative behavior of the Migdal’s +ratio (see Figs. 2 (B)). This is to say the electron- +phonon coupling cannot be always considered to +be improved in graphene-based superconductors by +the non-adiabatic effects as previously suggested in +[22, 28]. +To sum up, the perspectives for future research can be +given. In details, to provide better understating of the su- +perconducting state in LiC6 the discussed non-adiabatic +effects should be considered beyond the isotropic ap- +proximation. +Note that such preliminary analysis can +be already found in [28], while the adiabatic anisotropic +investigations are available in [5]. +The present study +can be also extended further toward other experimen- +tally observable thermodynamic parameters such as the +free energy or the critical thermodynamic field, accord- +ing to their importance in discussing the non-adiabatic +effects [40]. Finally, recent discussion given in [27] sug- +gest strongly metallic behavior of LiC6 despite its high +value of the Migdal’s ratio. In other words, the super- +conducting state in LiC6 may appear to be more peculiar +than previously anticipated and additional investigations +in this directions should be of great interest. +[1] H. Y. Lu, Y. Yang, L. Hao, W. S. Wang, L. Geng, +M. Zheng, Y. Li, N. Jiao, P. Zhang, and C. S. Ting, +Phys. Rev. 101, 214514 (2020). +[2] E. Thingstad, A. Kamra, J. W. Wells, and A. Sudbø, +Phys. Rev. B 101, 214513 (2020). +[3] A. P. Durajski, K. M. Skoczylas, and R. Szcz¸e´sniak, Su- +percond. Sci. Technol. 32, 125005 (2019). +[4] T. Uchihashi, +Supercond. Sci. Technol. 30, +013002 +(2017). +[5] J. J. Zheng and E. R. Margine, Phys. Rev. B 94, 064509 +(2016). +[6] J. Zhou, Q. Sun, Q. Wang, and P. Jena, Phys. Rev. B +92, 064505 (2015). +[7] J. Peˇsi´c, R. Gaji´c, K. Hingerl, and M. Beli´c, EPL 108, +67005 (2014). +[8] D. M. Guzman, H. M. Alyahyaei, and R. A. Jishi, 2D +Mater. 1, 021005 (2014). +[9] T. P. Kaloni, A. V. Balatsky, and U. Schwingenschl¨ogl, +EPL 104, 47013 (2013). +[10] G. Profeta, M. Calandra, and F. Mauri, Nat. Phys. 8, +131 (2012). +[11] M. Einenkel and K. B. Efetov, Phys. Rev. B 84, 214508 +(2011). +[12] G. Savini, A. C. Ferrari, and F. Giustino, Phys. Rev. +Lett. 105, 037002 (2010). +[13] R. Gholami, R. Moradian, S. Moradian, and W. E. Pick- +ett, Sci. Rep. 8, 13795 (2018). +[14] C. Bao, H. Zhang, T. Zhang, X. Wu, L. Luo, S. Zhou, +Q. Li, Y. Hou, W. Yao, L. Liu, et al., Phys. Rev. Lett. +126, 206804 (2021). + +7 +[15] C. Guti´errez, C. J. Kim, L. Brown, T. Schiros, D. Nord- +lund, E. B. Lochocki, K. M. Shen, J. Park, and A. N. +Pasupathy, Nat. Phys. 12, 950 (2016). +[16] B. M. Ludbrook, G. Levy, P. Nigge, M. Zonno, M. Schnei- +der, D. J. Dvorak, C. N. Veenstra, S. Zhdanovich, +D. Wong, P. Dosanjh, et al., PNAS 112, 11795 (2015). +[17] D. Szcz¸e´sniak, A. P. Durajski, and R. Szcz¸e´sniak, J. +Phys.: Condens. Matter 26, 255701 (2014). +[18] D. Szcz¸e´sniak and R. Szcz¸e´sniak, Phys. Rev. B 99, +224512 (2019). +[19] L. Pietronero, S. Str¨assler, and C. Grimaldi, Phys. Rev. +B 52, 10516 (1995). +[20] C. Grimaldi, L. Pietronero, and S. Str¨assler, Phys. Rev. +B 52, 10530 (1995). +[21] C. Grimaldi, L. Pietronero, and S. Str¨assler, Phys. Rev. +B 75, 1158 (1995). +[22] S. Q. Hu, X. B. Liu, D. Q. Chen, C. Lian, E. G. Wang, +and S. Meng, Phys. Rev. B 105, 224311 (2022). +[23] E. Cappelluti and L. Pietronero, J. Phys. Chem. Solids +67, 1941–1947 (2006). +[24] P. Paci, E. Cappelluti, C. Grimaldi, L. Pietronero, and +S. Str¨assler, Phys. Rev. B 69, 024507 (2004). +[25] R. Szcz¸e´sniak, Acta Phys. Pol. A 179-186, 509 (2006). +[26] L. P. Gor’kov, PNAS 113, 4646 (2016). +[27] E. F. Talantsev, Nanomaterials 13, 71 (2023). +[28] F. Schrodi, P. M. Oppeneer, and A. Aperis, Phys. Rev. +B 102, 024503 (2020). +[29] A. B. Migdal, Sov. Phys. JETP 34 (7), 996 (1958). +[30] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. +Rev. 106, 162 (1957). +[31] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. +Rev. 108, 1175 (1957). +[32] G. M. Eliashberg, Sov. Phys. JETP 11, 696 (1960). +[33] J. P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990). +[34] J. K. Freericks, E. J. Nicol, A. Y. Liu, and A. A. Quong, +Phys. Rev. B 55, 11651 (1997). +[35] D. +Szcz¸e´sniak, +A. +Z. +Kaczmarek, +E. +A. +Drzazga- +Szcz¸e´sniak, and R. Szcz¸e´sniak, Phys. Rev. B 104, 094501 +(2021). +[36] J. Bauer, J. E. Han, and O. Gunnarsson, Phys. Rev. B +87, 054507 (2013). +[37] W. Wan, Y. Ge, F. Yang, and Y. Yao, EPL 104, 36001 +(2013). +[38] Y. Ge, W. Wan, F. Yang, and Y. Yao, New J. Phys. 17, +035008 (2015). +[39] D. Szcz¸e´sniak and E. Drzazga, EPL 135, 67002 (2021). +[40] P. Miller, J. K. Freericks, and E. J. Nicol, Phys. Rev. B +58, 14498 (1998). + diff --git a/Y9FST4oBgHgl3EQfADiR/content/tmp_files/load_file.txt b/Y9FST4oBgHgl3EQfADiR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70db38c463f33472a121833766dc1f16a098eaca --- /dev/null +++ b/Y9FST4oBgHgl3EQfADiR/content/tmp_files/load_file.txt @@ -0,0 +1,616 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf,len=615 +page_content='Scalability of non-adiabatic effects in lithium-decorated graphene superconductor Dominik Szcz¸e´sniak Department of Theoretical Physics, Faculty of Science and Technology, Jan D�lugosz University in Cz¸estochowa, 13/15 Armii Krajowej Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=', 42200 Cz¸estochowa, Poland (Dated: February 1, 2023) The analysis is conducted to unveil how the non-adiabatic effects scale within the superconducting phase of lithium-decorated graphene (LiC6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Based on the Eliashberg formalism it is shown that the non-adiabatic effects notably reduce essential superconducting parameters in LiC6 and arise as a significant oppressor of the discussed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moreover, nonadiabaticity is found to scale with the strength of superconductivity, proportionally to the phonon energy scale and inversely with respect to the electron-phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' These findings are partially in contrast to other theoretical studies and show that superconductivity in LiC6 is more peculiar than previously anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In this context, the guidelines for enhancing superconducting phase in LiC6 and sibling materials are also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' INTRODUCTION In the literature, a number of scenarios exist on how to potentially induce conventional superconductivity in graphene, promising novel applications of this intrigu- ing carbon allotrope [1–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' These strategies are mainly aimed at the structural modifications of graphene to enhance number of charge carriers at the Fermi level and alter intrinsic semimetallic character of this mate- rial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Among the resulting structures, lithium-decorated graphene (LiC6) is here of particular interest since it is suggested to host phonon-mediated superconducting state generated via process analogous to the intercalation of graphite [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This approach relies on the introduction of adatoms that break chiral symmetry of graphene and lifts Fermi level to the van Hove singularity [10, 13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' However, superconductivity in LiC6 not only derives from the well-established method of synthesis but also appears to be actually feasible, as partially confirmed within the experiment [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' According to the above, LiC6 may be considered as a proving ground for the phonon-mediated superconduc- tivity at low-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Indeed, in recent years the superconducting phase in LiC6 received notable atten- tion in terms of its fundamental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The re- lated studies were devoted, but not limited to, the role of strong electron-phonon coupling [17], character of super- conducting gap [5], symmetry-breaking effects [13], ways of enhancing superconducting state [7], or the substrate ievmpact on superconductivity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Beside listed direc- tions of research, LiC6 appears also to be a perfect exam- ple of superconducting material for studying the influence of non-adiabatic pairing [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This aspect is particularly intriguing since non-adiabatic effects tend to manifest themselves relatively rarely in conventional superconduc- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The reason for that relates to the non-comparable electronic and phononic energy scales in most supercon- ducting materials with the electron-phonon pairing mech- anism [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In other words, it can be often assumed that electrons follow adiabatically ionic oscillations and that the corresponding superconducting state can be de- scribed in a self-consistent manner [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' However, this is not the case when superconducting materials ex- hibit shallow conduction band such as the fullerenes [24], fullerides [23], bismuthates [25], transition-metal-oxides [26] or the discussed LiC6 [18] (see [27] for the review of nonadiabatic superconductors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' So far, the characteristic energy scales are one of the few signatures of non-adiabatic superconductivity in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Other than that recent theoretical studies show non-adiabatic effects to notably reduce magnitude of de- pairing interaction in the discussed material, in compar- ison to the adiabatic regime [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' These findings are also supplemented by the considerations suggesting that non- adiabaticity contributes to the electron-phonon coupling (λ) and modulates the transition temperature (TC) in doped graphene [22] or two-dimensional superconductors in general [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Still, little is known about the scalability of non-adiabatic effects in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In the first approxima- tion, it can be only qualitatively argued that their impact changes according to the Migdal’s ratio (known also as the expansion ratio) given by m = λωD/EF , where EF is the Fermi energy and ωD denotes Debye’s frequency [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In fact, although m-ratio can provide some in- formation on the scalability problem it is mostly used to determine whether or not given material can be described within the Migdal’s theorem [29] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' within adiabatic or non-adiabatic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Hence, the measurable and direct role of the above parameters and their variations in shap- ing non-adiabatic superconductivity in LiC6 is somewhat hindered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In details, it is unknown how changes in the m- ratio components modify experimentally observable ther- modynamic properties such as the superconducting gap or the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This is to say, what trends in thermodynamics can be expected due to the strength of the non-adiabatic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As a result, the relevancy of the energy scales and the electron-phonon coupling in the non-adiabatic limit is also not well-estimated yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Therefore, addressing these aspects would be of great im- portance to the better understanding of superconductiv- ity in LiC6 and potentially other sibling low-dimensional materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moreover, it should also help in assessing im- pact of the external factors that can be applied to mod- ify the aforementioned properties and ultimately enhance arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='13697v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='supr-con] 31 Jan 2023 2 the superconducting state even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' To provide deeper insight into the scalability of non- adiabatic effects in LiC6, the present study analyzes be- havior of the discussed material in the adiabatic and non-adiabatic limit when the expansion ratio parame- ters vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This is done within the Eliashberg formalism that generalizes conventional Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [30, 31] by incorporat- ing the strong-coupling, retardation and non-adiabatic effects [19–21, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As a result it allows to consider both regimes of interest and relate predictions on the piv- otal thermodynamics to the potential factors responsible for the m-ratio variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Here the latter is modeled in reference to [7], by recalling the fact that the deforma- tion potential is able to simultaneously influence energy scales and the electron-phonon coupling in a given su- perconducting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This effect is captured via the percentage change in graphene lattice constant given as δ = |a − a0| /a0 × 100%, where a0(a) is the unmodified (modified) lattice constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In what follows, sev- eral levels of δ are considered allowing for tracing changes in the m-ratio components on the same footing and in the direct relation to the experimentally observable case (δ = 0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Based on that it is possible to unveiled how the non-adiabatic effects scale in LiC6 and what can be done to eliminate their potentially negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' METODOLOGY The theoretical formalism of choice is provided here by following the study of Freericks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [34], where con- venient form of the generalized Eliashberg equations for considering the non-adiabatic superconductivity is pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This theoretical approach is based on the per- turbative theory introduced originally by Pietronero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' in [19–21], which incorporates non-adiabatic effects via vertex corrections to the electron-phonon interac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' However, the theoretical scenario given in [34] in- cludes specific computational techniques for better ac- curacy and efficiency, such as the perturbative theory on the imaginary-axis and the high-frequency resum- mation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In this manner, the resulting equa- tions provide compromise between predictive capabili- ties and computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that such ap- proach was already proved successful in describing non- adiabatic superconductivity not only in LiC6 but also other phonon-mediated superconductors such as lead [34] or bismuthates [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In respect to the above, inital approximations are as- sumed in accordance to [34] and the character of the superconducting phase in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (i) the direct dependence on momentum is neglected for the electron-phonon matrix elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' in correspondence to the isotropic nature of superconducting gap in LiC6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (ii) the depairing correlations are modeled only by the first-order Coulomb pseudopotential terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' due to the fact that higher-order contributions are negligibly small for phonon-mediated superconductors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (iii) sim- ilarly only the lowest-order vertex corrections to the electron-phonon interaction are considered to describe the non-adiabatic effects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' since the Fermi liquid picture in LiC6 appears to be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As a results, it is possi- ble to derive self-consistent Eliashberg equations beyond Midal’s theorem within perturbation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In details, their form on the imaginary axis for the order parameter function (φn = φ (iωn)) and the wave function renormal- ization factor (Zn = Z (iωn)) is following: φn = πkBT M � m=−M Kn,m − µ⋆ m � ω2mZ2m + φ2m φm − Vφ, (1) Zn = 1 + πkBT ωn M � m=−M Kn,m � ω2mZ2m + φ2m ωmZm − VZ, (2) where, kBT is the inverse temperature, with kB denot- ing the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In what follows, ωn = πkBT (2n + 1) is the n-th Matsubara frequency with the cutoff M = 1100 for numerical stability above T = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moreover, Kn,m stands for the electron-phonon pairing kernel given as: Kn,m = 2 � ωD 0 dω ω ω2 + 4π2 (kBT)2 (n − m)2 α2F (ω) , (3) with ω being the phonon frequency, α describing the av- erage electron-phonon coupling and F (ω) denoting the phonon density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that the product of the two latter is known as the electron-phonon spectral func- tion which provides most important information about a physical system within the Eliashberg formalism [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Here, several α2F (ω) functions are considered, each of them corresponding to the different δ-value in order to analyze behavior of LiC6 when the Migdal’s parameter and its components very.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' For this purpose, the exact forms of the α2F (ω) functions are assumed after [7] for δ ∈ ⟨0, 3, 5, 7, 10⟩ %, where the first case corresponds to the experimentally observed superconducting phase of LiC6 whereas the remaining functions describe poten- tial variations from its pristine form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The remaining information is given via the Coulomb pseudopotential µ⋆ n = µ⋆θ (ωc − |ωn|), with θ standing for the the Heavi- side function and ωc for the cut-off frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' To consider all the δ cases on equal footing the conventional value of µ⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='1 [36] which is close to the magnitude of Coulomb depairing interaction predicted for LiC6 at δ = 0% [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Vφ and VZ are the lowest-order vertex correc- 3 tion terms of the following form: Vφ = π3 (kBT)2 4EF M � m=−M M � m′=−M Kn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='mKn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='m′ × 1 � (ω2mZ2m + φ2m) (ω2 m′Z2 m′ + φ2 m′) × 1 � (ω2 m′′Z2 m′′ + φ2 m′′) × (φmφm′φm′′ + 2φmωm′Zm′ωm′′Zm′′ − ωmZmωm′Zm′φm′′) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (4) and VZ = π3 (kBT)2 4EF ωn M � m=−M M � m′=−M Kn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='mKn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='m′ × 1 � (ω2mZ2m + φ2m) (ω2 m′Z2 m′ + φ2 m′) × 1 � (ω2 m′′Z2 m′′ + φ2 m′′) × (ωmZmωm′Zm′ωm′′Zm′′ + 2ωmZmφm′φm′′ − φmφm′ωm′′Zm′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (5) Based on the above, when vertex corrections are consid- ered within the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (1) and (2) they are refereed here to as the non-adiabatic Eliashberg equations (N-E), other- wise, when the corrections are neglected, the formalism is reduced to the adiabatic Eliashberg equations (A-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In what follows, by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (1) and (2) it is pos- sible to obtain estimates on the most important ther- modynamic properties of superconducting state in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Specifically, the central role in such analysis is played by the order parameter function that is obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (1) and (2) as: ∆n(T) = φn/Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Here of special interest is the maximum value (m = 1) of ∆n(T) which contains information on the transition temperature and the superconducting gap half-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The former is de- termined based on the relation ∆m=1(TC) = 0, whereas the latter is given by ∆m=1(T0), with T0 = 2 K being the lowest temperature assumed for calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Since the aforementioned α2F (ω) function is dependent on the characteristic energy scales and the electron-phonon cou- pling constant, the described solutions of the Eliashberg equations also inherit such dependence, allowing for the analysis of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' THE RESULTS AND DISCUSSION In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1, the main numerical results are presented as obtained by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (1) and (2) iteratively with respect to the temperature (see [25] and [34] for more details on the computational methods used here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In de- tails, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1 depicts the behavior of ∆m=1(T) function for T ∈ ⟨T0, TC⟩ at the assumed levels of lattice constant 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 ∆m=1 (mev) T (K) A-E: δ=0% δ=3% δ=5% δ=7% δ=10% N-E: δ=0% δ=3% δ=5% δ=7% δ=10% FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1: The maximum value of order parameter (∆m=1(T)) as a function of temperature in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The results are de- picted for the selected values of lattice constant deviation in graphene (δ) as obtained within the adiabatic (closed sym- bols) and non-adiabatic (open symbols) regime of the Eliash- berg equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Solid lines constitute the guides for an eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' deviation denoted by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that the 0% case corre- sponds to the unaltered LiC6 material, hosting the ex- perimentally observable superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' On the other hand, the remaining cases describe situation when crystal lattice of graphene changes according to the al- ready mentioned expression: δ = |a − a0| /a0 × 100%, with a0(a) standing for the unmodified (modified) lattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moreover, as allowed by the employed formal- ism, the discussed thermal behavior of ∆m=1(T) func- tion is plotted for the adiabatic (open symbols) and non- adiabatic (closed symbols) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In all figures, symbols relate to the exact numerical results of the Eliashberg equations and solid lines constitute guides for an eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The result depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1 reveal several general as- pects of the superconducting state in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In particular, it can be observed that for δ > 3% the increase of the δ value causes notable increase of the ∆m=1(T) in the entire temperature range for both considered regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This trend is not conserved only when comparing re- sult obtained at the two lowest levels of δ, as caused by the δ-driven increase of charge transfer that emp- ties interlayer states and notably reduces the electron- phonon coupling constant at δ = 3% with respect to the 0% case [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Nonetheless, the observed effect means that above some level of δ the superconducting state in LiC6 is clearly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This observation can be quantified by deducing the transition temperature values from the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In particular, TC = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='78 K at δ = 0% and TC ∈ ⟨8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='48, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='61⟩ K for δ ∈ ⟨3, 10⟩ % within the A-E limit, whereas TC = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='29 K at δ = 0% and TC ∈ ⟨6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='59, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='46⟩ K for δ ∈ ⟨3, 10⟩ % when consid- ering the N-E equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that these observations are in qualitative agreement with the previous studies conducted within the adiabatic limit by using the Allen- 4 Dynes formula in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The difference between these data sets and results given in [7] is due to the fact that the as- sumed Eliashberg equations incorporate strong-coupling, retardation, and non-adiabatic effects which are missing in the Allen-Dynes formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' At this point, it is also in- structive to note that the TC value estimated at δ = 0% is slightly higher in comparison to the predictions made within the Eliashberg formalism for the experimentally derived electron-phonon spectral function, as presented in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This discrepancy is obviously caused by the as- sumed value of µ⋆, smaller than the one suggested in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The reason to make such assumption is to allow for better comparison not only with the BCS-derived re- sults given in [7] but also other two-dimensional super- conductors, which are often still hypothetical structures and their superconducting state is described by µ⋆ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [37, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that even if µ⋆ would be as- sumed here after [18], the main outcomes and findings of the present analysis would not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This includes es- timates on the superconducting gap half-width that can be made based on the results plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This is to say, the general behavior of ∆m=1(T0) parameter is the same as in the case of TC and it will not change qualita- tively when assuming other µ⋆ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Specifically, in the present study ∆m=1(T0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='39 meV at δ = 0% whereas ∆m=1(T0) ∈ ⟨1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='30, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='55⟩ meV for δ ∈ ⟨3, 10⟩ % in the A-E regime, while the N-E equations yield ∆m=1(T0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='22 meV at δ = 0% and ∆m=1(T0) ∈ ⟨1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='11, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='84⟩ meV for δ ∈ ⟨3, 10⟩ %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that results on TC and ∆m=1(T0) can be supplemented by introducing their characteristic ratio, familiar in the BCS theory and given by [30, 31, 33]: R = 2∆m=1(T0) kBTC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (6) The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (6) not only allows for additional insight into the considered problem but also provides yet another ob- servable for future comparisons with the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As it can be expected, the obtained values of R follow the same trends like the TC and ∆m=1(T0) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The values of R base on the A-E equations are R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='67 at δ = 0% and R ∈ ⟨3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='55, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='72⟩ for δ ∈ ⟨3, 10⟩ %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' On the other hand, the N-E equations give R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='89 at δ = 0% and R ∈ ⟨3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='92, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='98⟩ for δ ∈ ⟨3, 10⟩ %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Still, both sets present values higher than the level suggested within the BCS theory and equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' It means that the strong- coupling and retardation effects play relatively impor- tant role in shaping the superconducting state in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This observation is in agreement with the strength of the electron-phonon coupling reported in [7] and previ- ous findings given in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Beside the above observations, it is also crucial to note that the reported results suggest simultaneous changes in the energy scales and the electron-phonon coupling constant due to the variations of δ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Indeed, all of these characteristic parameters exhibit increasing or decreasing trends along with the growing δ value (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' For convenience, their cumulative behavior is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (B) in terms of already introduced 120 140 160 180 1000 1200 1400 1600 1800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='14 0 2 4 6 8 10 4 8 12 16 20 24 28 ωD (meV) EF (meV) (A) λ m m=ωD/EF m=λωD/EF (B) Dx (%) x=1 x=2 x=3 δ (%) (C) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2: The behavior of (A) the Fermi energy (EF ), Debye’s frequency (ωD) and electron-phonon coupling constant (λ), (B) the dressed (m = λωD/EF ) and bare (m = ωD/EF ) Migdal’s ratio as well as (C) the percentage differences be- tween adiabatic and non-adiabatic estimates for the critical temperature (D1), superconducting gap half-width (D2) and their cumulative ratio (D3) at the selected values of the lat- tice deviation in graphene (δ) in the LiC6 superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The closed symbols depicts exact results, the solid lines are the guides for an eye and the color arrows points to the cor- responding axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' dressed Migdal’s ratio (m = λωD/EF ) but also its bare 5 TABLE I: The parameters of superconducting state in LiC6 for the selected values of the lattice deviation in graphene (δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In a consecutive order, the parameters are: the electron-phonon coupling constant (λ), the Debye’s frequency (ωd), the Fermi energy (EF ), the bare (ωd/EF ) and dressed (λωd/EF ) Migdal’s ratio, the transition temperature (TC), the superconducting gap half-width (∆m=1(T0)) as well as the thermodynamic ratio for the two last ones (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that, where necessary, the results are presented for the in terms of the adiabatic (A − E) and non-adiabatic (N − E) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moreover, the percentage differences between estimates in these two limits for TC (D1), ∆m=1(T0) (D2) and R (D3) are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' A-E N-E δ λ ωd EF ωd/EF λωd/EF TC ∆m=1(T0) R TC ∆m=1(T0) R D1 D2 D3 (%) (meV) (meV) (K) (meV) (K) (meV) (%) (%) (%) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='61 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='21 1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='078 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='78 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='98 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='20 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='75 counterpart (m = ωD/EF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In what follows, it is argued here that the scalability of non-adiabatic effects in LiC6 can be traced with respect to the pivotal parameters en- tering Migdal’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In this context, first it should be noted that for each considered δ value the non-adiabatic equations yield lower ∆m=1(T) values than their adia- batic counterparts (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This directly relates to the fact that the transition temperature values as well as the estimates of the superconducting gap are lower in the non-adiabatic regime when comparing to the adiabatic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As a results, the percentage difference between esti- mates made in two considered regimes can be introduced as a measure of non-adiabatic effects impact on super- conducting phase in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This new measure is depicted for all considered thermodynamic parameters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In details, the percentage difference between TC val- ues determined in the adiabatic and non-adiabatic limits is D1 = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='54% at δ = 0% and D1 = ⟨25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='08, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='20⟩% for δ ∈ ⟨3, 10⟩ %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Similarly, the same percentage mea- sure but for the ∆m=1(T0) parameter is D2 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='03% for δ = 0% and D2 = ⟨15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='77, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='67⟩% when δ ∈ ⟨3, 10⟩ %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Finally, the cumulative ratio gives the corresponding per- centage D3 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='37% for δ = 0% and D3 = ⟨9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='91, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='75⟩% when again δ ∈ ⟨3, 10⟩ %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Based on these results, it is clear that the critical temperature is the most influenced by the non-adiabatic effects from all three considered pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' It also presents the most visible signature of the charge transfer from interlayer states at δ ∈ 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' On the contrary, the cumulative ratio shows the smallest dis- crepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Still all three parameters exhibit the same qualitative behavior i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' the percentage difference be- tween adiabatic and non-adiabatic results increases up to δ ∈ 3% and then starts to almost linearly decrease as the δ takes higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The final observations can be made when comparing results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (C) with the estimates de- picted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (A) and (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In particular, it can quali- tatively argued that the behavior of percentage measures does not fully resemble the Migdal’s ratio dependence on the δ value, although the latter is considered to be the first approximation approach to provide information on the scalability of non-adiabatic effects in a superconduc- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Precisely speaking, only the bare ratio can be con- sidered somewhat similar in behavior to the percentage measures, whereas its dressed value presents almost in- verse character with respect to the parameters given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' To inspect these discrepancies even further it is instructive to compare the percentage difference mea- sures with the characteristic component parameters of the Migdal’s ratio, as plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The out- come is that only the Debye’s energy scales qualitatively the same as the percentage difference measures, while the electron-phonon coupling gives inverse characteristic and the electronic energy scale is practically nowhere similar to the results given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS In summary, the presented analysis provides new in- sight into the superconducting properties of LiC6 in terms of its non-adiabatic characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' It shows how the non-adiabatic effects scale in LiC6 with respect to the deviation of lattice constant in graphene, which can be considered as an exemplary factor that modifies strength of the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In details, the discussed scalability is expressed here in terms of the percentage difference between estimates of pivotal thermodynamic parameters (the transition temperature, superconduct- ing gap and their ratio) obtained within the adiabatic and non-adiabatic regime, allowing for further compari- son with the Migdal’s expansion ratio that characterizes nonadiabaticity in the first approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' For conve- nience, all the obtained numerical results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Based on the above findings it is possible to draw sev- eral conclusions related, but no limited to, the scalability of non-adiabatic effects in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In details: (i) The introduction of vertex corrections to the electron-phonon interaction causes notable changes in the pivotal thermodynamic parameters of the su- perconducting state in LiC6, in particular their de- crease in comparison to the adiabatic limit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 6 2 (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This is to say, the superconducting state appears to have strongly non-adiabatic character, where non-adiabatic effects act as an important op- pressor of superconductivity in LiC6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Notably the superconducting state sustains its non-adiabatic character even when superconductivity in LiC6 is strongly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Still the non-adiabatic effects are observed to visibly vary with the strength of superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moreover, it is found that nonadiabaticity is supplemented by the strong- coupling and retardation effects, meaning that LiC6 is a somewhat unorthodox phonon-mediated super- conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (ii) The considered thermodynamic parameters present the same qualitative behavior under the influence of non-adiabatic effects when the strength of super- conductivity is varied (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Nonetheless, the critical temperature is suggested to be partic- ularly sensitive to nonadiabaticity, while supercon- ducting gap is showing much smaller dependence on the variation of the discussed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As a re- sult, this opens new prospect for increasing tran- sition temperature value in LiC6 according to the presented here findings, saying that smaller mag- nitude of non-adiabatic effects leads to the higher transition temperature (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that this trend is not conserved when including results for the unaltered LiC6, due to the decreased charge transfer from the interlayer states in comparison to other considered cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' It can be additionally argued that such trend may be considered general for other graphene-based superconductor which ex- hibit similar dependence on nonadiabaticity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' recent study on the electron-doped graphene [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' (iii) The deeper inspection of the obtained results shows that the non-adiabatic effects scale with the strength of superconductivity, proportionally to the phonon energy scale and inversely to the electron-phonon coupling magnitude (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (A) and (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that, while the former observa- tion agrees with the predictions of both considered forms of the Migdal’s ratio, the latter is in contrast to what can be expected based on the dressed pa- rameter and previous theoretical studies consider- ing superconductivity in two-dimensional systems [22, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' However, the mentioned trends does not take into account existing interplay between all components of the Migdal’s ratio and the fact that the electron-phonon coupling constant increases al- most linearly with the electronic energy scale (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' As a results, strong electron-phonon cor- relations correspond to the relatively wide conduc- tion band that causes suppression of nonadiabatic- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This argument is confirmed qualitatively by the observed here cumulative behavior of the Migdal’s ratio (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 2 (B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' This is to say the electron- phonon coupling cannot be always considered to be improved in graphene-based superconductors by the non-adiabatic effects as previously suggested in [22, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' To sum up, the perspectives for future research can be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In details, to provide better understating of the su- perconducting state in LiC6 the discussed non-adiabatic effects should be considered beyond the isotropic ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Note that such preliminary analysis can be already found in [28], while the adiabatic anisotropic investigations are available in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' The present study can be also extended further toward other experimen- tally observable thermodynamic parameters such as the free energy or the critical thermodynamic field, accord- ing to their importance in discussing the non-adiabatic effects [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Finally, recent discussion given in [27] sug- gest strongly metallic behavior of LiC6 despite its high value of the Migdal’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' In other words, the super- conducting state in LiC6 may appear to be more peculiar than previously anticipated and additional investigations in this directions should be of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Hao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Geng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Jiao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zhang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Ting, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 101, 214514 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Thingstad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Kamra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wells, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Sudbø, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 101, 214513 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Durajski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Skoczylas, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, Su- percond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 32, 125005 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Uchihashi, Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 30, 013002 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zheng and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Margine, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 94, 064509 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Sun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Jena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 92, 064505 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Peˇsi´c, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Gaji´c, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Hingerl, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Beli´c, EPL 108, 67005 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Guzman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Alyahyaei, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Jishi, 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 1, 021005 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Kaloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Balatsky, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Schwingenschl¨ogl, EPL 104, 47013 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Profeta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Calandra, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Mauri, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 8, 131 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Einenkel and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Efetov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 84, 214508 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Savini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Ferrari, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Giustino, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 105, 037002 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Gholami, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moradian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Moradian, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pick- ett, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 8, 13795 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Bao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Luo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Hou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 126, 206804 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 7 [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Guti´errez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Kim, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Schiros, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Nord- lund, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Lochocki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Shen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Park, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pasupathy, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 12, 950 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Ludbrook, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Levy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Nigge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zonno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Schnei- der, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Dvorak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Veenstra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Zhdanovich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Dosanjh, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=', PNAS 112, 11795 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Durajski, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Matter 26, 255701 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 99, 224512 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pietronero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Str¨assler, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Grimaldi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 52, 10516 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Grimaldi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pietronero, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Str¨assler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 52, 10530 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Grimaldi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pietronero, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Str¨assler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 75, 1158 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Lian, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Meng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 105, 224311 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Cappelluti and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pietronero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Solids 67, 1941–1947 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Paci, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Cappelluti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Grimaldi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pietronero, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Str¨assler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 69, 024507 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, Acta Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' A 179-186, 509 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Gor’kov, PNAS 113, 4646 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Talantsev, Nanomaterials 13, 71 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Schrodi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Oppeneer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Aperis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 102, 024503 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Migdal, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' JETP 34 (7), 996 (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Bardeen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Cooper, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Schrieffer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 106, 162 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Bardeen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Cooper, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Schrieffer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 108, 1175 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [32] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Eliashberg, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' JETP 11, 696 (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Carbotte, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 62, 1027 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Freericks, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Nicol, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Liu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Quong, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 55, 11651 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Kaczmarek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Drzazga- Szcz¸e´sniak, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 104, 094501 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Bauer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Han, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Gunnarsson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 87, 054507 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [37] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Ge, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Yang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Yao, EPL 104, 36001 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [38] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Ge, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Wan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Yang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Yao, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' 17, 035008 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Szcz¸e´sniak and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Drzazga, EPL 135, 67002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' [40] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Freericks, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Nicol, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} +page_content=' B 58, 14498 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FST4oBgHgl3EQfADiR/content/2301.13697v1.pdf'} diff --git a/YdFAT4oBgHgl3EQf3R5S/content/tmp_files/2301.08719v1.pdf.txt b/YdFAT4oBgHgl3EQf3R5S/content/tmp_files/2301.08719v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d7508ce4250cbf02f89e7b03088a379350dbc34 --- /dev/null +++ b/YdFAT4oBgHgl3EQf3R5S/content/tmp_files/2301.08719v1.pdf.txt @@ -0,0 +1,2120 @@ +The stochastic digital human is now enrolling for in +silico imaging trials – Methods and tools for +generating digital cohorts +A Badano1, M Lago1, E Sizikova1, JG Delfino1, S Guan1 +and MA Anastasio2 and B Sahiner1 +1Division of Imaging, Diagnostics, and Software Reliability, Office of Science and +Engineering Laboratories, Center for Devices and Radiological Health, +U. S. Food and Drug Administration, Silver Spring, MD 20993 +2Department of Bioengineering, The Grainger College of Engineering, University +of Illinois, Urbana, IL 61801 +E-mail: aldo.badano@fda.hhs.gov +23 January 2023 +Abstract. +Randomized clinical trials, +while often viewed as the highest +evidentiary bar by which to judge the quality of a medical intervention, are +far from perfect. +In silico imaging trials are computational studies that seek +to ascertain the performance of a medical device by collecting this information +entirely via computer simulations. The benefits of in silico trials for evaluating new +technology include significant resource and time savings, minimization of subject +risk, the ability to study devices that are not achievable in the physical world, +allow for the rapid and effective investigation of new technologies and ensure +representation from all relevant subgroups. +To conduct in silico trials, digital +representations of humans are needed. +We review the latest developments in +methods and tools for obtaining digital humans for in silico imaging studies. First, +we introduce terminology and a classification of digital human models. Second, +we survey available methodologies for generating digital humans with healthy and +diseased status, and examine briefly the role of augmentation methods. Finally, +we discuss the trade-offs of four approaches for sampling digital cohorts and the +associated potential for study bias with selecting specific patient distributions. +Social media blur (100-w): From digital twins to other digital humans for +in silico trials: we review methods and tools for obtaining stochastic humans for +digital cohorts [LINK] +Submitted to: PRGB +arXiv:2301.08719v1 [cs.AI] 20 Jan 2023 + +CONTENTS +2 +Contents +1 +Introduction +3 +2 +Terminology +4 +3 +Representations +4 +4 +Individual models +5 +4.1 +Personalized models +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +4.2 +Family models +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +5 +Population models +6 +5.1 +Image-based models +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +5.1.1 +Image-based parametric models . . . . . . . . . . . . . . . . . . +6 +5.1.2 +Image-based generative models . . . . . . . . . . . . . . . . . . +7 +5.2 +Knowledge-based models . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +6 +Modeling disease +9 +6.1 +Image-based models of disease . . . . . . . . . . . . . . . . . . . . . . . +9 +6.2 +Knowledge-based models of disease . . . . . . . . . . . . . . . . . . . . +10 +7 +Role of augmentation methods +10 +8 +Considerations for sampling digital cohorts +11 +9 +Summary and conclusions +12 + +CONTENTS +3 +1. Introduction +Two decades ago, in the epilogue of their seminal +textbook on image science [1], Barrett and Myers +pointed out that in the future, sport games might be +played with simulated athletes. The advancement of +computer graphics and simulation technologies sparked +the notion that perhaps the excitement of a real-life +sports event could be conducted in the simulation +space with digital models of athletes. +Since then, +continuous advances in computer processing power and +modeling techniques have taken place, driven primarily +by entertainment applications [2] and quickly becoming +a significant component of research and development +(R&D) efforts in a variety of industries‡. +Industries +that have widely adopted computational modeling and +in silico methods throughout the product life-cycle +include automotive [3] and manufacturing [4] among +others [5]. Medicine lags considerably behind [6] due, +in part, to model complexity, challenging validation, +associated potential risks for new devices and drugs, +and lack of consensus and regulatory standards. +Randomized clinical trials, while often viewed +as the highest evidentiary bar by which to judge +the quality of a medical intervention, are far from +perfect. Common causes of failure include safety issues, +difficulties with patient recruitment, enrollment, and +retention [7]. In addition, clinical trials can suffer from +under-representation of rare subpopulations [8]. These +limitations represent a unique opportunity to develop +in silico trials that are completed as planned, safely, +and that include digital cohorts with a representative +distribution of subject characteristics and numbers +large enough for appropriate statistical power. +As +pointed out in [9], in silico data has the potential to +address lack of data availability, sharing mechanisms +and privacy challenges associated with the use of +medical information. +In silico imaging trials are computational studies +that seek to ascertain the performance of a medical +device for the intended population, collecting this +information entirely in the digital world via computer +simulations. The benefits of in silico imaging trials for +evaluating new technology include significant resource +and time savings, minimization of subject risk, and +ethical considerations [10, 11]. +Moreover, in silico +trials can be used to study devices that do not +yet exist or are not practically attainable in the +(limited) physical world, allow for the rapid and +effective investigation of new technologies [11, 12, +13], and facilitate representation from all relevant +subpopulations. +Each one of these benefits is an +‡ To date, Superbowl games are played with physical-world +athletes, in part due to the difficulty of conveying real-life +personal struggle, an essential component of the entertainment +context for sport players and teams (see, for instance, here). +essential consideration within the context of the +regulatory evaluation of medical technology [11]. +The realization that computational models of +humans would take center stage in medical imaging +system assessment is not new. +Full optimization of +imaging systems for specific medical tasks requires +objects +(physical +or +digital) +that +represent +the +variability seen in patients. +For many decades, +scientists have relied on practical and simpler versions +of patients [14]. However, recent advances in computer +processing power and simulation methods are now +facilitating the development of more detailed and +realistic patient models that are based on digital +stochastic descriptions of the model components. For +instance, a recent report demonstrated the feasibility of +an in silico trial, the Virtual Imaging Clinical Trial for +Regulatory Evaluation (VICTRE), as an alternative +approach to establish regulatory evidence in support +of medical imaging products [15]. +There are numerous parallels between digital- +and physical-world trials. +Fundamentally, in silico +trials must include the same essential elements of +well-designed physical-world clinical trials. +Firstly, +the population of subjects for whom the new device +or technology is intended must be defined. +The +study design must contain clear rules for selection and +rejection of subjects from a distribution of healthy and +diseased subjects. +However, in silico trials are not +subject to effects from covariates in patient selection. +For +instance, +a +common +problem +in +evaluating +screening tests meant for asymptomatic subjects is +that a portion of the enrolled population might be +symptomatic [16] with the potential for verification +bias [17]. Secondly, when there are two technologies +that are being compared, i.e., a new, yet unproven +technology and a comparator technology currently in +clinical use, both must be unambiguously defined. +A good choice for comparator technology should be +associated with accurate representations of the device +characteristics as supported by validation studies [18]. +Thirdly, the study requires a definition of the users +of the device’s outcome (i.e., images in the case of an +imaging device trial). +These first three components +reflect the physical intended use of the device under +investigation, i.e., the intended populations of subjects, +the intended device comparison, and the intended +image interpreters that will be using the device in the +physical world. Finally, whether physical or digital, the +trial design must provide a definition of the primary +outcome to be evaluated, a protocol and statistical +analysis associated with the trial, and an analysis of +the risk and benefits introduced by the device under +investigation. +Both +physical +and +in +silico +studies +require +enrollment +of +representative +subjects. +In +this + +CONTENTS +4 +review, we survey the latest developments in methods +and +tools +for +generating +the +cohorts +of +digital +humans +for +imaging +studies +that +represent +the +variability of physical-world subject populations. We +refer to the digital cohorts consisting of digital +humans (realizations of the digital human models) as +“stochastic humans”. Assessment of new technology +and the regulatory evaluation of that technology +requires establishing performance levels for intended +populations and, therefore, necessitates computational +models that allow sampling of the parameter space +defining the subject population in the physical world. +We propose to name these models digital humans as +opposed to digital replicas or twins to avoid confusion. +The review is organized as follows. +First, +we introduce terminology and representation models +regarding +the +different +types +of +digital +humans +described throughout the article. Second, we survey +available methodologies for generating digital humans +with healthy status and for generating diseased cases. +Then, we briefly discuss the role of augmentation +methods and conclude with an analysis of sampling +techniques that may be used to generate the digital +cohorts for evaluating the performance of imaging +devices. +2. Terminology +A variety of terminologies are being used or proposed +for describing digital representations of humans in +medicine and other fields. +In the literature, some +of these are often used without the benefit of a +clear definition and, +in some instances, +wrongly +interchangeably. +We propose to use the term stochastic digital +human to denote digital representations of humans (or +human body parts) generated from multiple random +outputs by sampling known distributions for the +model characteristics matching the variability observed +in human populations. +In contrast, non-stochastic +representations are deterministic digital versions of a +single physical exemplar (e.g., a model of a human +body at a given time) or a group (or family) of +physical exemplars which are differentiated by varying +physical parameters. +Contrary to other terms and +concepts currently being discussed including digital +families, avatars, chimeras, and digital twins, the +concept of a stochastic digital human represents an +approach for in silico trials and regulatory evaluation +that estimates the performance of an imaging device for +a population of subjects rather than for an individual +patient, thus incorporating the variability observed in +the population. +We propose to classify all digital humans as +either individual or population models (see Figure 1). +Individual models are necessarily image-based while +population models can be derived either from images +or from knowledge of the fundamental characteristics +that define the relevant features of a human. +Note +that we will use the term digital human to refer to the +models even if the represented object is a part of the +body or the whole body of a subject. +3. Representations +Physical objects (including humans) can be repre- +sented using continuous variables. +We consider the +models of humans as continuous in space (r) and time +(t) and described by a coefficient vector affecting a set +of model characteristics: +fm(r, t) ≈ +N +� +n=1 +θnφn(r, t). +(1) +Here, N is the dimension of the approximate finite- +dimensional representation of the object, and the +subscript m indicates the modeling approximation to +differentiate from the actual object f(r, t). +The collection of expansion functions {φn(r, t)}N +n=1 +is employed to form fm(r, t), and θn denotes the n-th +component of the N-dimensional expansion coefficient +vector θ. The quantity fm(r, t) constitutes a discrete +representation of a digital human that can be readily +displayed on a computer or digitally processed. For the +case where the expansion functions are defined as in- +dicator functions that describe non-overlapping space- +time voxels, θ can sometimes be interpreted as a digital +image whose components θn represent the integrated +value of the object over the support of the voxel. +More generally, a digital human model can be es- +tablished by integrating the continuous representation +fm(r, t) over a collection of N voxels as +fn = +� +vn +fm(r, t) d3r dt, +n = 1, · · · , N, +(2) +where vn denotes the support of the n-th spatial- +temporal voxel and fn denotes the n-th component of +a N-dimensional vector f that represents the digital +human. +As discussed below, the choice of the expansion +functions and associated expansion coefficients can be +specified in different ways, with the general goal of +making fm(r, t) an accurate approximation of f(r, t). +The expansion functions can depict geometry (e.g., +size, morphology), material properties (e.g., x-ray +interaction cross-sections, elasticity) or other relevant +features (e.g., radioactivity, blood oxygenation levels). +For simplicity, we will consider that the stochastic +human does not vary with time and proceed only with + +CONTENTS +5 +the spatial dimension r. However, the concepts that +follow can readily be generalized to model time-varying +descriptions.[19] +In practice, the coefficient vector θ can be modeled +as a random vector and the expansion functions +{φn(r)}N +n=1 as random processes. +Methodologies for +generating large cohorts of digital stochastic models of +humans for in silico imaging trials, including models +for organs and tissues with appropriate variability, can +rely on either sampling θ, φn or both from appropriate +distributions representing the intended population. We +can denote the cohort of digital stochastic humans as +follows, +{fs}S +s=1 = +� +n +θs +nφn(r), +(3) +where +s +denotes +a +particular +state +or +random +realization of a digital human in a cohort of size S. +When φn are known, analytically or numerically, +the stochastic models are referred to as procedural. +In this case, the modeler is left with choosing the +coefficient vector defining the object (θ). In cases for +which the defining characteristics are unknown, θn and +φn can be estimated from imaging data. +In the following sections, we review available +methods and tools for generating digital human models +and digital cohorts. +We present a classification of +available approaches in Figure 1. +4. Individual models +Individual models attempt to create a digital replica +of a specific physical object. Individual models can be +categorized as personalized and family models. These +models are not stochastic since they are meant to +represent individual subjects with as much detail and +accuracy as achievable from the image data. In this +respect, the representation introduced in Section 3 +applies only with S = 1 resulting in a single coefficient +vector (θn) defining the individual. +The +digital +representation +in +these +cases +is +typically a multidimensional voxelized array that +can be segmented into structures such as tissues +and organs. +Early attempts relied on geometrical +volumes represented by analytical expressions altered +to generate a wide variety of sizes and shapes. In other +words, φn are described by quadrics and θn represent +properties of the volumes defined by the surfaces (e.g., +x-ray attenuation and scattering properties). +These +computational models have proved useful in areas of +quality control of imaging systems [20, 21] and in +radiation dosimetry [22]. Even with more sophisticated +geometrical structures [23, 24, 25] and more spatial +detail, these approaches lack the ability to accurately +represent the statistical variability found in humans, +organs and tissues. While these simpler models remain +practical and useful for some tasks, the lack of realism +and variability makes them unsuitable for generating +digital humans for in silico imaging trials. +4.1. Personalized models +Personalized models aim to capture patient-specific +information in a digital representation [26]. Medical +digital +replicas +of +human +subjects +are +in +silico +representations of an individual in terms of anatomy +and physiology. +Sometimes referred to as digital +twins [27], these replicas are designed to simulate +parts or the whole body of a subject for prognostic +or predictive assessments. +These models including digital twins can be +continuously updated from multimodal medical if +the data characteristics change over time§. +Digital +twins are of interest in the context of evaluating and +selecting optimal medical treatments [28] or imaging +procedures [29] within clinical practice, and can also +be incorporated into other in silico applications [30]. +For instance, Wang [31] suggested three applications +in the areas of medical imaging: optimal selection of +scanning techniques (so called “virtual comparative +scanning”), data sharing from in silico scanning of +the digital replica to the open source community, +and improvement of the regulatory process of image +reconstruction algorithms. Patient image datasets can +also be used to generate models of specific tissues and +organs. For instance, the Visible Human project [32] +was first made available in 1994 by the National +Library of Medicine (NIH) to facilitate anatomy +visualization applications and includes a detailed data +set of cross-sectional photographs of the human body. +4.2. Family models +Personalized models of a small number of subjects can +be assembled into families to generate a collection of +a small number of digital humans spanning a common +set of parameters, such as subjects’ body size and age. +These models are based on image acquisitions using +different modalities including computed tomography +(CT), magnetic resonance imaging (MRI) and chest +radiographs (CXR). +An example of a family model is the Virtual +Family [33], released by FDA ∥ in 2012. The Virtual +Family consists of a set of detailed, anatomically +correct whole-body models of an adult male, an adult +female, and two children based on high-resolution +MRI data of healthy volunteers. Organs and tissues +§ A related concept is an avatar, an artistic and sometimes +aspirational digital representation of the human in the digital +world for interactivity purposes. +∥ https://www.fda.gov/about-fda/cdrh-offices/ +virtual-family + +CONTENTS +6 +Digital +humans for +in silico trials +Population +models +(stochastic) +Knowledge- +based +Image-based +Generative +Parametric +Individual +models (non- +stochastic) +Family +Personalized +Figure 1. Classification of ethods to generate digital humans for in silico clinical trials. +are represented using computer-aided design (CAD) +techniques where each component is a high-resolution, +non self-intersecting mesh. +In this case, the models +are used for electromagnetic, thermal and acoustic +simulations in the safety assessment of active and +passive medical implants [34]. +Safety evaluations do +not require full sampling of the intended population +and can be performed with a small number of +exemplars, provided the exemplars adequately cover +the needed parameter space. +Similar approaches are utilized in efforts to +provide models of patient anatomy using patient +images +as +the +basis +for +development +of +cohorts +including using MRI and CT images for modeling +lungs [35] and torso [14]. More recently, image-derived +digital and physical models of the breast have been +proposed by Kiarashi [36] and Bliznakova [37]. +In +this approach, a voxelized breast model is derived +from patient images through image segmentation for +determining the composition of each voxel [38, 39, 40, +41, 42, 43, 44]. Patient-derived models are limited to +the imaging characteristics of the acquisition system +and are also affected by the imperfections of the +segmentation methods. The resulting models can also +be augmented with physiological features to facilitate +imaging studies involving contrast agents [45]. +5. Population models +Testing new imaging devices, however, requires the +availability of large digital cohorts of stochastic digital +humans that can be assembled to properly power +a study not only on the aggregate (i.e., for the +entire population), but also to analyze for specific +subgroups +with +notable +characteristics, +including +under-represented populations. +In this section, we +focus our attention on models suited for the generation +of large cohorts of digital humans to be enrolled within +in silico imaging trials. +5.1. Image-based models +Image-based +models +estimate +and +sample +model +components from relevant characteristics within the +acquired +medical +images. +Image-based +models +estimate model components φn and θn in Eq. 3 +from within the acquired medical images. +Whether +parametric or generative, +all image-based models +are +limited +by +the +quality +of +the +source +data +(i.e. +medical images), +including noise, +artifacts, +and contrast constraints, +and do not provide an +unequivocal mapping to the underlying tissues. +In +practice, the use of image-based models should also +acknowledge the limitation arising from the existence +of a null space of the imaging system [46]. +The +null space, which typically arises from the mapping +of a continuous object to discrete data with an +imperfect image acquisition system, results in an +unavoidable loss of information regarding the object. +Given that the imaging system operator is only +partially known for most imaging systems and cannot +represent information obscured by the null space of +the imaging transformation, image-based models are +limited even when imaging system models include noise +measurement. +5.1.1. +Image-based parametric models +In image- +based parametric models, the generation of cohorts is +achieved by creating models based on available sets +of patient imaging data and model modification tech- +niques including parametric deformation, morphing, +and registration. +Parametric models (also known as +stylized phantoms [47]) capture a population cohort by +a set of mathematical equations representing a series +of surfaces (e.g., splines) defining organs that are later + +CONTENTS +7 +voxelized into a volumetric model. The popular 4D ex- +tended cardiac-torso (XCAT) phantom [48] is an exam- +ple of an image-based parametric model, and a survey +of other representations can be found in Kainz [49]. +One limitation of this approach is that model +development is typically performed on a small number +of available patient images. For instance, Erickson [39] +presented a methodology to create a database of +anatomically variable 3D digital breast models from +dedicated breast CT images using a tissue classification +and segmentation algorithm and a fuzzy C-means +segmentation algorithm. +The study provided a +population of 224 breast phantoms incorporating +a range of breast types, +volumes, +densities, +and +parenchymal patterns. +However, using hundreds of +images might be insufficient to properly characterize +a population for statistically powered in silico imaging +trials across patient variability. +Some recently released image datasets include +a +larger +number +of +cases. +For +example, +the +Medical Information Mart for Intensive Care (MIMIC) +CXR dataset [50] contains 227,835 imaging studies +from 65,379 patients presenting to the Beth Israel +Deaconess Medical Center Emergency Department +between 2011–2016. +Similarly, the Medical Imaging +and Data Resource Center (MIDRC) effort [51] is +undertaking a large, multi-year, systematic effort to +collect high-quality COVID data, and over 100,000 +imaging studies have been made public after 2 years +of work and with significant funding from the NIH. +However, data sets collected in these well-defined +areas are likely still insufficient to capture the total +variability in patient images and the large number of +subgroups one may find interesting to study ¶. This +limitation precludes the use of image-based parametric +models for accurately creating digital cohorts for large +scale in silico trials. +Generation of multiple realizations of humans to +constitute a cohort can be obtained by extending +image-derived models to create populations in a +statistical manner. +For instance, +Sturgeon [52] +developed synthetic breast models using principal +component analysis (PCA) to describe a small training +set of patient images. +In this approach, +each +existing patient breast CT volume was compactly +represented by the mean image plus a weighted sum of +eigenbreasts. The distribution of weights was sampled +to create synthesized breast phantoms that matched +fibroglandular density and noise power law exponent +distributions in real images. Hence, the distribution +of the synthetic model is determined by that of the +training data, and, therefore, might suffer from a lack +¶ “I cannot breed them. So help me, I have tried. We need more +. . . than can ever be assembled. Millions, so we can be trillions +more,” Niander Wallace in Blade Runner 2049 (see https: +//www.imdb.com/title/tt1856101/characters/nm0001467). +of appropriate representations of cases at the tails of +the distribution (e.g., very large or very small, very +dense or very glandular breasts). +A related concept +from the computer vision and graphics community is +the statistical human body model, in which a vertex- +based model of the body surface is learned, typically +via PCA, from subjects’ input. The techniques rely on +linear blend skinning (LBS) to constrain the surface +vertex deformation with respect to a template bone +skeleton [53]. Created for non-medical purposes, these +parametric models are typically learned from training +examples of lower resolution than what is common in +medical imaging. +One alternative approach is to add deformation +morphing using an anatomic template [26]. Lee [47] +introduce a hybrid, +non-uniform rational B-spline +surface (NURBS) based phantom of an infant by +combining the expressiveness of a voxel phantom +with the flexibility of geometric manipulation and +organ positioning in a parametric phantom. Another +example is the XCAT Warp [54], where AI-assisted +unsupervised registration is used to warp XCAT to +patient CT images to capture a more broad set +of variations, compared to the existing organ and +model scaling offered by XCAT. These methods are +suitable for investigating digital-twin approaches where +individual models reflecting the characteristics of a +single individual are needed. +5.1.2. Image-based generative models +Image-based +generative models attempt to synthesize a population +of stochastic digital humans from information con- +tained in medical images. Ideally this population cap- +tures the variability in the anatomy and tissue prop- +erties within a specified cohort of to-be-imaged sub- +jects. Consider a collection of N-dimensional digital +humans {fs}S +s=1 that represents the cohort of interest +as described by Eq. 3. This setting corresponds to a +practical situation in which an in silico study employs +a fully discrete representation of an imaging system in +which a finite-dimensional approximation of an object +is mapped to discrete image data. +As mentioned in +Section 3, each digital human fs can be interpreted as +a realization of a random vector f that is characterized +by an unknown probability density function pr(f). The +ability to sample from pr(f) to generate large ensem- +bles of objects for use in in silico imaging trials is, at +least conceptually, the ultimate objective of a stochas- +tic digital human model. Emerging generative methods +that utilize neural networks are being actively devel- +oped for this purpose [55]. We refer to these methods +as generative models. +A generative adversarial net- +work (GAN) is a type of generative model that has +recently been very popular for high-resolution image +synthesis [56], image translation [57, 58] and a number + +CONTENTS +8 +of generative image applications [59]. Instead of explic- +itly modelling pr(f), which is difficult due to the high +dimensionality of f, GANs seek to define a stochas- +tic process for drawing samples. As such, GANs are +categorized as implicit generative models. Specifically, +GANs operate by mapping samples from an analyti- +cally tractable, low-dimensional distribution pr(z) to +the sought after samples of the high-dimensional dis- +tribution pr(f). Typically, pr(z) is specified as an in- +dependent and identically distributed (i.i.d.) standard +normal distribution, and therefore, samples of the ran- +dom vector z can be readily generated. The mapping +is usually implemented via a deep neural network re- +ferred to as the generator. Simultaneously with gen- +erator training, a discriminator network is trained to +discriminate between the real and generated examples. +Therefore, the training process is adversarial and is +approximately solving a min-max optimization prob- +lem [60]. In this case, a collection of training data (typ- +ically images) are utilized to learn how to sample from +an empirical distribution that approximates pr(f). An +excellent review of GAN applications for medical im- +age generation can be found in [61]. The adversarial +training process for GANs is inherently unstable and +can result in a phenomenon known as mode collapse, +in which the model fails to sample from certain re- +gions of probability space. In addition, the generated +samples are often of low resolution. A number of alter- +native generative models [62, 63] have been developed +to address these challenges in applications to medical +imaging [64]. For example, generative latent optimiza- +tion (GLO) [62] trains deep convolutional generators +by minimizing a simple reconstruction loss, improving +on GAN training instabilities. Diffusion models [63, 65] +learn a Markov chain of diffusion steps incrementally +adding and subtracting noise from data, significantly +outperforming GANs in output image quality [66]. To +date, almost all studies of deep generative models have +focused on synthesizing images rather than object rep- +resentations. +Limitations +There are several significant challenges to +employing GANs or other types of deep generative +models to establish stochastic human models. +A +fundamental and potentially limiting issue is the fact +that a collection of objects {fs}S +s=1 is generally not +available. Medical images are degraded by the presence +of measurement noise and/or reconstruction artifacts +which are a limitation of the imaging system and +not representative of the true underlying objects. As +such, conventional GANs that are directly trained +on degraded images will not learn how to sample +from the true distribution of objects. +In essence, +there is a “chicken and egg problem” when seeking to +establish stochastic human models via deep generative +models. There are two possible ways to circumvent this +limitation. First, one can utilize high-quality medical +images as surrogates of the objects. +For example, +in certain tomographic imaging modalities and under +specific conditions, images of object properties can +be reconstructed and accurately approximate the true +object properties. +In this case, GANs are trained +in the conventional manner, with images representing +the training data. If these images are representative +of the desired subject cohort, the GAN has the +opportunity to accurately capture object variability. +Second, one can modify the GAN training process +to incorporate the image degradation process in +training. +This approach, referred to as an ambient +GAN (AmGAN) [67], utilizes a generator network +that is augmented with a measurement operator. +Objects produced by the generator are mapped to +degraded image data, which are then compared with +experimental images by the discriminator network. +This permits establishment of an implicit generative +model that describes object randomness to be learned +from indirect and noisy measurements of the objects +themselves. In a preliminary study, the AmGAN was +explored for establishing stochastic object models from +imaging measurements for use in optimizing imaging +systems [67]. +While promising, +the use of deep generative +models for in silico clinical trials is nascent and +there remain important topics for future investigation. +The objective assessment of these technologies is +largely lacking, and there is no consensus regarding +what statistical information can be reliably learned. +Additionally, current models have largely been applied +on 2D images and their extension to three-dimensions +is an ongoing topic of research. +Finally, as with +any data-driven method for establishing stochastic +human models, the presence of an imaging system null +space will fundamentally limit the ability of GANs +to describe certain components of the to-be-imaged +objects. +The extent to which the null space can be +mitigated also remains a topic of ongoing research [67]. +5.2. Knowledge-based models +Knowledge-based (also known as procedural) models +are constructed by sampling a set of φn and θn in +Eq. 3 from distributions representing the relevant +characteristics of the models. +The characteristics of +the distributions are often derived from physical or +biological measurements. +Procedural models allow +for an unlimited number of random realizations of +the object, leading to the possibility of creating large +cohorts of digital humans including the representation +of +rare +cases, +and +at +varying +spatial +resolution +which can properly account for small structures +that +might +be +relevant +for +the +specific +imaging + +CONTENTS +9 +task being studied. +However, +they are usually +computationally intensive and require a large number +of parameters to be defined and estimated based +on prior knowledge. +Their accuracy and realism +depend on the parameter combinations and they can +sometimes generate completely unrealistic outputs. +Knowledge-based, procedural models are common +in modeling breast anatomy for imaging studies. +Graff [68] proposed a detailed model that begins +with defining an outside surface using a quadratic +hemisphere +shell +with +a +skin +layer +and +nipple +area overlaid. +The shape of the shell is then +adjusted for the overall breast volume and surface +curvature. +Using a Voronoi segmentation approach, +the interior is randomly divided into regions of +fat or glandular components, with each glandular +component containing a ductal network with terminal +duct +lobular +units. +The +volume +is +then +filled +with Cooper’s ligaments, chest muscle, and blood +vessels. +For the VICTRE trial [15], +the breast +model was sampled with a 50-µm voxel size. +The +implementation is initiated with a set of random seeds +and creates random voxelized breast anatomy objects +segmented into nine different tissue types. +Several +different modeling techniques are employed including +a non-isotropic Voronoi segmentation, recursive tree +branching algorithms to generate a ductal tree and +vascular network, and Perlin-noise perturbed random +spheroids to create fat lobules. +A similar effort by Bliznakova [37] describes a 3D +breast software model for x-ray breast imaging simula- +tions based on a breast external shape, ductal lobular +system, Cooper’s ligaments and pectoralis muscle. In +this approach, a mammographic background texture +is added to the tissue regions. Blood vessels, nerves +and lymphatics were not modeled explicitly. A simi- +lar, more simplistic approach, was developed by Bakic +[69] based on two ellipsoidal regions of large scale tissue +elements: predominantly adipose tissue and predomi- +nantly fibro-glandular tissue. Internal tissue structures +within these regions are approximated by a distribution +of elements including shells, blobs, and a ductal tree. +Similar approaches have been reported for full-body +models [47]. +6. Modeling disease +Disease states can be incorporated into digital cohorts +using image-based methods or object-space models of +the condition. +The analogy between digital human +models and disease models can be established if we +consider lesions as continuous variables in space (r) +and time (t), described by a coefficient vector affecting +a set of lesion model characteristics. For simplicity, we +will consider the disease independent (of the underlying +anatomy where the disease is located) and additive. +This assumption allows us to represent the disease +cases as a sum of the stochastic human model and +the disease component, an addition that is typically +performed in the voxelized object model or directly +within the model images. We recognize this approach +is a known simplification, as disease processes often +have significant impact on underlying tissues. +Analogously to the description provided by Eq. 3, +we can generate a set of disease models {ds} defined +by: +{ds}S +s=1 = +� +n +λs +nψn(r, t), +(4) +where λs +n is a disease characteristics coefficient vec- +tor described by the function ψn over N parameters. +Characteristics that define lesions can include geomet- +ric functions (e.g., size, morphology), material proper- +ties (e.g., x-ray interaction cross-sections, elasticity) or +other relevant features (e.g., radioactivity, blood oxy- +genation levels). +Methodologies for generating and incorporating +disease into cohorts of digital stochastic models rely +on sampling λn and ψn from appropriate distributions +representing the intended population. In some cases, +disease models are specific to a given anatomical +location or physiology corresponding to a digital +human exemplar. In other cases, disease models are +independent of the digital healthy human and are +simply added or inserted multiple times into models +of healthy anatomy. In both cases, diseased subjects +are denoted by a cohort of digital stochastic humans +with added disease components: +{fs}S +s=1 = +� +n +θs +nφn(r) + +� +n +λs +nψn(r), +(5) +where {fs}S +s=1 is a cohort of diseased digital humans +(for simplicity, +and similarly as in the previous +section, +we choose to omit the time dimension). +Similarly to normal models, when ψn are unknown, +models of disease can be obtained relying on imaging. +Alternatively, when ψn are known, analytically or +numerically, the stochastic disease models are referred +to as knowledge-based (also known as procedural). +6.1. Image-based models of disease +Similar to image-based models of the human body, +image-based models of disease rely on imaging data +for extracting lesion information. Various techniques +for capturing disease characteristics, particularly for +breast +lesions, +have +recently +been +explored +[70, +71]. +Image-based neural network models for disease +modelling have also been explored. +For instance, +Kadia [72] proposed a method to generate synthetic, +infection-like patterns in the lung to create large + +CONTENTS +10 +collections of 2D and 3D training examples for deep +segmentation models. +While image-based models +contain features from actual patient data and thus +may look more realistic at first glance, they suffer +from limited resolution of the tumor model, largely +determined by the imaging acquisition characteristics +and limited number of available lesion morphologies, +shapes, and sizes. In addition, image-based methods +require an institutional review board (IRB) approval +for obtaining and utilizing the diseased case data +for research and development, which could delay or +disadvantage some analysis efforts. +6.2. Knowledge-based models of disease +Knowledge-based models of disease are constructed +by sampling a set of known (or assumed known) +ψn and λn in Eq. 4 from distributions representing +the relevant characteristics of the disease, +where +distributions +are +often +derived +from +physical +or +biological measurements. In contrast to image-based +models, knowledge-based models enable the generation +of unlimited numbers of lesion shapes with variable +resolution. +Examples of knowledge-based models +include de Sisternes [73] spiculated breast cancer mass +model and Sengupta [74] growing breast mass models. +In [74], a breast lesion growth method based on +biological and physiological phenomena accounting for +the stiffness of surrounding anatomical structures was +introduced. Breast ligaments were considered as rigid +structures with elastic moduli in the range of 8x104- +4x105 kPa, while fat (elastic modulus varying from +0.5 to 25 kPa) and glandular tissues (elastic modulus +varying from 7.5 to 66 kPa) constituting the more +elastic regions of the breast. In this approach, tumor +cells are less likely to grow through stiffer structures +and instead, preferentially proliferate through the more +elastic regions of the breast. Depending on the breast +local anatomical structures, a range of unique lesion +morphologies can be realized, allowing lesions to blend +naturally into the anatomical regions. +A common simplifying assumption is to define the +disease model independent from other human model +components. +For example, in VICTRE [15] and in +Sengupta [75], breast cancer mass lesions are added +to the normal breast models by replacing voxels in +the breast with voxels of the lesion model, without +modification to adjacent voxels. This approach, while +practical, does not account for the significant effect +of the growing tumors on its surrounding tissues, +typically visible in x-ray images as architectural +distortions suggestive of abnormalities. +To consider +these effects, Eq. 5 needs to be modified to account for +the interaction between normal and disease models. +7. Role of augmentation methods +Augmentation methods start with an already-defined +object, image or a set of defined objects, and generate +new examples based on properties of inputs, +as +well as pre-defined or data-driven transformations (in +contrast, digital human models start with only an +object description, such as that given in Eq. +1). +GAN-based models (see Section 5.1.2) are similar to +augmentation methods in that they employ complex +transformations derived with the help of training +data sets. +Augmentation methods typically employ +analytically-defined or stochastic operators that do +not require the use of neural networks, and can be +applied both in the object domain and in the acquired +image domain. Techniques in the latter group generate +examples that could be obtained through an imaging +system applied to an object with an accompanying +degradation (e.g., smoothing, noise, reconstruction +artifacts). +Geometric transformations, intensity operations, +and spatial filtering are among the most basic types +of augmentation methods. Geometric transformations +redefine the spatial relationships among voxels or +geometrical locations in an object, and include affine +(scaling, rotation, translation, reflection and shearing), +as well as non-affine transformations, such as non- +linear warping and morphing [76]. Intensity operations +modify intensity values in a grayscale image or +channel values (e.g., RGB or CMYK) in a color +image. Examples include operations such as a family +of gamma corrections, linear contrast adjustments, +and remapping voxel values using a pre-defined or +pseudo-random remapping curve [77, 78]. +Spatial +filtering (using a filter mask) is another possibility for +generating a new image or object based on an existing +one. Spatial filtering can be linear (in which case it can +be implemented by a convolution operation) or non- +linear (e.g., median filtering), and can be implemented +to smooth or sharpen to emphasize certain features. +Finally, +all three types of augmentations can be +combined +using +a +continuous +mapping +from +the +parameter space of transformations to the image or +object space [79]. +Noise injection is an image augmentation method +that enhances robustness of machine learning models +and belongs to the family of domain randomization +(DR) methods [80]. +Although noise injection after +data acquisition does not generate a new member +of a patient population, it can generate a different +representation of an object in the image domain, and +can be useful for augmenting patient cohorts obtained +with in silico modeling. Some earlier and non-medical +applications of noise injection in machine learning +sought to augment the image data sets without +regard to the physics of image acquisition [81, 82]. + +CONTENTS +11 +Other works used physics-based techniques for noise +modeling and addition, improving realism of the noise +appearance in the augmented images [83, 84]. +The +main benefit of noise injection in the image domain +for in silico trials is that it may allow for the rapid +generation of different representations of the same +object at different noise levels, leading to comparisons +that may require less computational power compared +to a full implementation of image acquisition physics +applied to a digital stochastic object model. Addition +of texture to a model in the object domain has +similarities to noise injection in the image domain +in that both techniques aim at producing noise-like +properties (e.g., using a noise power spectrum in +modeling), but are different in that addition of texture +in the object domain does not attempt to model the +noise from data acquisition [85]. +Combination of objects or images is another +popular augmentation technique. +In the object +domain, +combination +of +an +object +model +for +a +normal (non-diseased) patient with a lesion model +(as described in Section 6) can be thought of as an +example of this type of augmentation. +Generating +new members of a patient population based on an +eigenspace analysis of existing patient objects, as was +done in [52] and described in Section 5.1.1 is another +example of augmentation in the object domain. +In +the image domain, researchers investigated tools for +the extraction of image parts from one clinical image +and then their insertion into a new location on the +same or different image. Pezeshk [86] used an image +blending technique based on Poisson image editing to +insert pulmonary nodules extracted from one chest CT +exam into another. +Augmenting a training data set +for a machine learning model using this technique can +improve the model performance on independent, real +test datasets [87]. Likewise, Ghanian [88] used a similar +technique to insert microcalcification clusters extracted +from one mammogram into another mammogram, +and showed that experienced observers cannot reliably +distinguish +between +computationally +inserted +and +native clusters. Besides the ability to convince experts, +desirable properties for such combination techniques +include acceptable noise properties in the combined +image, plausible lesion-background combinations (that +might require the intervention of an operator during +the augmentation process), and a sufficient range +of +variation +in +the +combined +images +that +can +be generated, +which are often difficult to satisfy +simultaneously. +The +main +advantage +of +data +augmentation +methods is their practicality. +For example, existing +images or models both for normal and diseased +patients can be manipulated (with relative ease) with +geometric transformations leading to expanded patient +representations. +When implemented in the image +domain, augmentation methods are fast, bypassing the +stage where a model for the imaging system is applied +to the object to yield an image. However, important +shortcomings accompany these advantages. +Unless +deliberate attention is paid, augmentation methods +may yield objects or images that are biologically or +physically implausible. An extreme example may be +an intensity transformation that results in bones with +lower Hounsfield units than soft tissue. +Although +this can be avoided easily by using an intensity +transformation that is monotonically increasing, most +augmentation +methods +and +transformations +need +careful planning to avoid such inconsistencies, and it +may not be possible to avoid all inconsistencies. The +consequences of such implausible images or objects +on the results of an in silico imaging trial should be +carefully considered. In addition, many augmentation +techniques do not result in an independent, new +representation from the population, but rather in +representations that are highly dependent on the +original objects or images used as inputs to the +augmentation method. For example, lesion insertion +methods described in the previous paragraph do not +increase the number of lesions in the augmented +data set, but only the lesion-background combinations +that are generated. +Again, the consequences of this +limitation in the range of variation of generated images +should be an important consideration in an in silico +imaging trial that uses augmentation. +8. Considerations for sampling digital cohorts +In silico studies require careful study planning and +good +clinical +trial +design. +Even +if +and +when +methodologies for developing digital stochastic models +of humans for imaging studies become widely available, +generating digital cohorts needs an understanding of +the trade-offs and potential for bias associated with +selecting a specific distribution of study subjects. At +the start of the design of an in silico imaging trial +is the challenging task of scoping the population +of the digital humans to be included in the study. +For instance, a number of previous computational +studies in breast imaging using procedural models used +a uniform sampling with a desired average of 50% +adipose and 50% fibroglandular voxels [89] with an +uncompressed breast size of 14 cm. Another example +of enrollment strategy can be found in the OpenVCT +platform, where a range of size and glandularity is +specified and then uniformly randomly sampled [90]. +A more recent in silico imaging study used sampling +from a multi-class distribution identifying 4 different +breast densities resulting in the characteristics of the +intended population [15]. + +CONTENTS +12 +������������������������ = 0.79 +������������������������ = 0.89 +∆������������ = 0.10 +������������������������ = 0.84 +������������������������ = 0.90 +∆������������ = 0.06 +������������������������ = 0.77 +������������������������ = 0.88 +∆������������ = 0.11 +������������������������ = 0.99 +������������������������ = 1.00 +∆������������ = 0.01 +Figure 2. Effect of sampling strategies on performance assessment. Sampling is from a bimodal distribution of subjects (seen in +3D insert in the second panel from the left) described by 2 random parameters: (from left to right) uniform, matched, simpler, and +narrow. Only 20 samples are shown here for ease of visualization. The gray shading depicts the distribution from which samples +are taken in each of the 4 cases. AM, AT , and ∆A refer to the lesion detection average AUC for mammography, average AUC for +digital breast tomosynthesis, and the average AUC difference, respectively. . +Through in silico enrollment, +digital cohorts +{fs}S +s=1 are generated. +We denote the distribution +of the population of digital humans as fd, where d +represents the digital world, and the distribution of +subjects in the intended population as fi. +In this +context, the goal of the in silico enrollment is to +minimize the difference ∆f = |fd − fi| between the +digital (d) and physical-world intended distributions, +where +|.| +denotes +a +statistical +distance +measure. +Clinical trial enrollment programs in the physical world +require strategies to ensure a reasonable ∆f given +available sampling resources. At first approximation, +the +in +silico +enrollment +should +approximate +the +intended distribution to a greater extent than the +corresponding physical clinical trial enrollment. +Analysis of ∆f corresponding to a given in silico +enrollment strategy may be needed to understand how +the difference across study subject distributions could +affect the outcome of the trial. Here, we discuss a test +case (see Figure 2) that compares different enrollment +strategies for an in silico trial comparing digital +mammography (DM) and digital breast tomosynthesis +(DBT) derived from the VICTRE [15] project. +We +assume the populations (digital and physical) consist +of normal and diseased subjects with a prevalence +of 0.5. +These two classes of patients are therefore +sampled with equal probability. +We calculate the +difference of performance (measured using the area +under the receiver operating characteristic curve, or +AUC, in the task of differentiating between normal +and disease subjects) between mammography and +digital breast tomosynthesis. We consider the following +four sampling approaches. +In the first approach +(uniform), fi is unknown and subjects are sampled +uniformly within a range of interest, from all possible +combinations of the input parameters that define +f. +In the second approach (matched), fi is known +and subjects are sampled from the true underlying +distribution. +In the third approach (simpler), fi +is unknown, but can be approximated by another, +simpler distribution from which samples are obtained. +Finally, in the fourth approach (narrow), fi is known +to be a narrow, well-defined subset of the general +population of subjects of particular interest (e.g., rare +diseases or very obese subjects). +For this simplified example, let fi be a bimodal +distribution defined by two parameters (e.g., breast +size and glandularity). +Using Eq. 3, we can express +the model through two expansion functions φ1,2, each +associated with one of the two random variables +affected by a random parameter set given by θ1,2. As +seen in Figure 2, one of the modes of the distribution +has twice the amplitude and half the variance of the +other. +The four density plots illustrate a top view +of the distribution contour plot with the individual +samples drawn using the four different sampling +strategies. The results demonstrate that the choice of +sampling strategy can have a significant effect on the +difference in AUC, which for this example case, ranges +from a difference of 0.01 (almost zero) to 0.11 in terms +of device performance. +9. Summary and conclusions +In silico trials are an emerging area of regulatory +research that offer the ability to capture highly +diverse patient distributions at a significant time and +cost savings, compared to traditional physical clinical +trials. +To conduct in silico trials, realistic digital +representations of humans are needed. In this paper, +we reviewed and discussed existing techniques for +generating digital humans, including disease models, +for in silico imaging trials. +Digital humans can +be created using image-based or knowledge-based +techniques. +In summary, we favor techniques with +object-based representations (rather than images of + +80 +60■CONTENTS +13 +objects) in order to decouple the characteristics of the +image acquisition system from the characteristics of +the object (true representation of the physical-world +human). +In generating digital humans for in silico +trials, one should consider the quality and quantity of +the source data or knowledge used, and whether the +models represent a single patient, a small cohort, or a +sizable population with realistic patient variability. +It remains a crucial next step to evaluate the +quality of the digital human models and the images +that can be generated with them. In particular, it is +essential to carefully identify the patient distribution +that the particular digital human model can and +cannot capture, in order to prevent misuse and ensure +patient safety. +We need to study to what extent +model-derived data contributes to our understanding of +performance levels for populations with rare diseases or +for populations underrepresented in traditional clinical +trials. +Future work should examine the ethical and +safety considerations of relying on digital humans for +clinical trials. Overall, the use of in silico imaging trials +and in silico trials in medicine is a rapidly developing +field and has the potential to address many of the +emerging challenges in the regulatory evaluation of +medical devices. +References +[1] H. H. Barrett and K. J. Myers, Foundations of image +science. John Wiley & Sons, 2013. +[2] N. Magnenat-Thalmann and D. Thalmann, Handbook of +virtual humans. John Wiley & Sons, 2005. +[3] A. Dosovitskiy, G. Ros, F. Codevilla, A. Lopez, and +V. Koltun, “Carla: An open urban driving simulator,” +in Conference on robot learning, pp. 1–16, PMLR, 2017. +[4] C. Cimino, +E. Negri, +and L. Fumagalli, +“Review of +digital twin applications in manufacturing,” Computers +in Industry, vol. 113, p. 103130, 2019. +[5] F. Tao, H. Zhang, A. Liu, and A. Y. Nee, “Digital twin +in industry: +State-of-the-art,” IEEE Transactions on +industrial informatics, vol. 15, no. 4, pp. 2405–2415, +2018. +[6] A. Thelen, X. Zhang, O. Fink, and et al., “A comprehensive +review of digital twin - part 1: modeling and twinning +enabling technologies,” Struct Multidisc Optim, 2022. +[7] J. Fan, X. Liu, Y. Li, H. Xia, R. Yang, J. Li, and Y. Zhang, +“Quality problems of clinical trials in china: evidence +from quality related studies,” Trials, vol. 23, no. 1, pp. 1– +11, 2022. +[8] U. Food, D. Administration, et al., “Diversity plans to im- +prove enrollment of participants from underrepresented +racial and ethnic populations in clinical trials; draft guid- +ance for industry; availability,” 2022. +[9] J.-F. Rajotte, R. Bergen, D. L. Buckeridge, K. El Emam, +R. Ng, and E. Strome, “Synthetic data as an enabler +for machine learning applications in medicine,” Iscience, +vol. 25, no. 11, 2022. +[10] E. Abadi, W. P. Segars, B. M. Tsui, P. E. Kinahan, +N. Bottenus, A. F. Frangi, A. Maidment, J. Lo, and +E. Samei, “Virtual clinical trials in medical imaging: +a review,” Journal of Medical Imaging, vol. 7, no. 4, +p. 042805, 2020. +[11] A. Badano, “In silico imaging clinical trials: cheaper, faster, +better, safer, and more scalable,” Trials, vol. 22, no. 1, +pp. 1–7, 2021. +[12] E. Abadi, W. P. Segars, G. M. Sturgeon, J. E. Roos, C. E. +Ravin, and E. Samei, “Modeling lung architecture in the +XCAT series of phantoms: Physiologically based airways, +arteries and veins,” IEEE Trans. Med. Imaging, vol. 37, +pp. 693–702, Mar. 2018. +[13] L. Wedlund and J. Kvedar, “Simulated trials: +in silico +approach adds depth and nuance to the rct gold- +standard,” NPJ digital medicine, vol. 4, no. 1, 2021. +[14] W. P. Segars and B. M. Tsui, “Mcat to xcat: The evolution +of 4-d computerized phantoms for imaging research,” +Proceedings of the IEEE, vol. 97, no. 12, pp. 1954–1968, +2009. +[15] A. +Badano, +C. +G. +Graff, +A. +Badal, +D. +Sharma, +R. Zeng, F. W. Samuelson, S. J. Glick, and K. J. +Myers, “Evaluation of digital breast tomosynthesis as +replacement of full-field digital mammography using an +in silico imaging trial,” JAMA network open, vol. 1, +pp. e185474–e185474, 11 2018. +[16] M. Pepe, “Evaluating technologies for classification and +prediction in medicine,” Statistics in medicine, vol. 24, +no. 24, pp. 3687–3696, 2005. +[17] W. N. Arifin and U. K. Yusof, “Correcting for partial +verification bias in diagnostic accuracy studies: +A +tutorial using r,” Statistics in Medicine, vol. 41, no. 9, +pp. 1709–1727, 2022. +[18] F. Berti, L. Antonini, G. Poletti, C. Fiuza, T. J. Vaughan, +F. Migliavacca, +L. Petrini, +and G. Pennati, +“How +to validate in silico deployment of coronary stents: +strategies and limitations in the choice of comparator,” +Frontiers in Medical Technology, p. 37, 2021. +[19] H. H. Barrett and L. Caucci, +“Stochastic models for +objects and images in oncology and virology: application +to PI3K-Akt-mTOR signaling and COVID-19 disease,” +Journal of Medical Imaging, vol. 8, no. S1, p. S16001, +2020. +[20] L. A. Shepp and B. F. Logan, “The fourier reconstruction of +a head section,” IEEE Transactions on Nuclear Science, +vol. 21, no. 3, pp. 21–43, 1974. +[21] J. Martin, M. Ruthven, R. Boubertakh, and M. E. Miquel, +“Realistic dynamic numerical phantom for mri of the +upper vocal tract,” Journal of Imaging, vol. 6, no. 9, +2020. +[22] W. S. Snyder, M. R. Ford, G. G. Warner, and H. Fisher Jr, +“Estimates of absorbed fractions for monoenergetic +photon sources uniformly distributed in various organs +of a heterogeneous phantom.,” tech. rep., Oak Ridge +National Lab., Tenn., 1969. +[23] M. Caon, “Voxel-based computational models of real human +anatomy: +a review,” +Radiation and environmental +biophysics, vol. 42, no. 4, pp. 229–235, 2004. +[24] X. G. Zu, “The vip-man model-a digital human testbed for +radiation siimulations,” SAE transactions, pp. 779–787, +2005. +[25] X. George Xu, “Computational phantoms for organ dose +calculations in radiation protection and imaging,” The +Phantoms of Medical and Health Physics, pp. 225–262, +2014. +[26] W. Fu, S. Sharma, E. Abadi, A.-S. Iliopoulos, Q. Wang, +J. Y. Lo, X. Sun, W. P. Segars, and E. Samei, “iphantom: +a framework for automated creation of individualized +computational phantoms and its application to ct organ +dosimetry,” IEEE Journal of Biomedical and Health +Informatics, vol. 25, no. 8, pp. 3061–3072, 2021. +[27] M. +N. +Kamel +Boulos +and +P. +Zhang, +“Digital +twins: +from personalised medicine to precision public health,” +Journal of Personalized Medicine, vol. 11, no. 8, p. 745, +2021. +[28] M. +Kamel +Boulos +and +Z. +P., +“Digital +twins: +From + +CONTENTS +14 +personalised medicine to precision public health,” J Pers +Med, vol. 11, no. 8, p. 745, 2021. +[29] F. +Pesapane, +A. +Rotili, +S. +Penco, +L. +Nicosia, +and +E. Cassano, “Digital twins in radiology,” Journal of +Clinical Medicine, vol. 11, no. 21, p. 6553, 2022. +[30] T. Erol, A. F. Mendi, and D. Do˘gan, “The digital twin +revolution in healthcare,” in 2020 4th International +Symposium on Multidisciplinary Studies and Innovative +Technologies (ISMSIT), pp. 1–7, IEEE, 2020. +[31] G. Wang, A. Badal, X. Jia, J. S. Maltz, J. Mueller, K. J. +Myers, C. Niu, M. Vannier, P. Yan, Z. Yu, and R. Zeng, +“Development of metaverse for intelligent healthcare,” +Nature Machine Intelligence, vol. 4, p. 9220929, 2022. +[32] V. Spitzer, +M. J. Ackerman, +A. L. Scherzinger, +and +D. Whitlock, “The visible human male: +a technical +report,” Journal of the American Medical Informatics +Association, vol. 3, no. 2, pp. 118–130, 1996. +[33] A. Christ, W. Kainz, E. G. Hahn, K. Honegger, M. Zefferer, +E. Neufeld, W. Rascher, R. Janka, W. Bautz, J. Chen, +et al., “The virtual family, development of surface- +based anatomical models of two adults and two children +for dosimetric simulations,” Physics in Medicine and +Biology, vol. 55, no. 2, p. N23, 2009. +[34] K. Fujimoto, T. A. Zaidi, D. Lampman, J. W. Guag, +S. Etheridge, H. Habara, and S. S. Rajan, “Comparison +of sar distribution of hip and knee implantable devices +in 1.5t conventional cylindrical-bore and 1.2t open-bore +vertical mri systems,” Magnetic Resonance in Medicine, +vol. 87, no. 3, pp. 1515–1528, 2022. +[35] A. +Duetschler, +G. +Bauman, +O. +Bieri, +P. +C. +Cattin, +S. Ehrbar, G. Engin-Deniz, A. Giger, M. Josipovic, +C. Jud, M. Krieger, et al., “Synthetic 4dct (mri) lung +phantom generation for 4d radiotherapy and image +guidance investigations,” Medical physics, vol. 49, no. 5, +pp. 2890–2903, 2022. +[36] N. Kiarashi, A. Nolte, G. M. Sturgeon, W. P. Segars, +S. V. Ghate, L. W. Nolte, E. Samei, and J. Y. Lo, +“Development and application of a suite of 4-d virtual +breast phantoms for optimization and evaluation of +breast imaging systems,” IEEE Trans Med Imaging, +vol. 33, no. 7, pp. 1401–9, 2014. +[37] K. Bliznakova, Z. Bliznakov, V. Bravou, Z. Kolitsi, and +N. Pallikarakis, “A three-dimensional breast software +phantom for mammography simulation,” +Physics in +Medicine & Biology, vol. 48, no. 22, p. 3699, 2003. +[38] C. M. Li, W. P. Segars, G. D. Tourassi, J. M. Boone, +and J. T. Dobbins III, “Methodology for generating a +3d computerized breast phantom from empirical data,” +Medical physics, vol. 36, no. 7, pp. 3122–3131, 2009. +[39] D. W. Erickson, J. R. Wells, G. M. Sturgeon, E. Samei, +J. T. Dobbins, W. P. Segars, and J. Y. Lo, “Population of +224 realistic human subject-based computational breast +phantoms,” Medical physics, vol. 43, no. 1, pp. 23–32, +2016. +[40] C. M. Hsu, M. L. Palmeri, W. P. Segars, A. I. Veress, and +J. T. Dobbins III, “Generation of a suite of 3d computer- +generated breast phantoms from a limited set of human +subject data,” Medical physics, vol. 40, no. 4, p. 043703, +2013. +[41] P. Elangovan, A. Mackenzie, D. R. Dance, K. C. Young, +V. Cooke, L. Wilkinson, R. M. Given-Wilson, M. G. +Wallis, and K. Wells, “Design and validation of realistic +breast models for use in multiple alternative forced choice +virtual clinical trials,” Physics in Medicine & Biology, +vol. 62, no. 7, p. 2778, 2017. +[42] E. Garc´ıa, C. Fedon, M. Caballo, R. Mart´ı, I. Sechopoulos, +and O. Diaz, “Realistic compressed breast phantoms +for medical physics applications,” in 15th International +Workshop on Breast Imaging (IWBI2020), vol. 11513, +pp. 30–37, SPIE, 2020. +[43] A. +Sarno, +G. +Mettivier, +F. +di +Franco, +A. +Varallo, +K. Bliznakova, A. M. Hernandez, J. M. Boone, and +P. Russo, “Dataset of patient-derived digital breast +phantoms +for in silico studies in +breast computed +tomography, digital breast tomosynthesis, and digital +mammography,” +Medical +Physics, +vol. +48, +no. +5, +pp. 2682–2693, 2021. +[44] M. Caballo, +C. Rabin, +C. Fedon, +A. Rodr´ıguez-Ruiz, +O. Diaz, J. M. Boone, D. R. Dance, and I. Sechopou- +los, “Patient-derived heterogeneous breast phantoms for +advanced dosimetry in mammography and tomosynthe- +sis,” Medical physics, vol. 49, no. 8, pp. 5423–5438, 2022. +[45] T. J. Sauer, E. Abadi, P. Segars, and E. Samei, “Anatomi- +cally and physiologically informed computational model +of hepatic contrast perfusion for virtual imaging trials,” +Medical Physics, vol. 49, no. 5, pp. 2938–2951, 2022. +[46] L. K. Tam, J. P. Stockmann, G. Galiana, and R. T. +Constable, “Null space imaging: +nonlinear magnetic +encoding +fields +designed +complementary +to +receiver +coil sensitivities for improved acceleration in parallel +imaging,” Magnetic resonance in medicine, vol. 68, no. 4, +pp. 1166–1175, 2012. +[47] C. Lee, D. Lodwick, D. Hasenauer, J. L. Williams, C. Lee, +and W. E. Bolch, “Hybrid computational phantoms of +the male and female newborn patient: +Nurbs-based +whole-body models,” Physics in Medicine & Biology, +vol. 52, no. 12, p. 3309, 2007. +[48] W. P. Segars, G. Sturgeon, S. Mendonca, J. Grimes, and +B. M. Tsui, “4d xcat phantom for multimodality imaging +research,” Medical physics, vol. 37, no. 9, pp. 4902–4915, +2010. +[49] W. Kainz, E. Neufeld, W. E. Bolch, C. G. Graff, C. H. Kim, +N. Kuster, B. Lloyd, T. Morrison, P. Segars, Y. S. Yeom, +et al., “Advances in computational human phantoms and +their applications in biomedical engineering—a topical +review,” IEEE transactions on radiation and plasma +medical sciences, vol. 3, no. 1, pp. 1–23, 2018. +[50] A. E. Johnson, T. J. Pollard, S. J. Berkowitz, N. R. +Greenbaum, M. P. Lungren, C.-y. Deng, R. G. Mark, and +S. Horng, “Mimic-cxr, a de-identified publicly available +database of chest radiographs with free-text reports,” +Scientific data, vol. 6, no. 1, pp. 1–8, 2019. +[51] “Medical +imaging +and +data +resource +center +(midrc).” +https://www.midrc.org/. Accessed: 2023-01-10. +[52] G. M. Sturgeon, +S. Park, +W. P. Segars, +and J. Y. +Lo, “Synthetic breast phantoms from patient based +eigenbreasts,” Med Phys, vol. 44, no. 12, p. 6270–6279, +2017. +[53] J. P. Lewis, +M. Cordner, +and N. Fong, +“Pose space +deformation: a unified approach to shape interpolation +and skeleton-driven deformation,” +in Proceedings of +the 27th annual conference on Computer graphics and +interactive techniques, pp. 165–172, 2000. +[54] J. Chen, Y. Li, Y. Du, and E. C. Frey, “Generating +anthropomorphic phantoms using fully unsupervised +deformable image registration with convolutional neural +networks,” Medical physics, vol. 47, no. 12, pp. 6366– +6380, 2020. +[55] F. Galbusera, +F. Niemeyer, +M. Seyfried, +T. Bassani, +G. Casaroli, A. Kienle, and H.-J. Wilke, “Exploring +the potential of generative adversarial networks for +synthesizing radiological images of the spine to be used +in in silico trials,” +Frontiers in bioengineering and +biotechnology, vol. 6, p. 53, 2018. +[56] A. Brock, J. Donahue, and K. Simonyan, “Large scale gan +training for high fidelity natural image synthesis,” arXiv +preprint arXiv:1809.11096, 2018. +[57] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, “Unpaired +image-to-image translation using cycle-consistent adver- +sarial networks,” in Proceedings of the IEEE interna- + +CONTENTS +15 +tional conference on computer vision, pp. 2223–2232, +2017. +[58] P. +Isola, +J.-Y. +Zhu, +T. +Zhou, +and +A. +A. +Efros, +“Image-to-image translation with conditional adversarial +networks,” in Proceedings of the IEEE conference on +computer vision and pattern recognition, pp. 1125–1134, +2017. +[59] Z. Wang, Q. She, and T. E. Ward, “Generative adversarial +networks in computer vision: A survey and taxonomy,” +ACM Computing Surveys (CSUR), vol. 54, no. 2, pp. 1– +38, 2021. +[60] I. +Goodfellow, +J. +Pouget-Abadie, +M. +Mirza, +B. +Xu, +D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, +“Generative adversarial networks,” Communications of +the ACM, vol. 63, no. 11, pp. 139–144, 2020. +[61] N. K. Singh and K. Raza, “Medical image generation +using generative adversarial networks: A review,” Health +informatics: A computational perspective in healthcare, +pp. 77–96, 2021. +[62] P. Bojanowski, A. Joulin, D. Lopez-Paz, and A. Szlam, +“Optimizing the latent space of generative networks,” +arXiv preprint arXiv:1707.05776, 2017. +[63] J. Ho, +A. Jain, +and P. Abbeel, +“Denoising diffusion +probabilistic models,” Advances in Neural Information +Processing Systems, vol. 33, pp. 6840–6851, 2020. +[64] D. +Li, +A. +Kar, +N. +Ravikumar, +A. +F. +Frangi, +and +S. Fidler, “Federated simulation for medical imaging,” in +International Conference on Medical Image Computing +and +Computer-Assisted +Intervention, +pp. +159–168, +Springer, 2020. +[65] F.-A. Croitoru, V. Hondru, R. T. Ionescu, and M. Shah, +“Diffusion models in vision: A survey,” arXiv preprint +arXiv:2209.04747, 2022. +[66] P. Dhariwal and A. Nichol, “Diffusion models beat gans +on image synthesis,” Advances in Neural Information +Processing Systems, vol. 34, pp. 8780–8794, 2021. +[67] W. Zhou, S. Bhadra, F. J. Brooks, H. Li, and M. A. +Anastasio, +“Learning stochastic object models from +medical imaging measurements by use of advanced +ambient generative adversarial networks,” Journal of +Medical Imaging, vol. 9, no. 1, p. 015503, 2022. +[68] C. G. Graff, “A new, open-source, multi-modality digital +breast phantom,” in Medical Imaging 2016: Physics of +Medical Imaging, vol. 9783, pp. 72–81, SPIE, 2016. +[69] P. R. Bakic, K. J. Myers, S. J. Glick, and A. D. Maidment, +“Virtual tools for the evaluation of breast imaging: +state-of-the science and future directions,” pp. 518–524, +Springer, Cham, 6 2016. +[70] N. +Dukov, +K. +Bliznakova, +F. +Feradov, +I. +Buliev, +H. Bosmans, G. Mettivier, P. Russo, L. Cockmartin, +and Z. Bliznakov, “Models of breast lesions based on +three-dimensional x-ray breast images,” Physica Medica, +vol. 57, pp. 80–87, 2019. +[71] K. Bliznakova, N. Dukov, F. Feradov, G. Gospodinova, +Z. Bliznakov, P. Russo, G. Mettivier, H. Bosmans, +L. Cockmartin, +A. Sarno, +et al., +“Development of +breast lesions models database,” Physica Medica, vol. 64, +pp. 293–303, 2019. +[72] D. Kadia, T. V. Nguyen, and V. Asari, “Synthesis for +robust segmentation of infected lung region on small- +scale data,” SSRM, 2022. +[73] L. de Sisternes, J. G. Brankov, A. M. Zysk, R. A. Schmidt, +R. M. Nishikawa, and M. N. Wernick, “A computational +model to generate simulated three-dimensional breast +masses,” Medical physics, vol. 42, no. 2, pp. 1098–1118, +2015. +[74] A. Sengupta, D. Sharma, and A. Badano, “Computational +model of tumor growth for in silico trials,” in Medical +Imaging 2021: Physics of Medical Imaging, vol. 11595, +pp. 1262–1270, SPIE, 2021. +[75] A. Sengupta, D. Sharma, and A. Badano, “Computational +model of tumor growth for in silico trials,” 2021. +[76] G. Wolberg, +“Geometric transformation techniques for +digital images: A survey,” 1988. +[77] P. Chlap, H. Min, N. Vandenberg, J. Dowling, L. Holloway, +and A. Haworth, “A review of medical image data +augmentation techniques for deep learning applications,” +J. Med. Imaging Radiat. Oncol., vol. 65, pp. 545–563, +Aug. 2021. +[78] L. S. Hesse, G. Kuling, M. Veta, and A. L. Martel, +“Intensity +augmentation +to +improve +generalizability +of +breast +segmentation +across +different +MRI +scan +protocols,” IEEE Trans. Biomed. Eng., vol. 68, pp. 759– +770, Mar. 2021. +[79] K. Tian, C. Lin, S. N. Lim, W. Ouyang, P. Dokania, +and P. Torr, “A continuous mapping for augmentation +design,” in Advances in Neural Information Processing +Systems (M. Ranzato, A. Beygelzimer, Y. Dauphin, +P. Liang, and J. W. Vaughan, eds.), vol. 34, pp. 13732– +13743, Curran Associates, Inc., 2021. +[80] H. Noh, T. You, J. Mun, and B. Han, “Regularizing +deep neural networks by noise: Its interpretation and +optimization,” ArXiv, vol. abs/1710.05179, 2017. +[81] F. J. Moreno-Barea, F. Strazzera, J. M. Jerez, D. Urda, +and L. Franco, “Forward noise adjustment scheme for +data augmentation,” in 2018 IEEE Symposium Series on +Computational Intelligence (SSCI), pp. 728–734, 2018. +[82] H.-J. Bae, C.-W. Kim, N. Kim, B. Park, N. Kim, J. B. +Seo, and S. M. Lee, “A perlin noise-based augmentation +strategy for deep learning with small data samples of +HRCT images,” Sci. Rep., vol. 8, p. 17687, Dec. 2018. +[83] A. O. Omigbodun, F. Noo, M. McNitt-Gray, W. Hsu, +and S. S. Hsieh, “The effects of physics-based data +augmentation on the generalizability of deep neural +networks: +Demonstration +on +nodule +false-positive +reduction,” Med Phys, vol. 46, pp. 4563–4574, Oct 2019. +[84] Z. Fabian, +R. Heckel, +and M. Soltanolkotabi, +“Data +augmentation for deep learning based accelerated mri +reconstruction with limited data,” in Proceedings of +the 38th International Conference on Machine Learning +(M. Meila and T. Zhang, eds.), vol. 139 of Proceedings +of Machine Learning Research, pp. 3057–3067, PMLR, +18–24 Jul 2021. +[85] E. Abadi, W. P. Segars, G. M. Sturgeon, B. Harrawood, +A. Kapadia, and E. Samei, “Modeling “textured” bones +in virtual human phantoms,” IEEE Transactions on +Radiation and Plasma Medical Sciences, vol. 3, no. 1, +pp. 47–53, 2019. +[86] A. Pezeshk, B. Sahiner, R. Zeng, A. Wunderlich, W. Chen, +and +N. +Petrick, +“Seamless +insertion +of +pulmonary +nodules in chest ct images,” IEEE Transactions on +Biomedical Engineering, vol. 62, no. 12, pp. 2812–2827, +2015. +[87] A. Pezeshk, N. Petrick, W. Chen, and B. Sahiner, “Seamless +lesion insertion for data augmentation in cad training,” +IEEE Transactions on Medical Imaging, vol. 36, no. 4, +pp. 1005–1015, 2017. +[88] Z. Ghanian, A. Pezeshk, N. Petrick, and B. Sahiner, +“Computational insertion of microcalcification clusters +on mammograms: +reader differentiation from native +clusters +and +computer-aided +detection +comparison,” +Journal of Medical Imaging, vol. 5, p. 1, nov 2018. +[89] X. +Gong, +S. +J. +Glick, +B. +Liu, +A. +A. +Vedula, +and +S. Thacker, “A computer simulation study comparing +lesion detection accuracy with digital mammography, +breast tomosynthesis, and cone-beam ct breast imaging,” +Medical physics, vol. 33, no. 4, pp. 1041–1052, 2006. +[90] B. +Barufaldi, +D. +Higginbotham, +P. +R. +Bakic, +and +A. D. A. Maidment, “OpenVCT: a GPU-accelerated +virtual clinical trial pipeline for mammography and + +CONTENTS +16 +digital breast tomosynthesis,” in Medical Imaging 2018: +Physics of Medical Imaging (J. Y. Lo, T. G. Schmidt, and +G.-H. Chen, eds.), vol. 10573, p. 1057358, International +Society for Optics and Photonics, SPIE, 2018. + diff --git a/YdFAT4oBgHgl3EQf3R5S/content/tmp_files/load_file.txt b/YdFAT4oBgHgl3EQf3R5S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b391c551184b5f28566e6eb2d285e2369cd3be9 --- /dev/null +++ b/YdFAT4oBgHgl3EQf3R5S/content/tmp_files/load_file.txt @@ -0,0 +1,1531 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf,len=1530 +page_content='The stochastic digital human is now enrolling for in silico imaging trials – Methods and tools for generating digital cohorts A Badano1, M Lago1, E Sizikova1, JG Delfino1, S Guan1 and MA Anastasio2 and B Sahiner1 1Division of Imaging, Diagnostics, and Software Reliability, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Food and Drug Administration, Silver Spring, MD 20993 2Department of Bioengineering, The Grainger College of Engineering, University of Illinois, Urbana, IL 61801 E-mail: aldo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='badano@fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='hhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='gov 23 January 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Randomized clinical trials, while often viewed as the highest evidentiary bar by which to judge the quality of a medical intervention, are far from perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In silico imaging trials are computational studies that seek to ascertain the performance of a medical device by collecting this information entirely via computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The benefits of in silico trials for evaluating new technology include significant resource and time savings, minimization of subject risk, the ability to study devices that are not achievable in the physical world, allow for the rapid and effective investigation of new technologies and ensure representation from all relevant subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' To conduct in silico trials, digital representations of humans are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We review the latest developments in methods and tools for obtaining digital humans for in silico imaging studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' First, we introduce terminology and a classification of digital human models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Second, we survey available methodologies for generating digital humans with healthy and diseased status, and examine briefly the role of augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Finally, we discuss the trade-offs of four approaches for sampling digital cohorts and the associated potential for study bias with selecting specific patient distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Social media blur (100-w): From digital twins to other digital humans for in silico trials: we review methods and tools for obtaining stochastic humans for digital cohorts [LINK] Submitted to: PRGB arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='08719v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='AI] 20 Jan 2023 CONTENTS 2 Contents 1 Introduction 3 2 Terminology 4 3 Representations 4 4 Individual models 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1 Personalized models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8 6 Modeling disease 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1 Image-based models of disease .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 10 7 Role of augmentation methods 10 8 Considerations for sampling digital cohorts 11 9 Summary and conclusions 12 CONTENTS 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Introduction Two decades ago, in the epilogue of their seminal textbook on image science [1], Barrett and Myers pointed out that in the future, sport games might be played with simulated athletes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The advancement of computer graphics and simulation technologies sparked the notion that perhaps the excitement of a real-life sports event could be conducted in the simulation space with digital models of athletes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Since then, continuous advances in computer processing power and modeling techniques have taken place, driven primarily by entertainment applications [2] and quickly becoming a significant component of research and development (R&D) efforts in a variety of industries‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Industries that have widely adopted computational modeling and in silico methods throughout the product life-cycle include automotive [3] and manufacturing [4] among others [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Medicine lags considerably behind [6] due, in part, to model complexity, challenging validation, associated potential risks for new devices and drugs, and lack of consensus and regulatory standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Randomized clinical trials, while often viewed as the highest evidentiary bar by which to judge the quality of a medical intervention, are far from perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Common causes of failure include safety issues, difficulties with patient recruitment, enrollment, and retention [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In addition, clinical trials can suffer from under-representation of rare subpopulations [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These limitations represent a unique opportunity to develop in silico trials that are completed as planned, safely, and that include digital cohorts with a representative distribution of subject characteristics and numbers large enough for appropriate statistical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' As pointed out in [9], in silico data has the potential to address lack of data availability, sharing mechanisms and privacy challenges associated with the use of medical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In silico imaging trials are computational studies that seek to ascertain the performance of a medical device for the intended population, collecting this information entirely in the digital world via computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The benefits of in silico imaging trials for evaluating new technology include significant resource and time savings, minimization of subject risk, and ethical considerations [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Moreover, in silico trials can be used to study devices that do not yet exist or are not practically attainable in the (limited) physical world, allow for the rapid and effective investigation of new technologies [11, 12, 13], and facilitate representation from all relevant subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Each one of these benefits is an ‡ To date, Superbowl games are played with physical-world athletes, in part due to the difficulty of conveying real-life personal struggle, an essential component of the entertainment context for sport players and teams (see, for instance, here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' essential consideration within the context of the regulatory evaluation of medical technology [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The realization that computational models of humans would take center stage in medical imaging system assessment is not new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Full optimization of imaging systems for specific medical tasks requires objects (physical or digital) that represent the variability seen in patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For many decades, scientists have relied on practical and simpler versions of patients [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, recent advances in computer processing power and simulation methods are now facilitating the development of more detailed and realistic patient models that are based on digital stochastic descriptions of the model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, a recent report demonstrated the feasibility of an in silico trial, the Virtual Imaging Clinical Trial for Regulatory Evaluation (VICTRE), as an alternative approach to establish regulatory evidence in support of medical imaging products [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' There are numerous parallels between digital- and physical-world trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fundamentally, in silico trials must include the same essential elements of well-designed physical-world clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Firstly, the population of subjects for whom the new device or technology is intended must be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The study design must contain clear rules for selection and rejection of subjects from a distribution of healthy and diseased subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, in silico trials are not subject to effects from covariates in patient selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, a common problem in evaluating screening tests meant for asymptomatic subjects is that a portion of the enrolled population might be symptomatic [16] with the potential for verification bias [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Secondly, when there are two technologies that are being compared, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', a new, yet unproven technology and a comparator technology currently in clinical use, both must be unambiguously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A good choice for comparator technology should be associated with accurate representations of the device characteristics as supported by validation studies [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Thirdly, the study requires a definition of the users of the device’s outcome (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', images in the case of an imaging device trial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These first three components reflect the physical intended use of the device under investigation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', the intended populations of subjects, the intended device comparison, and the intended image interpreters that will be using the device in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Finally, whether physical or digital, the trial design must provide a definition of the primary outcome to be evaluated, a protocol and statistical analysis associated with the trial, and an analysis of the risk and benefits introduced by the device under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Both physical and in silico studies require enrollment of representative subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this CONTENTS 4 review, we survey the latest developments in methods and tools for generating the cohorts of digital humans for imaging studies that represent the variability of physical-world subject populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We refer to the digital cohorts consisting of digital humans (realizations of the digital human models) as “stochastic humans”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Assessment of new technology and the regulatory evaluation of that technology requires establishing performance levels for intended populations and, therefore, necessitates computational models that allow sampling of the parameter space defining the subject population in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We propose to name these models digital humans as opposed to digital replicas or twins to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The review is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' First, we introduce terminology and representation models regarding the different types of digital humans described throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Second, we survey available methodologies for generating digital humans with healthy status and for generating diseased cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Then, we briefly discuss the role of augmentation methods and conclude with an analysis of sampling techniques that may be used to generate the digital cohorts for evaluating the performance of imaging devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Terminology A variety of terminologies are being used or proposed for describing digital representations of humans in medicine and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the literature, some of these are often used without the benefit of a clear definition and, in some instances, wrongly interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We propose to use the term stochastic digital human to denote digital representations of humans (or human body parts) generated from multiple random outputs by sampling known distributions for the model characteristics matching the variability observed in human populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In contrast, non-stochastic representations are deterministic digital versions of a single physical exemplar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', a model of a human body at a given time) or a group (or family) of physical exemplars which are differentiated by varying physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Contrary to other terms and concepts currently being discussed including digital families, avatars, chimeras, and digital twins, the concept of a stochastic digital human represents an approach for in silico trials and regulatory evaluation that estimates the performance of an imaging device for a population of subjects rather than for an individual patient, thus incorporating the variability observed in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We propose to classify all digital humans as either individual or population models (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Individual models are necessarily image-based while population models can be derived either from images or from knowledge of the fundamental characteristics that define the relevant features of a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Note that we will use the term digital human to refer to the models even if the represented object is a part of the body or the whole body of a subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Representations Physical objects (including humans) can be repre- sented using continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We consider the models of humans as continuous in space (r) and time (t) and described by a coefficient vector affecting a set of model characteristics: fm(r, t) ≈ N � n=1 θnφn(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' (1) Here, N is the dimension of the approximate finite- dimensional representation of the object, and the subscript m indicates the modeling approximation to differentiate from the actual object f(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The collection of expansion functions {φn(r, t)}N n=1 is employed to form fm(r, t), and θn denotes the n-th component of the N-dimensional expansion coefficient vector θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The quantity fm(r, t) constitutes a discrete representation of a digital human that can be readily displayed on a computer or digitally processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For the case where the expansion functions are defined as in- dicator functions that describe non-overlapping space- time voxels, θ can sometimes be interpreted as a digital image whose components θn represent the integrated value of the object over the support of the voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' More generally, a digital human model can be es- tablished by integrating the continuous representation fm(r, t) over a collection of N voxels as fn = � vn fm(r, t) d3r dt, n = 1, · · · , N, (2) where vn denotes the support of the n-th spatial- temporal voxel and fn denotes the n-th component of a N-dimensional vector f that represents the digital human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' As discussed below, the choice of the expansion functions and associated expansion coefficients can be specified in different ways, with the general goal of making fm(r, t) an accurate approximation of f(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The expansion functions can depict geometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', size, morphology), material properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', x-ray interaction cross-sections, elasticity) or other relevant features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', radioactivity, blood oxygenation levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For simplicity, we will consider that the stochastic human does not vary with time and proceed only with CONTENTS 5 the spatial dimension r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, the concepts that follow can readily be generalized to model time-varying descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [19] In practice, the coefficient vector θ can be modeled as a random vector and the expansion functions {φn(r)}N n=1 as random processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Methodologies for generating large cohorts of digital stochastic models of humans for in silico imaging trials, including models for organs and tissues with appropriate variability, can rely on either sampling θ, φn or both from appropriate distributions representing the intended population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We can denote the cohort of digital stochastic humans as follows, {fs}S s=1 = � n θs nφn(r), (3) where s denotes a particular state or random realization of a digital human in a cohort of size S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' When φn are known, analytically or numerically, the stochastic models are referred to as procedural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this case, the modeler is left with choosing the coefficient vector defining the object (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In cases for which the defining characteristics are unknown, θn and φn can be estimated from imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the following sections, we review available methods and tools for generating digital human models and digital cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We present a classification of available approaches in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Individual models Individual models attempt to create a digital replica of a specific physical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Individual models can be categorized as personalized and family models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These models are not stochastic since they are meant to represent individual subjects with as much detail and accuracy as achievable from the image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this respect, the representation introduced in Section 3 applies only with S = 1 resulting in a single coefficient vector (θn) defining the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The digital representation in these cases is typically a multidimensional voxelized array that can be segmented into structures such as tissues and organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Early attempts relied on geometrical volumes represented by analytical expressions altered to generate a wide variety of sizes and shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In other words, φn are described by quadrics and θn represent properties of the volumes defined by the surfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', x-ray attenuation and scattering properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These computational models have proved useful in areas of quality control of imaging systems [20, 21] and in radiation dosimetry [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Even with more sophisticated geometrical structures [23, 24, 25] and more spatial detail, these approaches lack the ability to accurately represent the statistical variability found in humans, organs and tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' While these simpler models remain practical and useful for some tasks, the lack of realism and variability makes them unsuitable for generating digital humans for in silico imaging trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Personalized models Personalized models aim to capture patient-specific information in a digital representation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Medical digital replicas of human subjects are in silico representations of an individual in terms of anatomy and physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sometimes referred to as digital twins [27], these replicas are designed to simulate parts or the whole body of a subject for prognostic or predictive assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These models including digital twins can be continuously updated from multimodal medical if the data characteristics change over time§.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Digital twins are of interest in the context of evaluating and selecting optimal medical treatments [28] or imaging procedures [29] within clinical practice, and can also be incorporated into other in silico applications [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, Wang [31] suggested three applications in the areas of medical imaging: optimal selection of scanning techniques (so called “virtual comparative scanning”), data sharing from in silico scanning of the digital replica to the open source community, and improvement of the regulatory process of image reconstruction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Patient image datasets can also be used to generate models of specific tissues and organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, the Visible Human project [32] was first made available in 1994 by the National Library of Medicine (NIH) to facilitate anatomy visualization applications and includes a detailed data set of cross-sectional photographs of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Family models Personalized models of a small number of subjects can be assembled into families to generate a collection of a small number of digital humans spanning a common set of parameters, such as subjects’ body size and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These models are based on image acquisitions using different modalities including computed tomography (CT), magnetic resonance imaging (MRI) and chest radiographs (CXR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' An example of a family model is the Virtual Family [33], released by FDA ∥ in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The Virtual Family consists of a set of detailed, anatomically correct whole-body models of an adult male, an adult female, and two children based on high-resolution MRI data of healthy volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Organs and tissues § A related concept is an avatar, an artistic and sometimes aspirational digital representation of the human in the digital world for interactivity purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' ∥ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='gov/about-fda/cdrh-offices/ virtual-family CONTENTS 6 Digital humans for in silico trials Population models (stochastic) Knowledge- based Image-based Generative Parametric Individual models (non- stochastic) Family Personalized Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Classification of ethods to generate digital humans for in silico clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' are represented using computer-aided design (CAD) techniques where each component is a high-resolution, non self-intersecting mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this case, the models are used for electromagnetic, thermal and acoustic simulations in the safety assessment of active and passive medical implants [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Safety evaluations do not require full sampling of the intended population and can be performed with a small number of exemplars, provided the exemplars adequately cover the needed parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Similar approaches are utilized in efforts to provide models of patient anatomy using patient images as the basis for development of cohorts including using MRI and CT images for modeling lungs [35] and torso [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' More recently, image-derived digital and physical models of the breast have been proposed by Kiarashi [36] and Bliznakova [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this approach, a voxelized breast model is derived from patient images through image segmentation for determining the composition of each voxel [38, 39, 40, 41, 42, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Patient-derived models are limited to the imaging characteristics of the acquisition system and are also affected by the imperfections of the segmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The resulting models can also be augmented with physiological features to facilitate imaging studies involving contrast agents [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Population models Testing new imaging devices, however, requires the availability of large digital cohorts of stochastic digital humans that can be assembled to properly power a study not only on the aggregate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', for the entire population), but also to analyze for specific subgroups with notable characteristics, including under-represented populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this section, we focus our attention on models suited for the generation of large cohorts of digital humans to be enrolled within in silico imaging trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Image-based models Image-based models estimate and sample model components from relevant characteristics within the acquired medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Image-based models estimate model components φn and θn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3 from within the acquired medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Whether parametric or generative, all image-based models are limited by the quality of the source data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' medical images), including noise, artifacts, and contrast constraints, and do not provide an unequivocal mapping to the underlying tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In practice, the use of image-based models should also acknowledge the limitation arising from the existence of a null space of the imaging system [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The null space, which typically arises from the mapping of a continuous object to discrete data with an imperfect image acquisition system, results in an unavoidable loss of information regarding the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Given that the imaging system operator is only partially known for most imaging systems and cannot represent information obscured by the null space of the imaging transformation, image-based models are limited even when imaging system models include noise measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Image-based parametric models In image- based parametric models, the generation of cohorts is achieved by creating models based on available sets of patient imaging data and model modification tech- niques including parametric deformation, morphing, and registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Parametric models (also known as stylized phantoms [47]) capture a population cohort by a set of mathematical equations representing a series of surfaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', splines) defining organs that are later CONTENTS 7 voxelized into a volumetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The popular 4D ex- tended cardiac-torso (XCAT) phantom [48] is an exam- ple of an image-based parametric model, and a survey of other representations can be found in Kainz [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' One limitation of this approach is that model development is typically performed on a small number of available patient images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, Erickson [39] presented a methodology to create a database of anatomically variable 3D digital breast models from dedicated breast CT images using a tissue classification and segmentation algorithm and a fuzzy C-means segmentation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The study provided a population of 224 breast phantoms incorporating a range of breast types, volumes, densities, and parenchymal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, using hundreds of images might be insufficient to properly characterize a population for statistically powered in silico imaging trials across patient variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Some recently released image datasets include a larger number of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For example, the Medical Information Mart for Intensive Care (MIMIC) CXR dataset [50] contains 227,835 imaging studies from 65,379 patients presenting to the Beth Israel Deaconess Medical Center Emergency Department between 2011–2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Similarly, the Medical Imaging and Data Resource Center (MIDRC) effort [51] is undertaking a large, multi-year, systematic effort to collect high-quality COVID data, and over 100,000 imaging studies have been made public after 2 years of work and with significant funding from the NIH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, data sets collected in these well-defined areas are likely still insufficient to capture the total variability in patient images and the large number of subgroups one may find interesting to study ¶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' This limitation precludes the use of image-based parametric models for accurately creating digital cohorts for large scale in silico trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Generation of multiple realizations of humans to constitute a cohort can be obtained by extending image-derived models to create populations in a statistical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, Sturgeon [52] developed synthetic breast models using principal component analysis (PCA) to describe a small training set of patient images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this approach, each existing patient breast CT volume was compactly represented by the mean image plus a weighted sum of eigenbreasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The distribution of weights was sampled to create synthesized breast phantoms that matched fibroglandular density and noise power law exponent distributions in real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hence, the distribution of the synthetic model is determined by that of the training data, and, therefore, might suffer from a lack ¶ “I cannot breed them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' So help me, I have tried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We need more .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' than can ever be assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Millions, so we can be trillions more,” Niander Wallace in Blade Runner 2049 (see https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='imdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='com/title/tt1856101/characters/nm0001467).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' of appropriate representations of cases at the tails of the distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', very large or very small, very dense or very glandular breasts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A related concept from the computer vision and graphics community is the statistical human body model, in which a vertex- based model of the body surface is learned, typically via PCA, from subjects’ input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The techniques rely on linear blend skinning (LBS) to constrain the surface vertex deformation with respect to a template bone skeleton [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Created for non-medical purposes, these parametric models are typically learned from training examples of lower resolution than what is common in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' One alternative approach is to add deformation morphing using an anatomic template [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lee [47] introduce a hybrid, non-uniform rational B-spline surface (NURBS) based phantom of an infant by combining the expressiveness of a voxel phantom with the flexibility of geometric manipulation and organ positioning in a parametric phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Another example is the XCAT Warp [54], where AI-assisted unsupervised registration is used to warp XCAT to patient CT images to capture a more broad set of variations, compared to the existing organ and model scaling offered by XCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These methods are suitable for investigating digital-twin approaches where individual models reflecting the characteristics of a single individual are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Image-based generative models Image-based generative models attempt to synthesize a population of stochastic digital humans from information con- tained in medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ideally this population cap- tures the variability in the anatomy and tissue prop- erties within a specified cohort of to-be-imaged sub- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Consider a collection of N-dimensional digital humans {fs}S s=1 that represents the cohort of interest as described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' This setting corresponds to a practical situation in which an in silico study employs a fully discrete representation of an imaging system in which a finite-dimensional approximation of an object is mapped to discrete image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' As mentioned in Section 3, each digital human fs can be interpreted as a realization of a random vector f that is characterized by an unknown probability density function pr(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The ability to sample from pr(f) to generate large ensem- bles of objects for use in in silico imaging trials is, at least conceptually, the ultimate objective of a stochas- tic digital human model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Emerging generative methods that utilize neural networks are being actively devel- oped for this purpose [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We refer to these methods as generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A generative adversarial net- work (GAN) is a type of generative model that has recently been very popular for high-resolution image synthesis [56], image translation [57, 58] and a number CONTENTS 8 of generative image applications [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Instead of explic- itly modelling pr(f), which is difficult due to the high dimensionality of f, GANs seek to define a stochas- tic process for drawing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' As such, GANs are categorized as implicit generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Specifically, GANs operate by mapping samples from an analyti- cally tractable, low-dimensional distribution pr(z) to the sought after samples of the high-dimensional dis- tribution pr(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Typically, pr(z) is specified as an in- dependent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=') standard normal distribution, and therefore, samples of the ran- dom vector z can be readily generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The mapping is usually implemented via a deep neural network re- ferred to as the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Simultaneously with gen- erator training, a discriminator network is trained to discriminate between the real and generated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Therefore, the training process is adversarial and is approximately solving a min-max optimization prob- lem [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this case, a collection of training data (typ- ically images) are utilized to learn how to sample from an empirical distribution that approximates pr(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' An excellent review of GAN applications for medical im- age generation can be found in [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The adversarial training process for GANs is inherently unstable and can result in a phenomenon known as mode collapse, in which the model fails to sample from certain re- gions of probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In addition, the generated samples are often of low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A number of alter- native generative models [62, 63] have been developed to address these challenges in applications to medical imaging [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For example, generative latent optimiza- tion (GLO) [62] trains deep convolutional generators by minimizing a simple reconstruction loss, improving on GAN training instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Diffusion models [63, 65] learn a Markov chain of diffusion steps incrementally adding and subtracting noise from data, significantly outperforming GANs in output image quality [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' To date, almost all studies of deep generative models have focused on synthesizing images rather than object rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Limitations There are several significant challenges to employing GANs or other types of deep generative models to establish stochastic human models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A fundamental and potentially limiting issue is the fact that a collection of objects {fs}S s=1 is generally not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Medical images are degraded by the presence of measurement noise and/or reconstruction artifacts which are a limitation of the imaging system and not representative of the true underlying objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' As such, conventional GANs that are directly trained on degraded images will not learn how to sample from the true distribution of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In essence, there is a “chicken and egg problem” when seeking to establish stochastic human models via deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' There are two possible ways to circumvent this limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' First, one can utilize high-quality medical images as surrogates of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For example, in certain tomographic imaging modalities and under specific conditions, images of object properties can be reconstructed and accurately approximate the true object properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this case, GANs are trained in the conventional manner, with images representing the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' If these images are representative of the desired subject cohort, the GAN has the opportunity to accurately capture object variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Second, one can modify the GAN training process to incorporate the image degradation process in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' This approach, referred to as an ambient GAN (AmGAN) [67], utilizes a generator network that is augmented with a measurement operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Objects produced by the generator are mapped to degraded image data, which are then compared with experimental images by the discriminator network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' This permits establishment of an implicit generative model that describes object randomness to be learned from indirect and noisy measurements of the objects themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In a preliminary study, the AmGAN was explored for establishing stochastic object models from imaging measurements for use in optimizing imaging systems [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' While promising, the use of deep generative models for in silico clinical trials is nascent and there remain important topics for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The objective assessment of these technologies is largely lacking, and there is no consensus regarding what statistical information can be reliably learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Additionally, current models have largely been applied on 2D images and their extension to three-dimensions is an ongoing topic of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Finally, as with any data-driven method for establishing stochastic human models, the presence of an imaging system null space will fundamentally limit the ability of GANs to describe certain components of the to-be-imaged objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The extent to which the null space can be mitigated also remains a topic of ongoing research [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Knowledge-based models Knowledge-based (also known as procedural) models are constructed by sampling a set of φn and θn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3 from distributions representing the relevant characteristics of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The characteristics of the distributions are often derived from physical or biological measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Procedural models allow for an unlimited number of random realizations of the object, leading to the possibility of creating large cohorts of digital humans including the representation of rare cases, and at varying spatial resolution which can properly account for small structures that might be relevant for the specific imaging CONTENTS 9 task being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, they are usually computationally intensive and require a large number of parameters to be defined and estimated based on prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Their accuracy and realism depend on the parameter combinations and they can sometimes generate completely unrealistic outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Knowledge-based, procedural models are common in modeling breast anatomy for imaging studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Graff [68] proposed a detailed model that begins with defining an outside surface using a quadratic hemisphere shell with a skin layer and nipple area overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The shape of the shell is then adjusted for the overall breast volume and surface curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Using a Voronoi segmentation approach, the interior is randomly divided into regions of fat or glandular components, with each glandular component containing a ductal network with terminal duct lobular units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The volume is then filled with Cooper’s ligaments, chest muscle, and blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For the VICTRE trial [15], the breast model was sampled with a 50-µm voxel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The implementation is initiated with a set of random seeds and creates random voxelized breast anatomy objects segmented into nine different tissue types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Several different modeling techniques are employed including a non-isotropic Voronoi segmentation, recursive tree branching algorithms to generate a ductal tree and vascular network, and Perlin-noise perturbed random spheroids to create fat lobules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A similar effort by Bliznakova [37] describes a 3D breast software model for x-ray breast imaging simula- tions based on a breast external shape, ductal lobular system, Cooper’s ligaments and pectoralis muscle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this approach, a mammographic background texture is added to the tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Blood vessels, nerves and lymphatics were not modeled explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A simi- lar, more simplistic approach, was developed by Bakic [69] based on two ellipsoidal regions of large scale tissue elements: predominantly adipose tissue and predomi- nantly fibro-glandular tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Internal tissue structures within these regions are approximated by a distribution of elements including shells, blobs, and a ductal tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Similar approaches have been reported for full-body models [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Modeling disease Disease states can be incorporated into digital cohorts using image-based methods or object-space models of the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The analogy between digital human models and disease models can be established if we consider lesions as continuous variables in space (r) and time (t), described by a coefficient vector affecting a set of lesion model characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For simplicity, we will consider the disease independent (of the underlying anatomy where the disease is located) and additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' This assumption allows us to represent the disease cases as a sum of the stochastic human model and the disease component, an addition that is typically performed in the voxelized object model or directly within the model images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We recognize this approach is a known simplification, as disease processes often have significant impact on underlying tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Analogously to the description provided by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, we can generate a set of disease models {ds} defined by: {ds}S s=1 = � n λs nψn(r, t), (4) where λs n is a disease characteristics coefficient vec- tor described by the function ψn over N parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Characteristics that define lesions can include geomet- ric functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', size, morphology), material proper- ties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', x-ray interaction cross-sections, elasticity) or other relevant features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', radioactivity, blood oxy- genation levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Methodologies for generating and incorporating disease into cohorts of digital stochastic models rely on sampling λn and ψn from appropriate distributions representing the intended population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In some cases, disease models are specific to a given anatomical location or physiology corresponding to a digital human exemplar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In other cases, disease models are independent of the digital healthy human and are simply added or inserted multiple times into models of healthy anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In both cases, diseased subjects are denoted by a cohort of digital stochastic humans with added disease components: {fs}S s=1 = � n θs nφn(r) + � n λs nψn(r), (5) where {fs}S s=1 is a cohort of diseased digital humans (for simplicity, and similarly as in the previous section, we choose to omit the time dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Similarly to normal models, when ψn are unknown, models of disease can be obtained relying on imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Alternatively, when ψn are known, analytically or numerically, the stochastic disease models are referred to as knowledge-based (also known as procedural).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Image-based models of disease Similar to image-based models of the human body, image-based models of disease rely on imaging data for extracting lesion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Various techniques for capturing disease characteristics, particularly for breast lesions, have recently been explored [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Image-based neural network models for disease modelling have also been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, Kadia [72] proposed a method to generate synthetic, infection-like patterns in the lung to create large CONTENTS 10 collections of 2D and 3D training examples for deep segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' While image-based models contain features from actual patient data and thus may look more realistic at first glance, they suffer from limited resolution of the tumor model, largely determined by the imaging acquisition characteristics and limited number of available lesion morphologies, shapes, and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In addition, image-based methods require an institutional review board (IRB) approval for obtaining and utilizing the diseased case data for research and development, which could delay or disadvantage some analysis efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Knowledge-based models of disease Knowledge-based models of disease are constructed by sampling a set of known (or assumed known) ψn and λn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4 from distributions representing the relevant characteristics of the disease, where distributions are often derived from physical or biological measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In contrast to image-based models, knowledge-based models enable the generation of unlimited numbers of lesion shapes with variable resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Examples of knowledge-based models include de Sisternes [73] spiculated breast cancer mass model and Sengupta [74] growing breast mass models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In [74], a breast lesion growth method based on biological and physiological phenomena accounting for the stiffness of surrounding anatomical structures was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Breast ligaments were considered as rigid structures with elastic moduli in the range of 8x104- 4x105 kPa, while fat (elastic modulus varying from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='5 to 25 kPa) and glandular tissues (elastic modulus varying from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='5 to 66 kPa) constituting the more elastic regions of the breast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this approach, tumor cells are less likely to grow through stiffer structures and instead, preferentially proliferate through the more elastic regions of the breast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Depending on the breast local anatomical structures, a range of unique lesion morphologies can be realized, allowing lesions to blend naturally into the anatomical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A common simplifying assumption is to define the disease model independent from other human model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For example, in VICTRE [15] and in Sengupta [75], breast cancer mass lesions are added to the normal breast models by replacing voxels in the breast with voxels of the lesion model, without modification to adjacent voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' This approach, while practical, does not account for the significant effect of the growing tumors on its surrounding tissues, typically visible in x-ray images as architectural distortions suggestive of abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' To consider these effects, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5 needs to be modified to account for the interaction between normal and disease models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Role of augmentation methods Augmentation methods start with an already-defined object, image or a set of defined objects, and generate new examples based on properties of inputs, as well as pre-defined or data-driven transformations (in contrast, digital human models start with only an object description, such as that given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' GAN-based models (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='2) are similar to augmentation methods in that they employ complex transformations derived with the help of training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Augmentation methods typically employ analytically-defined or stochastic operators that do not require the use of neural networks, and can be applied both in the object domain and in the acquired image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Techniques in the latter group generate examples that could be obtained through an imaging system applied to an object with an accompanying degradation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', smoothing, noise, reconstruction artifacts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Geometric transformations, intensity operations, and spatial filtering are among the most basic types of augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Geometric transformations redefine the spatial relationships among voxels or geometrical locations in an object, and include affine (scaling, rotation, translation, reflection and shearing), as well as non-affine transformations, such as non- linear warping and morphing [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Intensity operations modify intensity values in a grayscale image or channel values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', RGB or CMYK) in a color image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Examples include operations such as a family of gamma corrections, linear contrast adjustments, and remapping voxel values using a pre-defined or pseudo-random remapping curve [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Spatial filtering (using a filter mask) is another possibility for generating a new image or object based on an existing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Spatial filtering can be linear (in which case it can be implemented by a convolution operation) or non- linear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', median filtering), and can be implemented to smooth or sharpen to emphasize certain features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Finally, all three types of augmentations can be combined using a continuous mapping from the parameter space of transformations to the image or object space [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Noise injection is an image augmentation method that enhances robustness of machine learning models and belongs to the family of domain randomization (DR) methods [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Although noise injection after data acquisition does not generate a new member of a patient population, it can generate a different representation of an object in the image domain, and can be useful for augmenting patient cohorts obtained with in silico modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Some earlier and non-medical applications of noise injection in machine learning sought to augment the image data sets without regard to the physics of image acquisition [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' CONTENTS 11 Other works used physics-based techniques for noise modeling and addition, improving realism of the noise appearance in the augmented images [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The main benefit of noise injection in the image domain for in silico trials is that it may allow for the rapid generation of different representations of the same object at different noise levels, leading to comparisons that may require less computational power compared to a full implementation of image acquisition physics applied to a digital stochastic object model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Addition of texture to a model in the object domain has similarities to noise injection in the image domain in that both techniques aim at producing noise-like properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', using a noise power spectrum in modeling), but are different in that addition of texture in the object domain does not attempt to model the noise from data acquisition [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Combination of objects or images is another popular augmentation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the object domain, combination of an object model for a normal (non-diseased) patient with a lesion model (as described in Section 6) can be thought of as an example of this type of augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Generating new members of a patient population based on an eigenspace analysis of existing patient objects, as was done in [52] and described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='1 is another example of augmentation in the object domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the image domain, researchers investigated tools for the extraction of image parts from one clinical image and then their insertion into a new location on the same or different image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pezeshk [86] used an image blending technique based on Poisson image editing to insert pulmonary nodules extracted from one chest CT exam into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Augmenting a training data set for a machine learning model using this technique can improve the model performance on independent, real test datasets [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Likewise, Ghanian [88] used a similar technique to insert microcalcification clusters extracted from one mammogram into another mammogram, and showed that experienced observers cannot reliably distinguish between computationally inserted and native clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Besides the ability to convince experts, desirable properties for such combination techniques include acceptable noise properties in the combined image, plausible lesion-background combinations (that might require the intervention of an operator during the augmentation process), and a sufficient range of variation in the combined images that can be generated, which are often difficult to satisfy simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The main advantage of data augmentation methods is their practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For example, existing images or models both for normal and diseased patients can be manipulated (with relative ease) with geometric transformations leading to expanded patient representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' When implemented in the image domain, augmentation methods are fast, bypassing the stage where a model for the imaging system is applied to the object to yield an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' However, important shortcomings accompany these advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Unless deliberate attention is paid, augmentation methods may yield objects or images that are biologically or physically implausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' An extreme example may be an intensity transformation that results in bones with lower Hounsfield units than soft tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Although this can be avoided easily by using an intensity transformation that is monotonically increasing, most augmentation methods and transformations need careful planning to avoid such inconsistencies, and it may not be possible to avoid all inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The consequences of such implausible images or objects on the results of an in silico imaging trial should be carefully considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In addition, many augmentation techniques do not result in an independent, new representation from the population, but rather in representations that are highly dependent on the original objects or images used as inputs to the augmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For example, lesion insertion methods described in the previous paragraph do not increase the number of lesions in the augmented data set, but only the lesion-background combinations that are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Again, the consequences of this limitation in the range of variation of generated images should be an important consideration in an in silico imaging trial that uses augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Considerations for sampling digital cohorts In silico studies require careful study planning and good clinical trial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Even if and when methodologies for developing digital stochastic models of humans for imaging studies become widely available, generating digital cohorts needs an understanding of the trade-offs and potential for bias associated with selecting a specific distribution of study subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' At the start of the design of an in silico imaging trial is the challenging task of scoping the population of the digital humans to be included in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For instance, a number of previous computational studies in breast imaging using procedural models used a uniform sampling with a desired average of 50% adipose and 50% fibroglandular voxels [89] with an uncompressed breast size of 14 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Another example of enrollment strategy can be found in the OpenVCT platform, where a range of size and glandularity is specified and then uniformly randomly sampled [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A more recent in silico imaging study used sampling from a multi-class distribution identifying 4 different breast densities resulting in the characteristics of the intended population [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' CONTENTS 12 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='79 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='89 ∆������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='10 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='84 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='90 ∆������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='06 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='77 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='88 ∆������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='11 ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='99 ������������������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='00 ∆������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='01 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Effect of sampling strategies on performance assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sampling is from a bimodal distribution of subjects (seen in 3D insert in the second panel from the left) described by 2 random parameters: (from left to right) uniform, matched, simpler, and narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Only 20 samples are shown here for ease of visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The gray shading depicts the distribution from which samples are taken in each of the 4 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' AM, AT , and ∆A refer to the lesion detection average AUC for mammography, average AUC for digital breast tomosynthesis, and the average AUC difference, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Through in silico enrollment, digital cohorts {fs}S s=1 are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We denote the distribution of the population of digital humans as fd, where d represents the digital world, and the distribution of subjects in the intended population as fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this context, the goal of the in silico enrollment is to minimize the difference ∆f = |fd − fi| between the digital (d) and physical-world intended distributions, where |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='| denotes a statistical distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Clinical trial enrollment programs in the physical world require strategies to ensure a reasonable ∆f given available sampling resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' At first approximation, the in silico enrollment should approximate the intended distribution to a greater extent than the corresponding physical clinical trial enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Analysis of ∆f corresponding to a given in silico enrollment strategy may be needed to understand how the difference across study subject distributions could affect the outcome of the trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Here, we discuss a test case (see Figure 2) that compares different enrollment strategies for an in silico trial comparing digital mammography (DM) and digital breast tomosynthesis (DBT) derived from the VICTRE [15] project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We assume the populations (digital and physical) consist of normal and diseased subjects with a prevalence of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' These two classes of patients are therefore sampled with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We calculate the difference of performance (measured using the area under the receiver operating characteristic curve, or AUC, in the task of differentiating between normal and disease subjects) between mammography and digital breast tomosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We consider the following four sampling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the first approach (uniform), fi is unknown and subjects are sampled uniformly within a range of interest, from all possible combinations of the input parameters that define f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the second approach (matched), fi is known and subjects are sampled from the true underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In the third approach (simpler), fi is unknown, but can be approximated by another, simpler distribution from which samples are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Finally, in the fourth approach (narrow), fi is known to be a narrow, well-defined subset of the general population of subjects of particular interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', rare diseases or very obese subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' For this simplified example, let fi be a bimodal distribution defined by two parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', breast size and glandularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, we can express the model through two expansion functions φ1,2, each associated with one of the two random variables affected by a random parameter set given by θ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' As seen in Figure 2, one of the modes of the distribution has twice the amplitude and half the variance of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The four density plots illustrate a top view of the distribution contour plot with the individual samples drawn using the four different sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' The results demonstrate that the choice of sampling strategy can have a significant effect on the difference in AUC, which for this example case, ranges from a difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='01 (almost zero) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='11 in terms of device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Summary and conclusions In silico trials are an emerging area of regulatory research that offer the ability to capture highly diverse patient distributions at a significant time and cost savings, compared to traditional physical clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' To conduct in silico trials, realistic digital representations of humans are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In this paper, we reviewed and discussed existing techniques for generating digital humans, including disease models, for in silico imaging trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Digital humans can be created using image-based or knowledge-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In summary, we favor techniques with object-based representations (rather than images of 80 60■CONTENTS 13 objects) in order to decouple the characteristics of the image acquisition system from the characteristics of the object (true representation of the physical-world human).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In generating digital humans for in silico trials, one should consider the quality and quantity of the source data or knowledge used, and whether the models represent a single patient, a small cohort, or a sizable population with realistic patient variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' It remains a crucial next step to evaluate the quality of the digital human models and the images that can be generated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' In particular, it is essential to carefully identify the patient distribution that the particular digital human model can and cannot capture, in order to prevent misuse and ensure patient safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' We need to study to what extent model-derived data contributes to our understanding of performance levels for populations with rare diseases or for populations underrepresented in traditional clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Future work should examine the ethical and safety considerations of relying on digital humans for clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Overall, the use of in silico imaging trials and in silico trials in medicine is a rapidly developing field and has the potential to address many of the emerging challenges in the regulatory evaluation of medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Barrett and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Myers, Foundations of image science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' John Wiley & Sons, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Magnenat-Thalmann and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Thalmann, Handbook of virtual humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' John Wiley & Sons, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dosovitskiy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ros, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Codevilla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lopez, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Koltun, “Carla: An open urban driving simulator,” in Conference on robot learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1–16, PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cimino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Negri, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fumagalli, “Review of digital twin applications in manufacturing,” Computers in Industry, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 113, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 103130, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Liu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nee, “Digital twin in industry: State-of-the-art,” IEEE Transactions on industrial informatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2405–2415, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Thelen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fink, and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “A comprehensive review of digital twin - part 1: modeling and twinning enabling technologies,” Struct Multidisc Optim, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Xia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhang, “Quality problems of clinical trials in china: evidence from quality related studies,” Trials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1– 11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [8] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Food, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Administration, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “Diversity plans to im- prove enrollment of participants from underrepresented racial and ethnic populations in clinical trials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' draft guid- ance for industry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' availability,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rajotte, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bergen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Buckeridge, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' El Emam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ng, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Strome, “Synthetic data as an enabler for machine learning applications in medicine,” Iscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Abadi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tsui, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kinahan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bottenus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Frangi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Maidment, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lo, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, “Virtual clinical trials in medical imaging: a review,” Journal of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 042805, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Badano, “In silico imaging clinical trials: cheaper, faster, better, safer, and more scalable,” Trials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1–7, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Abadi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sturgeon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Roos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ravin, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, “Modeling lung architecture in the XCAT series of phantoms: Physiologically based airways, arteries and veins,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 37, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 693–702, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [13] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wedlund and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kvedar, “Simulated trials: in silico approach adds depth and nuance to the rct gold- standard,” NPJ digital medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [14] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tsui, “Mcat to xcat: The evolution of 4-d computerized phantoms for imaging research,” Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 97, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1954–1968, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Badano, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Graff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Badal, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zeng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samuelson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Glick, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Myers, “Evaluation of digital breast tomosynthesis as replacement of full-field digital mammography using an in silico imaging trial,” JAMA network open, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' e185474–e185474, 11 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pepe, “Evaluating technologies for classification and prediction in medicine,” Statistics in medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 24, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3687–3696, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Arifin and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Yusof, “Correcting for partial verification bias in diagnostic accuracy studies: A tutorial using r,” Statistics in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1709–1727, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [18] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Berti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Antonini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Poletti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fiuza, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Vaughan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Migliavacca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Petrini, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pennati, “How to validate in silico deployment of coronary stents: strategies and limitations in the choice of comparator,” Frontiers in Medical Technology, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 37, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [19] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Barrett and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Caucci, “Stochastic models for objects and images in oncology and virology: application to PI3K-Akt-mTOR signaling and COVID-19 disease,” Journal of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S16001, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Shepp and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Logan, “The fourier reconstruction of a head section,” IEEE Transactions on Nuclear Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 21–43, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Martin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ruthven, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Boubertakh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Miquel, “Realistic dynamic numerical phantom for mri of the upper vocal tract,” Journal of Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Snyder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ford, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Warner, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fisher Jr, “Estimates of absorbed fractions for monoenergetic photon sources uniformly distributed in various organs of a heterogeneous phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=',” tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', Oak Ridge National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', Tenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Caon, “Voxel-based computational models of real human anatomy: a review,” Radiation and environmental biophysics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 229–235, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [24] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zu, “The vip-man model-a digital human testbed for radiation siimulations,” SAE transactions, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 779–787, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [25] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' George Xu, “Computational phantoms for organ dose calculations in radiation protection and imaging,” The Phantoms of Medical and Health Physics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 225–262, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sharma, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Abadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Iliopoulos, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, “iphantom: a framework for automated creation of individualized computational phantoms and its application to ct organ dosimetry,” IEEE Journal of Biomedical and Health Informatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3061–3072, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kamel Boulos and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhang, “Digital twins: from personalised medicine to precision public health,” Journal of Personalized Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 745, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kamel Boulos and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “Digital twins: From CONTENTS 14 personalised medicine to precision public health,” J Pers Med, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 745, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pesapane, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rotili, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Penco, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nicosia, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cassano, “Digital twins in radiology,” Journal of Clinical Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6553, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Erol, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mendi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Do˘gan, “The digital twin revolution in healthcare,” in 2020 4th International Symposium on Multidisciplinary Studies and Innovative Technologies (ISMSIT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1–7, IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [31] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Badal, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Jia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Maltz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mueller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Myers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Niu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Vannier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Yan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Yu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zeng, “Development of metaverse for intelligent healthcare,” Nature Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9220929, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [32] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Spitzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ackerman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Scherzinger, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Whitlock, “The visible human male: a technical report,” Journal of the American Medical Informatics Association, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 118–130, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Christ, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kainz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hahn, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Honegger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zefferer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Neufeld, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rascher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Janka, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bautz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Chen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “The virtual family, development of surface- based anatomical models of two adults and two children for dosimetric simulations,” Physics in Medicine and Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' N23, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [34] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fujimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zaidi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lampman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Guag, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Etheridge, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Habara, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rajan, “Comparison of sar distribution of hip and knee implantable devices in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='5t conventional cylindrical-bore and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='2t open-bore vertical mri systems,” Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 87, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1515–1528, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Duetschler, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bauman, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bieri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cattin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ehrbar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Engin-Deniz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Giger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Josipovic, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Jud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Krieger, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “Synthetic 4dct (mri) lung phantom generation for 4d radiotherapy and image guidance investigations,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2890–2903, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [36] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kiarashi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nolte, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sturgeon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ghate, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nolte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lo, “Development and application of a suite of 4-d virtual breast phantoms for optimization and evaluation of breast imaging systems,” IEEE Trans Med Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1401–9, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [37] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bravou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kolitsi, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pallikarakis, “A three-dimensional breast software phantom for mammography simulation,” Physics in Medicine & Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3699, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tourassi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Boone, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dobbins III, “Methodology for generating a 3d computerized breast phantom from empirical data,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3122–3131, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Erickson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wells, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sturgeon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dobbins, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lo, “Population of 224 realistic human subject-based computational breast phantoms,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 23–32, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hsu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Palmeri, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Veress, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dobbins III, “Generation of a suite of 3d computer- generated breast phantoms from a limited set of human subject data,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 043703, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [41] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Elangovan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mackenzie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dance, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Young, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cooke, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wilkinson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Given-Wilson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wallis, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wells, “Design and validation of realistic breast models for use in multiple alternative forced choice virtual clinical trials,” Physics in Medicine & Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2778, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [42] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Garc´ıa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fedon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Caballo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mart´ı, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sechopoulos, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Diaz, “Realistic compressed breast phantoms for medical physics applications,” in 15th International Workshop on Breast Imaging (IWBI2020), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11513, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 30–37, SPIE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sarno, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mettivier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' di Franco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Varallo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hernandez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Boone, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Russo, “Dataset of patient-derived digital breast phantoms for in silico studies in breast computed tomography, digital breast tomosynthesis, and digital mammography,” Medical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2682–2693, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Caballo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rabin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fedon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rodr´ıguez-Ruiz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Diaz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Boone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dance, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sechopou- los, “Patient-derived heterogeneous breast phantoms for advanced dosimetry in mammography and tomosynthe- sis,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5423–5438, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [45] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sauer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Abadi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, “Anatomi- cally and physiologically informed computational model of hepatic contrast perfusion for virtual imaging trials,” Medical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 49, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2938–2951, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [46] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Stockmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Galiana, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Constable, “Null space imaging: nonlinear magnetic encoding fields designed complementary to receiver coil sensitivities for improved acceleration in parallel imaging,” Magnetic resonance in medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1166–1175, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [47] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lodwick, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hasenauer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Williams, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lee, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bolch, “Hybrid computational phantoms of the male and female newborn patient: Nurbs-based whole-body models,” Physics in Medicine & Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3309, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [48] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sturgeon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mendonca, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Grimes, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tsui, “4d xcat phantom for multimodality imaging research,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4902–4915, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [49] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kainz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Neufeld, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bolch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Graff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kuster, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lloyd, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Morrison, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Yeom, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “Advances in computational human phantoms and their applications in biomedical engineering—a topical review,” IEEE transactions on radiation and plasma medical sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1–23, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [50] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Johnson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pollard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Berkowitz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Greenbaum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lungren, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Deng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mark, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Horng, “Mimic-cxr, a de-identified publicly available database of chest radiographs with free-text reports,” Scientific data, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1–8, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [51] “Medical imaging and data resource center (midrc).” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='midrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Accessed: 2023-01-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [52] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sturgeon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Park, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lo, “Synthetic breast phantoms from patient based eigenbreasts,” Med Phys, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6270–6279, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [53] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lewis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cordner, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fong, “Pose space deformation: a unified approach to shape interpolation and skeleton-driven deformation,” in Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 165–172, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Du, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Frey, “Generating anthropomorphic phantoms using fully unsupervised deformable image registration with convolutional neural networks,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6366– 6380, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [55] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Galbusera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Niemeyer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Seyfried, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bassani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Casaroli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kienle, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wilke, “Exploring the potential of generative adversarial networks for synthesizing radiological images of the spine to be used in in silico trials,” Frontiers in bioengineering and biotechnology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 53, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [56] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Brock, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Donahue, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Simonyan, “Large scale gan training for high fidelity natural image synthesis,” arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='11096, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [57] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Park, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Isola, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Efros, “Unpaired image-to-image translation using cycle-consistent adver- sarial networks,” in Proceedings of the IEEE interna- CONTENTS 15 tional conference on computer vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2223–2232, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [58] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Isola, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhou, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Efros, “Image-to-image translation with conditional adversarial networks,” in Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1125–1134, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [59] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' She, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ward, “Generative adversarial networks in computer vision: A survey and taxonomy,” ACM Computing Surveys (CSUR), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1– 38, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [60] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pouget-Abadie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mirza, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Warde-Farley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ozair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Courville, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bengio, “Generative adversarial networks,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 139–144, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [61] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Singh and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Raza, “Medical image generation using generative adversarial networks: A review,” Health informatics: A computational perspective in healthcare, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 77–96, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [62] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bojanowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Joulin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lopez-Paz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Szlam, “Optimizing the latent space of generative networks,” arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='05776, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [63] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Jain, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Abbeel, “Denoising diffusion probabilistic models,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 6840–6851, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [64] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ravikumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Frangi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fidler, “Federated simulation for medical imaging,” in International Conference on Medical Image Computing and Computer-Assisted Intervention, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 159–168, Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [65] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Croitoru, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hondru, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ionescu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Shah, “Diffusion models in vision: A survey,” arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='04747, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [66] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dhariwal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nichol, “Diffusion models beat gans on image synthesis,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8780–8794, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [67] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bhadra, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Brooks, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Anastasio, “Learning stochastic object models from medical imaging measurements by use of advanced ambient generative adversarial networks,” Journal of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 015503, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [68] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Graff, “A new, open-source, multi-modality digital breast phantom,” in Medical Imaging 2016: Physics of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 9783, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 72–81, SPIE, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [69] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bakic, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Myers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Glick, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Maidment, “Virtual tools for the evaluation of breast imaging: state-of-the science and future directions,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 518–524, Springer, Cham, 6 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [70] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dukov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakova, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Feradov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Buliev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bosmans, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mettivier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Russo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cockmartin, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakov, “Models of breast lesions based on three-dimensional x-ray breast images,” Physica Medica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 57, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 80–87, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [71] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakova, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dukov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Feradov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Gospodinova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bliznakov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Russo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mettivier, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bosmans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Cockmartin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sarno, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', “Development of breast lesions models database,” Physica Medica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 64, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 293–303, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [72] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kadia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nguyen, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Asari, “Synthesis for robust segmentation of infected lung region on small- scale data,” SSRM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [73] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' de Sisternes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Brankov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zysk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Schmidt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Nishikawa, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wernick, “A computational model to generate simulated three-dimensional breast masses,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1098–1118, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sengupta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sharma, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Badano, “Computational model of tumor growth for in silico trials,” in Medical Imaging 2021: Physics of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 11595, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1262–1270, SPIE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [75] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sengupta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sharma, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Badano, “Computational model of tumor growth for in silico trials,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [76] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wolberg, “Geometric transformation techniques for digital images: A survey,” 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [77] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Chlap, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Min, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Vandenberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dowling, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Holloway, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Haworth, “A review of medical image data augmentation techniques for deep learning applications,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Imaging Radiat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Oncol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 65, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 545–563, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [78] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hesse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kuling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Veta, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Martel, “Intensity augmentation to improve generalizability of breast segmentation across different MRI scan protocols,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 68, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 759– 770, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [79] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Tian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lim, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ouyang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dokania, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Torr, “A continuous mapping for augmentation design,” in Advances in Neural Information Processing Systems (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ranzato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Beygelzimer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Dauphin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Liang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Vaughan, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' ), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 13732– 13743, Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [80] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Noh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' You, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Mun, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Han, “Regularizing deep neural networks by noise: Its interpretation and optimization,” ArXiv, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' abs/1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='05179, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [81] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Moreno-Barea, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Strazzera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Jerez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Urda, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Franco, “Forward noise adjustment scheme for data augmentation,” in 2018 IEEE Symposium Series on Computational Intelligence (SSCI), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 728–734, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [82] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bae, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Park, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Seo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lee, “A perlin noise-based augmentation strategy for deep learning with small data samples of HRCT images,” Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 17687, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Omigbodun, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Noo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' McNitt-Gray, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hsu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Hsieh, “The effects of physics-based data augmentation on the generalizability of deep neural networks: Demonstration on nodule false-positive reduction,” Med Phys, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 46, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4563–4574, Oct 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [84] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Fabian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Heckel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Soltanolkotabi, “Data augmentation for deep learning based accelerated mri reconstruction with limited data,” in Proceedings of the 38th International Conference on Machine Learning (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Meila and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zhang, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' ), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 139 of Proceedings of Machine Learning Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3057–3067, PMLR, 18–24 Jul 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [85] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Abadi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Segars, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sturgeon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Harrawood, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Kapadia, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Samei, “Modeling “textured” bones in virtual human phantoms,” IEEE Transactions on Radiation and Plasma Medical Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 47–53, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [86] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pezeshk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sahiner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Zeng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Wunderlich, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Chen, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Petrick, “Seamless insertion of pulmonary nodules in chest ct images,” IEEE Transactions on Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 2812–2827, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [87] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pezeshk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Petrick, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Chen, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sahiner, “Seamless lesion insertion for data augmentation in cad training,” IEEE Transactions on Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1005–1015, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [88] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Ghanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Pezeshk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Petrick, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Sahiner, “Computational insertion of microcalcification clusters on mammograms: reader differentiation from native clusters and computer-aided detection comparison,” Journal of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1, nov 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [89] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Gong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Glick, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Vedula, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Thacker, “A computer simulation study comparing lesion detection accuracy with digital mammography, breast tomosynthesis, and cone-beam ct breast imaging,” Medical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1041–1052, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' [90] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Barufaldi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Higginbotham, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Bakic, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Maidment, “OpenVCT: a GPU-accelerated virtual clinical trial pipeline for mammography and CONTENTS 16 digital breast tomosynthesis,” in Medical Imaging 2018: Physics of Medical Imaging (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Lo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Schmidt, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' Chen, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' ), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 10573, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} +page_content=' 1057358, International Society for Optics and Photonics, SPIE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdFAT4oBgHgl3EQf3R5S/content/2301.08719v1.pdf'} diff --git a/_9FQT4oBgHgl3EQf8TZV/content/tmp_files/2301.13446v1.pdf.txt b/_9FQT4oBgHgl3EQf8TZV/content/tmp_files/2301.13446v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ada0d3e5e09678e5a16d10368dfc96d891b4192c --- /dev/null +++ b/_9FQT4oBgHgl3EQf8TZV/content/tmp_files/2301.13446v1.pdf.txt @@ -0,0 +1,6276 @@ +arXiv:2301.13446v1 [cs.LG] 31 Jan 2023 +Sharp Variance-Dependent Bounds in Reinforcement Learning: +Best of Both Worlds in Stochastic and Deterministic Environments +Runlong Zhou˚ +Zihan Zhang: +Simon S. Du; +February 1, 2023 +Abstract +We study variance-dependent regret bounds for Markov decision processes (MDPs). +Algorithms +with variance-dependent regret guarantees can automatically exploit environments with low variance +(e.g., enjoying constant regret on deterministic MDPs). The existing algorithms are either variance- +independent or suboptimal. We first propose two new environment norms to characterize the fine-grained +variance properties of the environment. For model-based methods, we design a variant of the MVP al- +gorithm [Zhang et al., 2021a] and use new analysis techniques show to this algorithm enjoys variance- +dependent bounds with respect to our proposed norms. In particular, this bound is simultaneously min- +imax optimal for both stochastic and deterministic MDPs, the first result of its kind. We further initiate +the study on model-free algorithms with variance-dependent regret bounds by designing a reference- +function-based algorithm with a novel capped-doubling reference update schedule. Lastly, we also provide +lower bounds to complement our upper bounds. +1 +Introduction +We consider episodic reinforcement learning (RL) on tabular Markov Decision Processes (MDPs). Existing +algorithms can be categorized into two classes: model-based methods whose space complexity scales quadrat- +ically with the number of states [Auer et al., 2008, Agrawal and Jia, 2017, Azar et al., 2017, Dann et al., +2017, 2019, Zanette and Brunskill, 2019, Zhang et al., 2021a] and model-free methods whose space complex- +ity scales linearly with the number of states [Jin et al., 2018, Bai et al., 2019, Zhang et al., 2020, Li et al., +2021]. +The MDPs in practice often enjoy benign structures, so problem-dependent regret bounds are of great +interest [Zanette and Brunskill, 2019]. RL algorithms often perform far better on these MDPs than what +their worst-case guarantees would suggest. Motivated by this observation, we want to systematically study +algorithms with regrets depending on quantities that characterizes the hardness of MDPs. Ideally, such +algorithms should automatically exploit the MDP instance without the prior knowledge of problem-dependent +quantities. +As a motivating example, for time-homogeneous MDPs with total reward bounde by 1, the +minimax regret bound for deterministic MDPs is OpSAq where S and A are number of states and actions, +respectively and the worst-case minimax optimal regret bound for stochastic MDPs is rO +`? +SAK +˘ +where +K is the number of episodes. Many problems can be formulated as deterministc MDPs, such as shortest +path (maze, real world navigation), combinatorial optimization, Atari games [Mnih et al., 2013] and many +games (mountain car, lunar lander, robotics, etc.) in OpenAI Gym [Brockman et al., 2016]. Deterministic +systems can also approximate stochastic systems well (see Section 2 and 6 in Bertsekas [2012]). We want +an algorithm designed for generic stochastic MDPs with worst-case minimax optimal regret bound while +enjoying the OpSAq bound when the MDP is deterministic. +˚University of Washington. Email: vectorzh@cs.washington.edu +:Email: zihan-zh17@mails.tsinghua.edu.cn +;University of Washington. Email: ssdu@cs.washington.edu +1 + +Zanette and Brunskill [2019] is a pioneer work which provides a model-based algorithm whose regret +scales with variance-depedent quantities. They defined a quantity, Q‹, named the maximum per-step con- +ditional variance to characterize the randomness of the MDP instance, and showed a regret bound of +rOp?HQ‹ ¨ SAK ` H5{2S2Aq, where H is the planning horizon. This bound is still not satisfactory because: +① There exist MDPs with Q‹ “ Ωp1q, so the regret reduces to rOp +? +HSAKq which does not match the min- +imax optimal bound rOp +? +SAKq. ② For deterministic MDPs (Q‹ “ 0), the regret reduces to rOpH5{2S2Aq, +which does not match the optimal OpSAq bound. +Contributions. +This paper makes the following contributions which significantly advance our understand- +ing of problem-dependent bounds in reinforcement learning. +‚ First, We introduce the total multi-step conditional variance, VarΣ +K and the maximum policy-value +variance, Var‹, to provide fine-grained characterizations of the variance in the MDP (see Section 4 for the +formal definitions). Importantly, regret bounds that depend on these quantities will reduce to the minimax +optimal bound in the worst case whereas the existing notion HQ‹ cannot. +‚ Second, for model-based methods, we identify the obstacles preventing the current state-of-the-art +minimax optimal algorithm, MVP Zhang et al. [2021a], from being variance-dependent. We make necessary +improvements and introduce a truncation method to bound the total variance. We show the regret bound of +the improved algorithm, MVP-V, scales with Var‹ or VarΣ +K. In particular, these bounds imply that, MVP-V is +minimax optimal for both the classes of stochastic and deterministic MDPs. To our knowledge, this is first +result of such kind. See Table 1 for comparions between model-based methods. +‚ Third, we initiate the study of model-free algorithms with variance-dependent regrets. We explain +why existings model-free algorithms cannot be variance-dependent. We futher propose a new model-free +algorithm, UCB-Advantage-V, which relies on a a capped-doubling manner of updates for reference values. +We further utilize a novel analysis technique which bounds value gaps directly from the existing uniform +convergence bound to give the first variance-dependent bound for model-free algorithms. Importantly, this +bound reduces to the minimax optimal bound for the worst-case MDPs. See Table 2 for comparisons between +model-free algorithms. +‚ Lastly, we prove minimax regret lower bounds for the class of MDPs with bounded variances. We show +that the main order terms of our regret upper bounds match these lower bounds, so our proposed algorithms +are minimax optimal for the class of variance-bounded MDPs. +1.1 +Technical Overview +For model-based algorithms, existing state-of-the-art work [Zhang et al., 2021a] fails to be variance-dependent. +It is hard to bound the total variance by its expectation using martingale concentration inequalities directly, +while avoiding an H-dependency. This is because the total variance within an episode can be as large as +ΩpHq. We introduce a novel analysis technique which truncates the total variance of each episode to a con- +stant and apply martingale concentration inequalities on this sequence, and show that with high probability +there is no truncation. We also apply a more refined concentration inequality to the transition model to +have a dependency on the maximum support instead of the size of the state space. This step is crucial in +obtaining the tight bound for deterministic MDPs. +For the model-free algorithm, existing work [Zhang et al., 2020] upper-bounds all the four bias terms in +their Equation (13) by variance-independent main order terms. We identify the problem incurred by the +large bias in reference values, and replace the update with a capped-doubling manner. Since too frequent +updates discard past data very often, this method balances between the summation of gaps of value functions +and the waste of data. We integrate directly over the error between the estimated value and the optimal +value to bound the total squared gaps between them, whereas Zhang et al. [2020] bound them with a coarse +binary gap of either H or the final gap. Combined with many other finer-grained analyses throughout the +proof, we can finally remove all the variance-independent main order terms except for the total variance. +2 + +Algorithm +Regret +Variance- +Dependent +Stochastic- +Optimal +Deterministic- +Optimal +Horizon- +Free +Euler +Zanette and Brunskill [2019] +rOp?HQ‹ ¨ SAK ` H5{2S2Aq +Yes +No +No +No +rOp +? +SAK ` H5{2S2Aq +No +Yes +No +No +MVP +Zhang et al. [2021a] +rOp +? +SAK ` S2Aq +No +Yes +No +Yes +MVP-V +This work +rOp +b +mintVarΣ +K, Var‹KuSA ` ΓSAq +Yes +Yes +Yes +Yes +Table 1: +Comparisons between model-based algorithms for time-inhomogeneous MDPs with total reward +bounded by 1. +rO hides logarithm factors. S, A, Γ, H and K are number of states, actions, maximum +support of the transition model, planning horizon and interaction episodes. Q‹, VarΣ +K and Var‹ are variance +notations in Section 4. Q‹ and VarΣ +K are upper bounded by 1 in the worst case and become 0 when the MDP +is deterministic. An “Yes” in each column means: Variance-Dependent: the regret has a main order term +scaling with any variance notation. Stochastic-Optimal: the regret has a main order term of rOp +? +SAKq +which matches the minimax lower bound. +Deterministic-Optimal: the regret is rOpSAq on deterministic +MDPs (with variance equal to 0). Horizon-Free: the regret has only logarithmic dependency on H. +Paper Overview. +The paper is organized as follows. We first list basic concepts of MDPs in Section 3, +then define variance quantities in Section 4. Our main results then come in three sections: Sections 5 and 6 +show the algorithms, theorems, corollaries and proof sketches of our model-based and model-free methods, +respectively. Section 7 shows our lower bounds for the class of variance-bounded MDPs. +2 +Related Works +Minimax optimal regret bounds. +Algorithms for regret minimization can be categorized into two +classes: model-based and model-free. Being model-free means the space complexity is OpHSAq, prohibiting +the estimation of the whole transition model Phps1|s, aq. +For model-based methods, there are previous +work [Zhang et al., 2021a, 2022] achieving a property called horizon-free, which allows only logarithmic +dependency on H for regrets. As explained in Jiang and Agarwal [2018], in many scenarios with a long +planning horizon, the interesting regime is K ! H. This underscores the importance of being horizon-free, +because for H-dependent bounds, only when K " H they become sub-linear in K. Being horizon-free is +challenging, because it requires utilizing transition data for the same state-action pair from different steps +and handling a spike in rewards. There are many works other than those we cite in Section 1 giving nearly +minimax optimal bounds: Bartlett and Tewari [2012], Osband et al. [2013], Osband and Van Roy [2017], +Fruit et al. [2018a], Talebi and Maillard [2018], Simchowitz and Jamieson [2019], Russo [2019], Zhang and Ji +[2019], Neu and Pike-Burke [2020], Xiong et al. [2021], Pacchiano et al. [2020]. We compare our results with +the state-of-the-art in Table 1 (model-based) and Table 2 (model-free). +Other problem-dependent results. +Most problem-dependent results prior to Zanette and Brunskill +[2019] focus on the infinite-horizon setting. Some depend on the range of value functions [Bartlett and Tewari, +2012, Fruit et al., 2018b]. Maillard et al. [2014] introduces a hardness measure called distribution norm. +Talebi and Maillard [2018] provides a problem-dependent regret bound that scales as a function of the vari- +ance of the next state distribution under strong assumptions on mixing time. There are gap-dependent +results for multi-armed bandits and RL [Even-Dar et al., 2006, Auer et al., 2008, Simchowitz and Jamieson, +2019, Xu et al., 2021, Yang et al., 2021]. +Jin et al. [2020] shows that with a slight modification, the algorithm in Zanette and Brunskill [2019] can +have a first-order regret, with the main order term depending on the optimal value function. Wagenmaker et al. +[2022] provides a first-order regret for linear MDPs. When the total reward is bounded by 1 almost surely, +for any policy its variance is not larger than this value. This means a first-order dependency is weaker than +a variance-dependency. +3 + +Algorithm +Regret +Variance- +Dependent +Stochastic- +Optimal +Q-learning (UCB-B) +Jin et al. [2018] +rOp +? +H4SAK ` H9{2S3{2A3{2q +No +No +UCB-Advantage +Zhang et al. [2020] +rOp +? +H3SAK ` +4? +H33S8A6Kq +No +Yes +Q-EarlySettled- +Advantage +Li et al. [2021] +rOp +? +H3SAK ` H6SAq +No +Yes +UCB-Advantage-V +This work +rOp +b +mintVarΣ +K, Var‹KuHSA +` +4? +H15S5A3Kq +Yes +Yes +Table 2: +Comparison between model-free algorithms for time-inhomogeneous MDPs with every reward +bounded by 1. An “Yes” in each column means: Variance-Dependent: the bound scales with the variance +term that characterizes the randomness of the environment; Stochastic-Optimal: in the wortt-case, the +regret’ dominating term becomes rOp +? +H3SAKq which matches the minimax lower bound. +3 +Preliminaries +Notations. +For any event E, we use +1rEs to denote the indicator function. +For any random variable +X, we use VpXq to denote its variance. For any set X, we use ∆pXq to denote the probability simplex +over X. +For any positive integer n, we denote rns :“ t1, 2, . . ., nu. +For any probability distribution P, +we use supppPq “ ř +x +1rPpxq ą 0s to denote the size of its support. Suppose x and y are n-dimensional +vectors, we denote xy :“ řn +i“1 xiyi and xq :“ pxq +1, xq +2, . . . , xq +nq for any number q. If x P ∆prnsq, we use +Vpx, yq “ ř +i xipyi ´ xyq2 “ xy2 ´ pxyq2 to denote the variance of y under x. We use 1k to denote a vector +with all 0 but an only 1 on the k-th position. +Finite-horizon MDPs. +A finite-horizon MDP can be described by a tuple M “ pS, A, P, R, Hq. S is the +finite state space with size S and A is the finite action space with size A. For any h P rHs, Ph : SˆA Ñ ∆pSq +is the transition function and Rh : SˆA Ñ ∆pr0, 1sq is the reward distribution with mean rh : SˆA Ñ r0, 1s. +H is the planning horizon, i.e., episode length. We denote Γ :“ maxh,s,a supppPhp¨|s, aqq as the maximum +support of the transition function, and Ps,a,h :“ Php¨|s, aq. +Conditions for MDPs. +We have two conditions more general than the ordinary setting. As explained +below them, getting tight regret bounds are harder when they are met. +Condition 1. For any policy π, the total reward in a single episode is upper-bounded by 1 almost surely. +For those MDPs not satisfying Condition 1, we can normalize all the rewards by 1{H. Such a conversion +usually multiplies a factor of 1{H to the regret, but cannot take into account a spike in rewards, e.g., some +rhps, aq “ 1. +Condition 2. The MDP is time-homogeneous. Namely, there exist P : SˆA Ñ ∆pSq, R : SˆA Ñ ∆pr0, 1sq +and r : S ˆ A Ñ r0, 1s such that for any ps, aq P S ˆ A, Php¨|s, aq “ Pp¨|s, aq, Rhps, aq “ Rps, aq and +rhps, aq “ rps, aq for any h P rHs. +For simplicity, we denote Ps,a :“ Pp¨|s, aq and Ps,a,s1 :“ Pps1|s, aq. Any time-inhomogeneous MDP can +be instantiated by a time-homogeneous one to satisfy Condition 2. Let a mega state space S “ YH +h“1Sh, +where each Sh corresponds to the state space of the time-inhomogeneous MDP. For any ph, s, aq, we construct +Pps1 +h`1|sh, aq “ Phps1|s, aq and Rpsh, aq “ Rhps, aq, where sh is the corresponding state of s in Sh. S is +4 + +multiplied by H while Γ remains the same, and the regret changes accordingly. This condition underscores +the algorithm’s ability to use information of the same state-action pair from different steps. +We will introduce quantities in Section 4 to quantify determinism, but a fully-deterministic MDP is +very worth studying because the regret lower bound is the well-known ΩpSAq (under Conditions 1 and 2). +Thus, we care about whether the algorithms can have a constant regret (up to logarithm factors) under the +assumption of determinism. +Assumption 3. The MDP is deterministic. Namely, for any ph, s, aq P rHsˆS ˆA, Rhps, aq and Php¨|s, aq +map to a single real number and a single state respectively. +Policies. +A history-independent deterministic policy π chooses an action based on the current state and +time step. Formally, π “ tπhuhPrHs where πh : S Ñ A maps a state to an action. We use Π to denote the +set of all such policies. +Episodic RL on MDPs. +Upon choosing action a at state s when it is the h-th step in an episode, the +agent observes a reward r „ Rhps, aq and the next state s1 „ Php¨|s, aq. When h “ H, the episode ends +after this observation. Thus, a policy π induces a (random) trajectory ptsh, ah, rhuhPrHs, sH`1q where s1 is +exogenously generated, ah “ πhpshq, rh „ Rhpsh, ahq and sh`1 „ Php¨|sh, ahq for h P rHs. The episodic RL +on MDPs proceeds in a total of K episodes. When one episode ends, a new initial state s1 is generated. The +agent should (adaptively) choose a policy πk for the k-th episode, put it into action and cannot change it +within an episode. +Value functions and Q-functions. +Given a policy π, we define its value function and Q-function as +V π +h psq :“ Eπ +« H +ÿ +t“h +rt +ˇˇˇˇˇ sh “ s +ff +, +Qπ +hps, aq :“ Eπ +« H +ÿ +t“h +rt +ˇˇˇˇˇ psh, ahq “ ps, aq +ff +. +It is easy to verify that Qπ +hps, aq “ rhps, aq ` Ps,a,hV π +h`1. +Performance measure. +The goal of episodic RL on MDPs is to find the policy which maximizes the total +reward collected in an episode, regardless of the initial state. Given M, such a goal can be achieved using +dynamic programming. Given this, we denote V ‹ :“ V π‹ and Q‹ :“ Qπ‹. We use cumulative regret as a +performance measure: +RegretpKq :“ +K +ÿ +k“1 +pV ‹ +1 psk +1q ´ V πk +1 +psk +1qq. +4 +Variance Quantities for MDPs +We use the notion of variance to quantify the degree of determinism of MDPs. +The first is called the +maximum per-step conditional variance [Zanette and Brunskill, 2019], which is only relevant to the optimal +value function. +Definition 4. The maximum per-step conditional variance for a particular MDP is defined as: +Q‹ :“ max +h,s,atVpRhps, aqq ` VpPs,a,h, V ‹ +h`1qu. +5 + +Zanette and Brunskill [2019] directly use HQ‹ to upper-bound the total per-step conditional variances +in an episode, a quantity which can be upper-bounded by a constant (see Lemmas 29 and 42). So when +Q‹ ě ΩpHq (or Ωp1{Hq under Condition 1), HQ‹ is not tight. In light of this, we define the total multi-step +conditional variance as a better notation in place of HQ‹. +Definition 5. The total multi-step conditional variance for a trajectory τ “ tsh, ahuhPrHs is defined as: +VarΣ +τ :“ +H +ÿ +h“1 +pVpRhpsh, ahqq ` VpPsh,ah,h, V ‹ +h`1qq. +During the learning process, let the trajectory of the k-th episode be τ k, then we denote VarΣ +pkq :“ VarΣ +τ k, and +VarΣ +K :“ řK +k“1 VarΣ +pkq. +We introduce another type of variance, called the maximum policy-value variance, which is novel in the +literature. +Definition 6. For any policy π P Π, its maximum value variance is defined as Varπ :“ maxsPS Varπ +1psq, +where +Varπ +1psq :“ Eπ +« H +ÿ +h“1 +` +VpRhpsh, ahqq ` VpPsh,ah,h, V π +h`1q +˘ +ff +. +The maximum policy-value variance for a particular MDP is defined as: +Var‹ :“ max +πPΠ Varπ. +Varπ +1psq is the variance of total reward of π starting from s, and the justification can be found in Ap- +pendix B.1. +Under Condition 1, by Lemma 20 we know that Varπ +1psq ď V π +1 psq ď V ‹ +1 psq. So Var‹ ď V ‹ +1 psq. This means +a variance-dependent regret is better than a first-order regret. +4.1 +Comparing VarΣ +pkq and Var‹ +We use this subsection to demonstrate that a small VarΣ +pkq does not imply a small Var‹, and vice versa. +Deterministic MDPs have VarΣ +pkq “ Var‹ “ 0. Trivially, Var‹ “ 0 ùñ VarΣ +pkq “ 0, while the reverse is +not ture. +When VarΣ +pkq “ 0 ă Var‹. +Consider the following time-homogeneous MDP with horizon H: For each state +s there is a good action a1 with a deterministic reward rps, a1q “ 1{H, and all other actions a1 have a +deterministic reward rps, a1q “ 0. For any state-action pair ps, aq, the transition is identically Ps,a,s1 “ 1{S. +The optimal policy always chooses a1 at any state s, so for any s and h, V ‹ +h psq “ pH ´ h ` 1q{H. For +any ph, s, aq, +VpRps, aqq ` VpPs,a, V ‹ +h`1q “ 0, +which means VarΣ +pkq “ 0. However, let π be a policy with πHps1q “ a1 for a certain state s1, and πhpsq “ a1 +for any other h or s. Then π yields cumulative rewards of 1 and 1 ´ 1{H, each with non-zero probabilities. +So Var‹ ą 0. +This example shows that deterministic MDPs are not the only MDPs satisfying VarΣ +pkq “ 0, and that +VarΣ +pkq “ 0 does not imply Var‹ “ 0. +6 + +!! +" +# $ " +!% +!& +#'( +#'( +!) +!* ++ , # +!"#$%&'$ +()*+$ +,-./*0#$ +1,23$*3*.0$ +2,(.24 +Figure 1: Example of Var‹ being arbitrarily smaller than VarΣ +pkq. +Dashed arrows represent probabilistic +transitions and solid arrows represent deterministic ones. The only reward comes at state s4 and on choosing +any action. +When VarΣ +pkq “ 1{4 ą Var‹. +Consider the time-homogeneous MDP in Figure 1: Ps1,a,s2 “ p for any a P A, +and the rest probability is into an MDP with no reward at all. +s2 is a state which we want to have a +high VarΣ +pkq: Ps2,a,s3 “ Ps2,a,s4 “ 1{2, where s3 and s4 are states with value 0 and 1 respectively. Thus at +s2, a, h “ 3, +VarΣ +pkq ě VpRps2, aqq ` VpPs2,a, V ‹ +3 q “ 1 +4. +We also have that for any policy π, V π +1 ps1q “ p{2, so by Lemma 20, Var‹ ď p{2. Taking p arbitrarily small +gives an arbitrarily large gap between VarΣ +pkq and Var‹. +This example shows that a small Var‹ does not imply a small VarΣ +pkq, so using only VarΣ +pkq is insufficient. +5 +Results of the Model-Based Algorithm +We propose MVP-V (Algorithm 1, where “V” stands for “Variance-dependent”), a model-based algorithm +with a variance-dependent regret bound. +Based on MVP [Zhang et al., 2021a] which is minimax optimal +under Conditions 1 and 2, we make necessary alterations to make the regret variance-dependent. +Common notations. +These are notations shared with our model-free algorithm. Let sk +h, ak +h and rk +h denote +the state, action and reward at the h-th step of the k-th episode. Let V k +h and Qk +h denote Vh and Qh at the +beginning of the k-th episode. rO hides polylogpH, S, A, K, 1{δq factors. +Algorithm description. +MVP-V re-plans whenever a state-action pair’s visitation is doubled. The bonus +function depends on the variance of the next-step value functions. It achieves variance-dependent regret by +using the empirical variances of rewards in the bonus, as opposed to using the empirical rewards themselves +in MVP. MVP-V is capable of handling Conditions 1 and 2 and Assumption 3. +Theorem 7. Assume that Conditions 1 and 2 hold. Let δ P p0, 1q be the confidence parameter and that +H, S, A, K, δ be known. With probability at least 1 ´ δ, the regret of MVP-V (Algorithm 1) run with +ι “ 99 +ˆ +ln +ˆ30002H5S7A5K5 +δ2 +˙ +` 1 +˙ +7 + +Algorithm 1 MVP-V +1: Input and initialize: Logarithm term ι; Trigger set L Ð t2i´1 | 2i ď KH, i “ 1, 2, . . .u. +2: for ps, a, s1, hq P S ˆ A ˆ S ˆ rHs do +3: +Nps, aq Ð 0, θps, aq Ð 0, φps, aq Ð 0, nps, aq Ð 0; +4: +Nps, a, s1q Ð 0, pPs,a,s1 Ð 0, Qhps, aq Ð 1, Vhpsq Ð 1. +5: end for +6: \\ Main algorithm begins +7: for k “ 1, 2, . . ., K do +8: +for h “ 1, 2, . . ., H do +9: +Observe sk +h; +10: +Take action ak +h “ arg maxa Qhpsk +h, aq; +11: +Receive reward rk +h and observe sk +h`1; +12: +Set ps, a, s1, rq Ð psk +h, ak +h, sk +h`1, rk +hq; +13: +Set Nps, aq Ð Nps, aq ` 1, θps, aq Ð θps, aq ` r, φps, aq Ð φps, aq ` r2, Nps, a, s1q Ð Nps, a, s1q ` 1. +14: +\\ Update empirical reward and transition probability +15: +if Nps, aq P L then +16: +Set prps, aq Ð θps, aq{Nps, aq; +17: +Set z +VarRps, aq Ð φps, aq{Nps, aq ´ prps, aq2; +18: +Set pPs,a,rs Ð Nps, a, rsq{Nps, aq for all rs P S; +19: +Set nps, aq Ð Nps, aq; +20: +Set TRIGGERED = TRUE. +21: +end if +22: +end for +23: +\\ Update Q-function +24: +if TRIGGERED then +25: +for h “ H, H ´ 1, ..., 1 do +26: +for ps, aq P S ˆ A do +27: +Set +bhps, aq Ð 4 +d +Vp pPs,a, Vh`1qι +maxtnps, aq, 1u ` 2 +d +z +VarRps, aqι +maxtnps, aq, 1u ` +21ι +maxtnps, aq, 1u; +Qhps, aq Ð mintprps, aq ` pPs,aVh`1 ` bhps, aq, 1u; +Vhpsq Ð max +a +Qhps, aq. +28: +end for +29: +end for +30: +Set TRIGGERED = FALSE. +31: +end if +32: end for +is bounded by +RegretpKq ď rOp +b +mintVarΣ +K, Var‹KuSA ` ΓSAq. +When Condition 1 holds, we have Var‹ ď 1. Thus, our results are better than the rOp +? +SAK ` S2Aq +regret of MVP, and achieve the horizon-free (only logarithm dependency on H) property. They are also +strictly better than the rOp?HQ‹ ¨ SAK ` H5{2S2Aqq regret in Zanette and Brunskill [2019]. +8 + +Proof sketch. +See Appendix B.2 for the rigorous proof. We follow the outline in Zhang et al. [2021a], +realizing that the total regret is upper-bounded by M1 ` M2 ` M3, where +M1 « +K +ÿ +k“1 +H +ÿ +h“1 +pPsk +h,ak +h ´ 1sk +h`1qV k +h`1, +M2 « +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ rpsk +h, ak +hq ´ Psk +h,ak +hV k +h`1q, +M3 « +K +ÿ +k“1 +˜ H +ÿ +h“1 +rpsk +h, ak +hq ´ V πk +1 +psk +1q +¸ +. +We expand rpsk +h, ak +hq by Bellman equation to derive a tighter bound for M3. This change is necessary to +remove a variance-independent rOp +? +Kq term. M1, M2, M3 can be then related to a crucial variance term +M4 « +K +ÿ +k“1 +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V k +h`1qq +so the regret is approximately rOp?SAM4q. The difference between VarΣ +K and M4 is of a lower order. To +upper bound M4 directly, we introduce bonus-correction terms +bck +hps, aq :“ V k +h psq ´ Ps,aV k +h`1 ´ rps, aq. +Let BCk +hpsq :“ bck +hps, aq ` Ps,aBCk +h`1 with a “ πk +hpsq, then it can be proven that BCk +hpsq “ V k +h psq ´ V πk +h psq. +Thus, M4 can be bounded by the sum of +Z « +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, BCk +h`1q +and +W “ +K +ÿ +k“1 +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V πk +h`1qq, +where Z is of a lower order and W ď rOpVar‹Kq. +However, the bound of W cannot be derived using +martingale concentration inequalities directly, because the summation of variances within an episode can +be of order ΩpHq, which will introduce a constant term of H, ruining the horizon-free property. We first +prove that the total variance in an episode is bounded by rOp1q with high probability, then the martingale +concentration inequality can be applied to terms truncated to rOp1q. To get the Γ-dependency in the lower +order term, we observe that Ps,a “ 0 ùñ pPs,a “ 0 and put this into concentration bounds. +Corollaries. +We study deterministic MDPs first. +Corollary 8. Assume that Conditions 1 and 2 and Assumption 3 hold. Let δ P p0, 1q be the confidence +parameter and that H, S, A, K, δ be known. With probability at least 1 ´ δ, the regret of MVP-V (Algorithm 1) +run with ι “ 99plnp30002H5S7A5K5{δ2q ` 1q is bounded by RegretpKq ď rOpSAq. +This is because Var‹ “ 0 and Γ “ 1 when the MDP is deterministic. With a more refined analysis, we can +totally eliminate the dependency on δ. Up to logarithm factors, MVP-V matches the lower bound of ΩpSAq. +So MVP-V is minimax optimal for the class of deterministic MDPs. +Another corollary arises when we remove Conditions 1 and 2. For MVP-V to work properly, we need to +apply the conversion methods written below the conditions. +9 + +Corollary 9. Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known. With probability at +least 1 ´ δ, the regret of MVP-V (Algorithm 1) run with ι “ 99plnp30002H12S7A5K5{δ2q ` 1q is bounded by +RegretpKq ď rOp +b +mintVarΣ +K, Var‹KuHSA ` H2ΓSAq. +Readers may notice that the scaling in main order terms are not typical. This is because when removing +Condition 1, VarΣ +K and Var‹ automatically scale by H2. +6 +Results of the Model-Free Algorithm +We propose UCB-Advantage-V (Algorithm 2) to initiate the study of model-free algorithms with variance- +dependent regrets. +Algorithm description. +In UCB-Advantage-V, the update of value functions is triggered by the beginning +of stages for each ps, a, hq triple separately, and the stage design approximately makes use of the last 1{H +fraction of data. Besides, the algorithm maintains reference values by assigning value functions to them at +another frequency. The update rule using the reference value decomposition can be illustrated as: +Qhps, aq Ð +{ +Ps,a,hV ref +h`1 ` +{ +Ps,a,hpVh`1 ´ V ref +h`1q ` prhps, aq ` bk +hps, aq, +where bk +hps, aq is the bonus, prhps, aq, +{ +Ps,a,hV ref +h`1 and +{ +Ps,a,hpVh`1 ´ V ref +h`1q are empirical estimates of rhps, aq, +Ps,a,hV ref +h`1 and Ps,a,hpVh`1 ´ V ref +h`1q respectively. In addition, a very simple update rule +Qhps, aq Ð +{ +Ps,a,hVh`1 ` prhps, aq ` bk +hps, aq +is also in use to provide a guarantee of uniform convergence of estimated value functions. +We make three major alterations to UCB-Advantage: ① We use empirical variances of rewards in bonuses. +② Due to a more refined analysis, we remove the rOpHpn´3{4 ` qn´3{4qq term in bonuses. ③ The reference +value functions are updated in a capped-doubling manner (cf. Line ??). +Alteration ③ is crucial to make the main order term variance-dependent, because there exist constant +gaps between reference values and optimal values, whose summation contributes to the regret as the main +order term in UCB-Advantage. By choosing an appropriate number of updates, we can balance between the +total constant gap and the deviation introduced by frequent updates, making the total variance the only factor +in the main order term. +Theorem 10. Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known. With probability +at least 1 ´ δ, the regret of UCB-Advantage-V (Algorithm 2) run with +ι “ 99 +ˆ +ln +ˆ70002pHSAKq5 +δ2 +˙ +` 1 +˙ +and +i‹ “ +R1 +2 log2 +ˆ +K +H5S3Aι2 +˙V +is bounded by +RegretpKq ď rOp +b +mintVarΣ +K, Var‹KuHSA ` +4? +H15S5A3Kq. +We have Var‹ ď H2, so our result is strictly better than the rOp +? +H3SAK ` +4? +H33S8A6Kq regret of +UCB-Advantage. +10 + +Algorithm 2 UCB-Advantage-V +1: Input and initialize: Logarithm term ι; Stage lengths e1 “ H, ei`1 “ tp1 ` 1{Hqeiu and stage trigger +set L Ð třj +i“1 ei | j “ 1, 2, . . .u; Reference trigger set R Ð t60000 ¨ 22iSAH3ι | i “ 1, 2, . . ., i‹u. +2: for ps, a, hq P S ˆ A ˆ rHs do +3: +Nhps, aq Ð 0, q +Nhps, aq Ð 0; +4: +θhps, aq Ð 0, φhps, aq Ð 0; +5: +Vhpsq Ð H ´ h ` 1, Qhps, aq Ð H ´ h ` 1, V ref +h ps, aq Ð H; +6: +qυhps, aq Ð 0; +7: +qµhps, aq Ð 0, qσhps, aq Ð 0; +8: +µref +h ps, aq Ð 0, σref +h ps, aq Ð 0. +9: end for +10: \\ Main algorithm begins +11: for k “ 1, 2, . . ., K do +12: +for h “ 1, 2, . . ., H do +13: +Observe sk +h; +14: +Take action ak +h “ arg maxa Qhpsk +h, aq; +15: +Receive reward rk +h and observe sk +h`1; +16: +Update accumulators: +n :“ Nhpsk +h, ak +hq ` +Ð 1, qn :“ q +Nhpsk +h, ak +hq ` +Ð 1; +θ :“ θhpsk +h, ak +hq ` +Ð rk +h, φ :“ φhpsk +h, ak +hq ` +Ð prk +hq2; +qυ :“ qυhpsk +h, ak +hq ` +Ð Vh`1psk +h`1q; +qµ :“ qµhpsk +h, ak +hq ` +Ð Vh`1psk +h`1q ´ V ref +h`1psk +h`1q, qσ :“ qσhpsk +h, ak +hq ` +Ð pVh`1psk +h`1q ´ V ref +h`1psk +h`1qq2; +µref :“ µref +h psk +h, ak +hq ` +Ð V ref +h`1psk +h`1q, σref :“ σref +h psk +h, ak +hq ` +Ð pV ref +h`1psk +h`1qq2. +17: +\\ Reaching the end of a stage, update Q-function +18: +if n P L then +19: +Set +prhpsk +h, ak +hq Ð θ +n, z +VarRpsk +h, ak +hq Ð φ +n ´ +ˆ θ +n +˙2 +; +¯b Ð 2 +c +H2ι +qn ; +νref Ð σref +n +´ +ˆµref +n +˙2 +, qν “ qσ +qn ´ +ˆ qµ +qn +˙2 +; +b Ð 4 +c +νrefι +n +` 4 +c +qνι +qn ` 2 +d +z +VarRhι +n +` 90Hι +qn +; +Qhpsk +h, ak +hq Ð min +" +prhpsk +h, ak +hq ` qυ +qn ` ¯b, prhpsk +h, ak +hq ` µref +n ` qµ +qn ` b, Qhpsk +h, ak +hq +* +; +Vhpsk +hq Ð max +a +Qhpsk +h, aq. +20: +\\ Reset intra-stage accumulators +21: +Set q +Nhpsk +h, ak +hq Ð 0, qµhpsk +h, ak +hq Ð 0, qυhpsk +h, ak +hq Ð 0, qσhpsk +h, ak +hq Ð 0. +22: +end if +23: +\\ Update reference value function +24: +if ř +aPA Nhpsk +h, aq P R then V ref +h psk +hq Ð Vhpsk +hq. +25: +end for +26: end for +11 + +Extra notations. +Let V ref,k +h +denote V ref +h +at the beginning of the k-th episode, and V REF +h +:“ V ref,K`1 +h +denote +the final reference value function. Let νref,k +h +, qνk +h, bk +h denote νref, qν, b for the value of Qk +hpsk +h, ak +hq. Let N k +hpsq +denote ř +a Nhps, aq at the beginning of the k-th episode. Let nk +h and qnk +h be the total number of visits to +psk +h, ak +h, hq prior to the current stage and during the stage immediately before the current stage with respect +to the same triple. +Proof sketch. +See Appendix B.3 for the rigorous proof. From Zhang et al. [2020], the regret is roughly +řK +k“1 +řH +h“1pψk +h`1 ` ξk +h`1 ` φk +h`1 ` bk +hq, where +ψk +h`1 « V ref,k +h`1 psk +h`1q ´ V REF +h`1psk +h`1q, +ξk +h`1 « pPsk +h,ak +h,h ´ 1sk +h`1qpV k +h`1 ´ V ‹ +h`1q, +φk +h`1 “ pPsk +h,ak +h,h ´ 1sk +h`1qpV ‹ +h`1 ´ V πk +h`1q. +All these four terms are bounded loosely in Zhang et al. [2020] such that they are all main order terms. +To establish a variance-dependent regret, we prove that only the b term is the main order term after our +aforementioned alterations. The φ term is a martingale and shown to be rOpH2q. For the rest terms, we need +the following argument: +N k +hpsq ě N0pǫq “ rO +ˆH5SA +ǫ2 +˙ +ùñ 0 ď V k +h psq ´ V ‹ +h psq ď ǫ. +Notice that the reference trigger set R in Algorithm 2 is composed of N0pβiq for i P ri‹s where βi :“ H{2i. +There is a constant gap of at least βi‹ between V REF +h +psq and V ‹ +h psq in the worst case, because the number of +updates is capped by i‹. This argument branches into two corollaries. The first one is we can bound value +gaps directly: +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq2 ď rOpH6SAq. +This can be utilized to bound the ξ term. The second one is that, we define +Bref,k +h +psq :“ +i‹ +ÿ +i“1 +βi´1 +1rN0pβi´1q ď N k +hpsq ă N0pβiqs, +then V ref,k +h +psq ´ V REF +h +psq ď Bref,k +h +psq, V ref,k +h +psq ´ V ‹ +h psq ď Bref,k +h +psq ` βi‹ and +ÿ +k,h +Bref,k +h +psk +hq ď rOpH5S2A2i‹q, +ÿ +k,h +pBref,k +h +psk +hqq2 ď rOpH6S2Ai‹q. +So we can directly bound the ψ term. We show that +νref,k +h +« rOpVpPsk +h,ak +h,h, V ‹ +h`1q ` pBref,k +h`1 psk +h`1qq2 ` β2 +i‹q, +qνk +h ď OppBref,k +h`1 psk +h`1qq2 ` β2 +i‹q. +The b term is hence bounded by +rOp +b +VarΣ +KHSA ` +a +H5SAK{22i‹q. +Analogous to the proof of Theorem 7, the difference between VarΣ +K and Var‹K is of a lower order. Finally, +the lower order terms are +rOp +a +H5SAK{22i‹ ` H5S2A2i‹q. +We derive Theorem 10 by choosing the optimal i‹. +12 + +Corollary. +We study deterministic MDPs. +Corollary 11. Assume that Assumption 3 holds. +Let δ P p0, 1q be the confidence parameter and that +H, S, A, K, δ be known. With probability at least 1 ´ δ, the regret of UCB-Advantage-V (Algorithm 2) run +with ι “ 99plnp70002pHSAKq5{δ2q ` 1q and i‹ “ +P +1{2 ¨ log2pK{H5S3Aι2q +T +is bounded by +RegretpKq ď rOp +4? +H15S5A3Kq. +Notice that the regret under Assumption 3 is not constant which we desire, this may be due to some +fundamental limit of model-free algorithms. However, since the research on model-free algorithms is still at +its nascent stage and there lack thorough understanding, our result provides the first look into the potential +of such algorithms. +Intuitively, for any algorithm to have a constant regret on deterministic MDPs, its value functions should +also converge in a constant steps. Previous model-free algorithms all use historical data to estimate value +functions. These data are biased because some of them are not up-to-date, making value functions hard to +converge in a constant steps. Here we identify difficulties for existing algorithms to be variance-dependent +for all K-related terms. +Q-learning (UCB-B) [Jin et al., 2018]. +In their proof of Lemma C.3, when bounding |P3 ´ P4|, there is a +variance-independent 1{ +? +t term in the gap between the estimations and true values. Notice that their result +is possible to be variance-dependent by not loosening +a +H7SAι{t ď H`H6SAι{t above their Equation(C.10) +while introducing a variance-independent K1{4 term. +UCB-Advantage [Zhang et al., 2020]. +There are biases in the reference value functions, because they are +updated for only finite times. If the update is not capped by a threshold, readers can easily verify that the +ψ term will become a variance-independent main order term. +Q-EarlySettled-Advantage [Li et al., 2021]. +There is a same issue about the constant gap between the +reference value and the optimal value when bounding R3 defined in their Equation(39c). +7 +Regret Lower Bounds +We show that for any algorithm and any variance V, there always exists an MDP such that the regret +main order terms of Theorem 7, Corollary 9 and Theorem 10 are tight. +This means that MVP-V and +UCB-Advantage-V are minimax optimal for the class of variance-bounded MDPs. The proofs for this section +are deferred to Appendix B.4. +Theorem 12. Assume S ě 6, A ě 2, H ě 3 tlog2pS ´ 2qu and 0 ă V ď Op1q. For any algorithm π, there +exists an MDP Mπ such that: +‚ It satisfies Conditions 1 and 2; +‚ VarΣ +τ , Var‹ “ ΘpVq for any possible trajectory τ; +‚ For K ě SA, the expected regret of π in Mπ after K episodes satisfies +E +« K +ÿ +k“1 +pV ‹ +1 psk +1q ´ V πk +1 +psk +1qq +ˇˇˇˇˇ Mπ, π +ff +“ Ωp +? +VSAKq. +Theorem 13. Assume S ě 6, A ě 2, H ě 3 tlog2pS ´ 2qu and 0 ă V ď OpH2q. For any algorithm π, there +exists an MDP Mπ such that: +‚ VarΣ +τ , Var‹ “ ΘpVq for any possible trajectory τ; +‚ For K ě HSA, the expected regret of π in Mπ after K episodes satisfies +E +« K +ÿ +k“1 +pV ‹ +1 psk +1q ´ V πk +1 +psk +1qq +ˇˇˇˇˇ Mπ, π +ff +“ Ωp +? +VHSAKq. +13 + +8 +Conclusion +We systematically study variance-dependent regret bounds for MDPs by introducing new notions of variances, +proposing model-based and model-free algorithms respectively, and providing regret lower bounds for the +class of variance-bounded MDPs. Our results improve upon the previous algorithms and achieves minimax +optimal regrets for the class of variance-bounded MDPs. Our model-based algorithm is minimax optimal for +deterministic MDPs. Finally, we identify some possible limit of current model-free algorithms. One possible +future direction is to find a new model-free algorithm with a constant regret for deterministic MDPs. +References +Shipra Agrawal and Randy Jia. Optimistic posterior sampling for reinforcement learning: worst-case regret +bounds. Advances in Neural Information Processing Systems, 30, 2017. +Peter Auer, Thomas Jaksch, and Ronald Ortner. Near-optimal regret bounds for reinforcement learning. +Advances in neural information processing systems, 21, 2008. +Mohammad Gheshlaghi Azar, Ian Osband, and R´emi Munos. Minimax regret bounds for reinforcement +learning. In ICML, 2017. +Yu Bai, Tengyang Xie, Nan Jiang, and Yu-Xiang Wang. Provably efficient q-learning with low switching +cost. Advances in Neural Information Processing Systems, 32, 2019. +Peter L Bartlett and Ambuj Tewari. Regal: A regularization based algorithm for reinforcement learning in +weakly communicating mdps. arXiv preprint arXiv:1205.2661, 2012. +Dimitri Bertsekas. Dynamic programming and optimal control: Volume I, volume 1. Athena scientific, 2012. +Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech +Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016. +Liyu Chen, Mehdi Jafarnia-Jahromi, Rahul Jain, and Haipeng Luo. Implicit finite-horizon approximation +and efficient optimal algorithms for stochastic shortest path. In NeurIPS, 2021. +Christoph Dann, Tor Lattimore, and Emma Brunskill. Unifying pac and regret: Uniform pac bounds for +episodic reinforcement learning. In NIPS, 2017. +Christoph Dann, Lihong Li, Wei Wei, and Emma Brunskill. Policy certificates: Towards accountable rein- +forcement learning. In International Conference on Machine Learning, pages 1507–1516. PMLR, 2019. +Omar Darwiche Domingues, Pierre M´enard, Emilie Kaufmann, and Michal Valko. Episodic reinforcement +learning in finite mdps: Minimax lower bounds revisited. In Vitaly Feldman, Katrina Ligett, and Sivan +Sabato, editors, Proceedings of the 32nd International Conference on Algorithmic Learning Theory, vol- +ume 132 of Proceedings of Machine Learning Research, pages 578–598. PMLR, 16–19 Mar 2021. URL +https://proceedings.mlr.press/v132/domingues21a.html. +Eyal Even-Dar, Shie Mannor, Yishay Mansour, and Sridhar Mahadevan. Action elimination and stopping +conditions for the multi-armed bandit and reinforcement learning problems. Journal of machine learning +research, 7(6), 2006. +Ronan Fruit, Matteo Pirotta, and Alessandro Lazaric. +Near optimal exploration-exploitation in non- +communicating markov decision processes. Advances in Neural Information Processing Systems, 31, 2018a. +Ronan Fruit, Matteo Pirotta, Alessandro Lazaric, and Ronald Ortner. +Efficient bias-span-constrained +exploration-exploitation in reinforcement learning. +In International Conference on Machine Learning, +pages 1578–1586. PMLR, 2018b. +14 + +Aur´elien Garivier, Pierre M´enard, and Gilles Stoltz. Explore first, exploit next: The true shape of regret in +bandit problems. Mathematics of Operations Research, 44, 02 2016. doi: 10.1287/moor.2017.0928. +Nan Jiang and Alekh Agarwal. Open problem: The dependence of sample complexity lower bounds on +planning horizon. In Conference On Learning Theory, pages 3395–3398. PMLR, 2018. +Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan. +Is q-learning provably efficient? +Advances in neural information processing systems, 31, 2018. +Chi Jin, Akshay Krishnamurthy, Max Simchowitz, and Tiancheng Yu. Reward-free exploration for reinforce- +ment learning. In International Conference on Machine Learning, pages 4870–4879. PMLR, 2020. +Tor Lattimore and Csaba Szepesv´ari. Bandit Algorithms. Cambridge University Press, 2020. doi: 10.1017/ +9781108571401. +Gen Li, Laixi Shi, Yuxin Chen, Yuantao Gu, and Yuejie Chi. Breaking the sample complexity barrier to +regret-optimal model-free reinforcement learning. Advances in Neural Information Processing Systems, +34:17762–17776, 2021. +Odalric-Ambrym Maillard, Timothy A Mann, and Shie Mannor. How hard is my mdp?” the distribution- +norm to the rescue”. Advances in Neural Information Processing Systems, 27, 2014. +Andreas Maurer and Massimiliano Pontil. Empirical bernstein bounds and sample-variance penalization. In +COLT, 2009. +Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and +Martin Riedmiller. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013. +Gergely Neu and Ciara Pike-Burke. A unifying view of optimism in episodic reinforcement learning. Advances +in Neural Information Processing Systems, 33:1392–1403, 2020. +Ian Osband and Benjamin Van Roy. Why is posterior sampling better than optimism for reinforcement +learning? In International conference on machine learning, pages 2701–2710. PMLR, 2017. +Ian Osband, Daniel Russo, and Benjamin Van Roy. (more) efficient reinforcement learning via posterior +sampling. Advances in Neural Information Processing Systems, 26, 2013. +Aldo Pacchiano, Philip Ball, Jack Parker-Holder, Krzysztof Choromanski, and Stephen Roberts. On opti- +mism in model-based reinforcement learning. arXiv preprint arXiv:2006.11911, 2020. +Daniel Russo. Worst-case regret bounds for exploration via randomized value functions. Advances in Neural +Information Processing Systems, 32, 2019. +Max Simchowitz and Kevin G Jamieson. Non-asymptotic gap-dependent regret bounds for tabular mdps. +Advances in Neural Information Processing Systems, 32, 2019. +Mohammad Sadegh Talebi and Odalric-Ambrym Maillard. Variance-aware regret bounds for undiscounted +reinforcement learning in mdps. In Algorithmic Learning Theory, pages 770–805. PMLR, 2018. +Jean Tarbouriech, Runlong Zhou, Simon Shaolei Du, Matteo Pirotta, Michael Valko, and Alessandro Lazaric. +Stochastic shortest path: Minimax, parameter-free and towards horizon-free regret. In Neural Information +Processing Systems, 2021. +Andrew J Wagenmaker, Yifang Chen, Max Simchowitz, Simon Du, and Kevin Jamieson. First-order regret in +reinforcement learning with linear function approximation: A robust estimation approach. In International +Conference on Machine Learning, pages 22384–22429. PMLR, 2022. +15 + +Zhihan Xiong, Ruoqi Shen, and Simon S Du. Randomized exploration is near-optimal for tabular mdp. +arXiv preprint arXiv:2102.09703, 2021. +Haike Xu, Tengyu Ma, and Simon Du. Fine-grained gap-dependent bounds for tabular mdps via adaptive +multi-step bootstrap. In Conference on Learning Theory, pages 4438–4472. PMLR, 2021. +Kunhe Yang, Lin Yang, and Simon Du. Q-learning with logarithmic regret. In International Conference on +Artificial Intelligence and Statistics, pages 1576–1584. PMLR, 2021. +Andrea Zanette and Emma Brunskill. Tighter problem-dependent regret bounds in reinforcement learning +without domain knowledge using value function bounds. In ICML, 2019. +Zihan Zhang and Xiangyang Ji. Regret minimization for reinforcement learning by evaluating the optimal +bias function. Advances in Neural Information Processing Systems, 32, 2019. +Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Almost optimal model-free reinforcement learning via reference- +advantage decomposition. Advances in Neural Information Processing Systems, 33:15198–15207, 2020. +Zihan Zhang, Xiangyang Ji, and Simon Shaolei Du. Is reinforcement learning more difficult than bandits? +a near-optimal algorithm escaping the curse of horizon. In COLT, 2021a. +Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Model-free reinforcement learning: from clipped pseudo-regret +to sample complexity. In Proceedings of the 38th International Conference on Machine Learning, pages +12653–12662. PMLR, 2021b. +Zihan Zhang, Xiangyang Ji, and Simon Shaolei Du. Horizon-free reinforcement learning in polynomial time: +the power of stationary policies. In Annual Conference Computational Learning Theory, 2022. +Runlong Zhou, Ruosong Wang, and Simon S Du. Horizon-free reinforcement learning for latent markov +decision processes. arXiv preprint arXiv:2210.11604, 2022. +16 + +Appendix +Table of Contents +A +Technical Lemmas +17 +B +Missing Proofs +18 +A +Technical Lemmas +Lemma 14 (Hoeffding’s Inequality). Let Z, Z1, . . . , Zn be i.i.d. random variables with values in r0, bs and +let δ ą 0. Then we have +P +«ˇˇˇˇˇErZs ´ 1 +n +nÿ +i“1 +Zi +ˇˇˇˇˇ ą b +c +lnp2{δq +2n +ff +ď δ. +Lemma 15 (Bennett’s Inequality, Theorem 3 in Maurer and Pontil [2009]). Let Z, Z1, . . . , Zn be i.i.d. ran- +dom variables with values in r0, bs and let δ ą 0. Define VrZs “ ErpZ ´ ErZsq2s. Then we have +P +«ˇˇˇˇˇErZs ´ 1 +n +n +ÿ +i“1 +Zi +ˇˇˇˇˇ ą +c +2VrZs lnp2{δq +n +` b lnp2{δq +n +ff +ď δ. +Lemma 16 (Theorem 4 in Maurer and Pontil [2009]). Let Z, Z1, . . . , Zn pn ě 2q be i.i.d. random variables +with values in r0, bs and let δ ą 0. Define ¯Z “ 1 +nZi and ˆVn “ 1 +n +řn +i“1pZi ´ ¯Zq2. Then we have +P +» +– +ˇˇˇˇˇErZs ´ 1 +n +n +ÿ +i“1 +Zi +ˇˇˇˇˇ ą +d +2 ˆVn lnp2{δq +n ´ 1 +` 7b lnp2{δq +3pn ´ 1q +fi +fl ď δ. +Lemma 17 (Lemma 11 in Zhang et al. [2021b]). Let pMnqně0 be a martingale such that M0 “ 0 and +|Mn ´ Mn´1| ď c for some c ą 0 and any n ě 1. Let Varn “ řn +k“1 ErpMk ´ Mk´1q2|Fk´1s for n ě 0, where +Fk “ σpM1, . . . , Mkq. Then for any positive integer n and any ǫ, δ ą 0, we have that +P +” +|Mn| ě 2 +a +2Varn lnp1{δq ` 2 +a +ǫ lnp1{δq ` 2c lnp1{δq +ı +ď 2 +ˆ +log2 +ˆnc2 +ǫ +˙ +` 1 +˙ +δ. +Lemma 18 (Lemma 10 in Zhang et al. [2022]). Let X1, X2, . . . be a sequence of random variables taking +values in r0, ls. Define Fk “ σpX1, X2, . . . , Xk´1q and Yk “ ErXk | Fks for k ě 1. For any δ ą 0, we have +that +P +« +Dn, +nÿ +k“1 +Xk ě 3 +n +ÿ +k“1 +Yk ` l lnp1{δq +ff +ď δ, +P +« +Dn, +nÿ +k“1 +Yk ě 3 +nÿ +k“1 +Xk ` l lnp1{δq +ff +ď δ. +Lemma 19 (Lemma 30 in Chen et al. [2021]). For any two random variables X, Y , we have +VpXY q ď 2VpXqpsup |Y |q2 ` 2pErXsq2VpY q. +Consequently, sup |X| ď C implies VpX2q ď 4C2VpXq. +Lemma 20 (Bhatia–Davis Inequality). For any random variable X, VpXq ď psup X ´ErXsqpErXs´inf Xq. +17 + +B +Missing Proofs +B.1 +Justification for Definition 6 +Let Xπ +hpsq denote the random variable of cumulative reward starting from s as the h-th step. +Clearly, +V π +h psq “ ErXπ +h psqs. We denote Varπ +hpsq :“ VpXπ +h psqq. Since π P Π is deterministic, let a “ πhpsq. Law of +total variance states that VpY q “ ErVpY |Xqs ` VpErY |Xsq, so +Varπ +hpsq “ Er„Rhps,aq,s1„Ps,a,hrVpr ` Xπ +h`1ps1qqs ` Vr„Rhps,aq,s1„Ps,a,hpErr ` Xπ +h`1ps1qsq +“ Er„Rhps,aq,s1„Ps,a,hrVpXπ +h`1ps1qqs ` Vr„Rhps,aq,s1„Ps,a,hpr ` ErXπ +h`1ps1qsq +“ Es1„Ps,a,hrVarπ +h`1ps1qs ` Vr„Rhps,aqprq ` Vs1„Ps,a,hpV π +h`1ps1qq +“ Ps,a,hVarπ +h`1 ` VpRhps, aqq ` VpPs,a,h, V π +h`1q. +Let dπ +h P ∆pSq denote the state visitation distribution at the h-th step, i.e., +dπ +hpsq :“ Pπrsh “ ss. +By induction, we can prove that (with ah “ πhpshq) +Varπ +1psq “ +H +ÿ +h“1 +Esh„dπ +hrVpRhpsh, ahqq ` VpPsh,ah,h, V π +h`1qs “ Eπ +« H +ÿ +h“1 +` +VpRhpsh, ahqq ` VpPsh,ah,h, V π +h`1q +˘ +ff +. +B.2 +Model-based Algorithm: MVP-V (Algorithm 1) +Summary of notations. +Let sk +h, ak +h and rk +h denote the state, action and reward at the h-th step of the k-th +episode. Let V k +h psq, Qk +hps, aq, nkps, aq and pP k +s,a,s1 denote Vhpsq, Qhps, aq, nps, aq and pPs,a,s1 at the beginning +of the k-th episode. +Let K be the set of indexes of episodes in which no update is triggered. +By the update rule, it is +obvious that +ˇˇKCˇˇ ď SAplog2pKHq ` 1q. Let h0pkq be the first time an update is triggered in the k-th +episode if there is an update in this episode and otherwise H ` 1. Define X0 :“ tpk, h0pkqq | k P KCu and +X :“ tpk, hq | k P KC, h0pkq ` 1 ď h ď Hu. +Let Ipk, hq :“ +1rpk, hq R Xs. We use the “check” notation to denote the original value timed with Ipk, hq, +e.g., qV k +h :“ V k +h Ipk, hq and qβk +h :“ βk +hIpk, hq. +Proof of Theorem 7. We first run MVP-V (Algorithm 1) with ι “ 99plnpHSAK{δq ` 1q which is large enough +for all the probabilistic inequalities to hold. This choice will make the success probability be 1´polypS, A, H, K, ιqδ. +The lemmas are also proved assuming this choice of ι at first. +Based on Lemma 7 in Zhang et al. [2021a], by Lemmas 23 and 25 we have that +RegretpKq ď +K +ÿ +k“1 +H +ÿ +h“1 +pPsk +h,ak +h qV k +h`1 ´ qV k +h`1psk +h`1qq +loooooooooooooooooooooomoooooooooooooooooooooon +“:M1 +` +K +ÿ +k“1 +H +ÿ +h“1 +qβk +hpsk +h, ak +hq +loooooooooomoooooooooon +“:M2 +` +K +ÿ +k“1 +˜ H +ÿ +h“1 +rpsk +h, ak +hqIpk, hq ´ V πk +1 +psk +1q +¸ +loooooooooooooooooooooooomoooooooooooooooooooooooon +“:M3 +` +ˇˇKCˇˇ . +We utilize Equation (39) in Zhang et al. [2021a]: for any non-negative sequence pwk +hqkPrKs,hPrHs, +K +ÿ +k“1 +H +ÿ +h“1 +Ipk, hq +nkpsk +h, ak +hq ď OpSAιq, +K +ÿ +k“1 +H +ÿ +h“1 +d +wk +hIpk, hq +nkpsk +h, ak +hq ď O +¨ +˝ +g +f +f +eSAι +K +ÿ +k“1 +H +ÿ +h“1 +wk +hIpk, hq ` SAι +˛ +‚. +18 + +Thus by Lemma 25 we have +M2 ď O +¨ +˚ +˚ +˚ +˚ +˚ +˝ +g +f +f +f +f +e +SA +K +ÿ +k“1 +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V k +h`1qqIpk, hq +looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon +«:M4 +ι ` +g +f +f +f +f +e +ΓSA +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, V k +h`1 ´ V ‹ +h`1qIpk, hq +loooooooooooooooooooooooomoooooooooooooooooooooooon +«:M5 +ι +` +g +f +f +f +f +e +SA +K +ÿ +k“1 +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V ‹ +h`1qqIpk, hq +looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon +«:M6 +ι ` ΓSAι2 +˛ +‹‹‹‹‹‚ +. +We need to substitute Ipk, hq with Ipk, h`1q to get the precise definition of M4, M5 and M6. Such substitution +only introduces an error of Op +ˇˇKCˇˇq. By VpX ` Y q ď 2VpXq ` 2VpY q, +M6 ď OpM4 ` M5q, +and +M4 ď OpM5 ` M6q. +• If we use the former relation, M2 ď Op?SAM4ι ` ?ΓSAM5ι ` ΓSAι2q. Plugging in Lemma 26 and +Lemma 28, we have +M2 ď O +¨ +˝a +ΓSAM2ι ` +g +f +f +eSA +K +ÿ +k“1 +Varπkι ` ΓSAι2 +˛ +‚. +Solving the inequality gives +M2 ď O +¨ +˝ +g +f +f +eSA +K +ÿ +k“1 +Varπkι ` ΓSAι2 +˛ +‚. +By Lemma 8 of Zhang et al. [2021a], M1 ď Op?M4ι ` ιq. Further plugging in Lemma 27 gives +RegretpKq ď M1 ` M2 ` M3 ` +ˇˇKCˇˇ ď O +¨ +˝ +g +f +f +eSA +K +ÿ +k“1 +Varπkι ` ΓSAι2 +˛ +‚ď Op +? +Var‹SAKι ` ΓSAι2q. +• If we use the latter relation, M2 ď Op?SAM6ι ` ?ΓSAM5ι ` ΓSAι2q. First plug in Lemma 28, we +have M2 ď Op?ΓSAM2ι ` ?SAM6ι ` ΓSAι2q, which implies +M2 ď Op +a +SAM6ι ` ΓSAι2q. +For the regret, we need M1 ď Op?M4ι ` ιq, Lemmas 27 and 29 and M4 ď OpM5 ` M6q, so +RegretpKq ď Op +a +SAM6ι ` ΓSAι2q ď Op +b +VarΣ +KSAι ` ΓSAι2q. +The above results hold with probability at least 1 ´ 19HS2AKιδ. To establish the final result, we need +to scale δ to make the success probability be 1 ´ δ1. We upper-bound ι by 100pHSAK{ +? +δ ` 1q and solve +the inequality: +1900HS2AK +ˆHSAK +? +δ +` 1 +˙ +δ ď δ1. +Take δ “ pδ1{3000H2S3A2K2q2. By lnpHSAK{pδ{3000H2S3A2K2q2q ď Opιq, we conclude the proof. +19 + +Lemma 21. Define the following events: +E1 :“ +$ +& +%@ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, +ˇˇˇp pP k +s,a,s1 ´ Ps,a,s1qV ‹ +h`1 +ˇˇˇ ď 2 +d +Vp pP k +s,a, V ‹ +h`1qι +nkps, aq +` +14ι +3nkps, aq +, +. +- , (1) +E2 :“ +$ +’ +& +’ +% +@ps, a, kq P S ˆ A ˆ rKs, +ˇˇprkps, aq ´ rps, aq +ˇˇ ď 2 +g +f +f +e z +VarR +kps, aqι +nkps, aq +` +14ι +3nkps, aq +, +/ +. +/ +- +, +(2) +E3 :“ +# +@ps, a, s1, kq P S ˆ A ˆ S ˆ rKs, +ˇˇˇ pP k +s,a,s1 ´ Ps,a,s1 +ˇˇˇ ď +d +2Ps,a,s1ι +nkps, aq ` +1rPs,a,s1 ą 0sι +nkps, aq ++ +, +(3) +E4 :“ +# +@ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, +ˇˇˇp pP k +s,a,s1 ´ Ps,a,s1qV ‹ +h`1 +ˇˇˇ ď +d +2VpPs,a, V ‹ +h`1qι +nkps, aq +` +ι +nkps, aq ++ +. +(4) +We have that +PrE1s ě 1 ´ HSAKιδ, +PrE2s ě 1 ´ SAKιδ, +PrE3s ě 1 ´ S2AKιδ, +PrE4s ě 1 ´ HSAKιδ. +Proof of Lemma 21. PrE1s and PrE2s are direct results by applying Lemma 16 and +1 +x´1 ď 2 +x, taking union +bounds over the mentioned quantifiers and that nkps, aq P t1, 2, . . . , tlog2pHKquu. PrE3s and PrE4s are direct +results by applying Lemma 15 and that Ps,a,s1 “ 0 ùñ +pP k +s,a,s1 “ 0, finally taking union bounds over the +mentioned quantifiers and that nkps, aq P t1, 2, . . . , tlog2pHKquu. +Lemma 22 (Adapted from Lemma 14 in Zhang et al. [2021a] and Lemma 16 in Tarbouriech et al. [2021]). +For any fixed dimension D, let Υ :“ tv P RD : v ě 0, }v}8 ď Bu. For any two constants c1, c2 satisfying +c2 +1 ď c2, let f : ∆prDsq ˆ Υ ˆ R ˆ R Ñ R with fpp, v, n, ιq “ pv ` max +" +c1 +b +Vpp,vqι +n +, c2 Bι +n +* +. Then for all +p P ∆prDsq, v P Υ and n, ι ą 0, +1. fpp, v, n, ιq is non-decreasing in v, i.e., +@pv, v1q P Υ2, v ď v1, it holds that fpp, v, n, ιq ď fpp, v1, n, ιq; +2. fpp, v, n, ιq ě pv ` c1 +2 +b +Vpp,vqι +n +` c2 +2 +Bι +n . +Lemma 23. Conditioned on the successful events of Lemma 21, we have that for any ps, a, h, kq P S ˆ A ˆ +rHs ˆ rKs, Qk +hps, aq ě Q‹ +hps, aq. +Proof of Lemma 23. Let k be fixed and omit it for simplicity. +The proof is conducted by induction in +the order of h “ H ` 1, H, . . . , 1. +QH`1ps, aq “ 0 ě 0 “ Q‹ +H`1ps, aq holds trivially for any ps, aq P +S ˆ A. Now assume Qh`1ps, aq ě Q‹ +h`1ps, aq for any ps, aq P S ˆ A, hence Vh`1psq “ maxaPA Qh`1ps, aq ě +maxaPA Q‹ +h`1ps, aq “ V ‹ +h`1psq for any s P S. +prps, aq ` pPs,aVh`1 ` bhps, aq +“ +¨ +˝prps, aq ` 2 +d +z +VarRps, aqι +nps, aq +` +5ι +nps, aq +˛ +‚` +¨ +˝ pPs,aVh`1 ` 4 +d +Vp pPs,a, Vh`1qι +nps, aq +` +16ι +nps, aq +˛ +‚ +(i) +ě rps, aq ` pPs,aVh`1 ` max +$ +& +%4 +d +Vp pPs,a, Vh`1qι +nps, aq +, +16ι +nps, aq +, +. +- +looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon +“fp pPs,a,Vh`1,ι,nps,aqq +20 + +(ii) +ě rps, aq ` pPs,aV ‹ +h`1 ` max +$ +& +%4 +d +Vp pPs,a, V ‹ +h`1qι +nps, aq +, +16ι +nps, aq +, +. +- +looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon +“fp pPs,a,V ‹ +h`1,ι,nps,aqq +(iii) +ě rps, aq ` pPs,aV ‹ +h`1 ` 2 +d +Vp pPs,a, V ‹ +h`1qι +nps, aq +` +8ι +nps, aq +(iv) +ě rps, aq ` Ps,aV ‹ +h`1 +“ Q‹ +hps, aq, +where (i) is by E2 (Equation (2)); (ii) is by recognizing the last part as the function in Lemma 22, pc1, c2, Bq “ +p4, 16, 1q satisfying that c2 +1 ď c2 and using the first property based on the induction that Vh`1 ě V ‹ +h`1; (iii) +is by the second property in Lemma 22; (iv) is E1 (Equation (1)). So Qhps, aq ě Q‹ +hps, aq. +Lemma 24. With probability at least 1 ´ 2SAKιδ, we have that for any ps, a, kq P S ˆ A ˆ rKs, +z +VarR +kps, aq ď O +ˆ +VpRps, aqq ` +ι +nkps, aq +˙ +. +Proof of Lemma 24. Let s, a, k be fixed and omit k for simplicity. Assume that all the nps, aq realizations of +Rps, aq are prpiq +s,aqnps,aq +i“1 +. We have that +z +VarRps, aq “ +1 +nps, aq +nps,aq +ÿ +i“1 +prpiq +s,aq2 ´ +¨ +˝ +1 +nps, aq +nps,aq +ÿ +i“1 +rpiq +s,a +˛ +‚ +2 +. +From Lemma 15, +P +» +– +ˇˇˇˇˇˇ +1 +nps, aq +nps,aq +ÿ +i“1 +prpiq +s,aq2 ´ ErRps, aq2s +ˇˇˇˇˇˇ +ą +d +2VpRps, aq2qι +nps, aq +` +ι +nps, aq +fi +fl ď δ, +P +» +– +ˇˇˇˇˇˇ +1 +nps, aq +nps,aq +ÿ +i“1 +rpiq +s,a ´ ErRps, aqs +ˇˇˇˇˇˇ +ą +d +2VpRps, aqqι +nps, aq +` +ι +nps, aq +fi +fl ď δ. +By Lemma 19, VpRps, aq2q ď 4VpRps, aqq. So +ˇˇˇz +VarRps, aq ´ VpRps, aqq +ˇˇˇ +ď +ˇˇˇˇˇˇ +1 +nps, aq +nps,aq +ÿ +i“1 +prpiq +s,aq2 ´ ErRps, aq2s +ˇˇˇˇˇˇ +` +ˇˇˇˇˇˇˇ +¨ +˝ +1 +nps, aq +nps,aq +ÿ +i“1 +rpiq +s,a +˛ +‚ +2 +´ ErRps, aqs2 +ˇˇˇˇˇˇˇ +ď 2 +d +2VpRps, aqqι +nps, aq +` +ι +nps, aq ` +ˇˇˇˇˇˇ +1 +nps, aq +nps,aq +ÿ +i“1 +rpiq +s,a ` ErRps, aqs +ˇˇˇˇˇˇ +ˇˇˇˇˇˇ +1 +nps, aq +nps,aq +ÿ +i“1 +rpiq +s,a ´ ErRps, aqs +ˇˇˇˇˇˇ +ď 2 +d +2VpRps, aqqι +nps, aq +` +ι +nps, aq ` 2 +d +2VpRps, aqqι +nps, aq +` +2ι +nps, aq +“ 4 +d +2VpRps, aqqι +nps, aq +` +3ι +nps, aq. +21 + +Using 2?xy ď x ` y, we have +z +VarRps, aq ď VpRps, aqq ` 4 +d +2VpRps, aqqι +nps, aq +` +3ι +nps, aq ď 2VpRps, aqq ` +11ι +nps, aq. +This completes the proof. +Lemma 25. Conditioned on the successful events of Lemmas 21 and 24, we have that for any ps, a, h, kq P +S ˆ A ˆ rHs ˆ rKs +Qk +hps, aq ´ rps, aq ´ Ps,aV k +h`1 ď βk +hps, aq, +where +βk +hps, aq “ O +¨ +˝ +d +VpPs,a, V k +h`1qι +nkps, aq +` +d +VpPs,a, V ‹ +h`1qι +nkps, aq +` +d +ΓVpPs,a, V k +h`1 ´ V ‹ +h`1qι +nkps, aq +` +d +VpRps, aqqι +nkps, aq +` +Γι +nkps, aq +˛ +‚. +Proof of Lemma 25. For any ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, +p pP k +s,a ´ Ps,aqpV k +h`1 ´ V ‹ +h`1q +“ +ÿ +s1PS +p pP k +s,a,s1 ´ Ps,a,s1qrV k +h`1ps1q ´ V ‹ +h`1ps1q ´ Ps,apV k +h`1 ´ V ‹ +h`1qs +(i) +ď +ÿ +s1PS +d +2Ps,a,s1ι +nkps, aq +ˇˇV k +h`1ps1q ´ V ‹ +h`1ps1q ´ Ps,apV k +h`1 ´ V ‹ +h`1q +ˇˇ ` +ÿ +s1PS +1rPs,a,s1 ą 0sι +nkps, aq +ď +d +2ι +nkps, aq +ÿ +s1PS +b +1rPs,a,s1 ą 0sPs,a,s1rV k +h`1ps1q ´ V ‹ +h`1ps1q ´ Ps,apV k +h`1 ´ V ‹ +h`1qs2 ` +Γι +nkps, aq +(ii) +ď +d +2ι +nkps, aq +d ÿ +s1PS +1rPs,a,s1 ą 0s +d ÿ +s1PS +Ps,a,s1rV k +h`1ps1q ´ V ‹ +h`1ps1q ´ Ps,apV k +h`1 ´ V ‹ +h`1qs2 ` +Γι +nkps, aq +“ +d +2ΓVpPs,a, V k +h`1 ´ V ‹ +h`1q +nkps, aq +` +Γι +nkps, aq, +where (i) is by E3 (Equation (3)); (ii) is by Cauchy-Schwarz inequality. While retaining most other steps in +Appendix C.1 of Zhang et al. [2021a] which require E4 (Equation (4)), we have +βk +hps, aq “ O +¨ +˚ +˝ +d +Vp pP k +s,a, V k +h`1qι +nkps, aq +` +d +VpPs,a, V ‹ +h`1qι +nkps, aq +` +d +ΓVpPs,a, V k +h`1 ´ V ‹ +h`1qι +nkps, aq +` +g +f +f +e z +VarR +kps, aqι +nkps, aq +` +Γι +nkps, aq +˛ +‹‚. +Similar as the steps above Equation(36) in Zhang et al. [2021a] which require E3 (Equation (3)), we have +that +Vp pP k +s,a, V k +h`1q ď O +ˆ +VpPs,a, V k +h`1q ` +Γι +nkps, aq +˙ +. +Combined with Lemma 24 we have the desired result. +Lemma 26. Conditioned on the successful events of Lemma 25, with probability at least 1 ´ 5Kιδ, we have +that +M4 ď O +˜ K +ÿ +k“1 +Varπk ` M2 ` SAι2 +¸ +ď OpVar‹K ` M2 ` SAι2q. +22 + +Proof of Lemma 26. Define +bck +hps, aq :“ V k +h psq ´ Ps,aV k +h`1 ´ rps, aq P r´1, 1s, +(5) +which stands for bonus-correction. By Lemma 25, bck +hps, aq ď βk +hps, aq. However, we make the distinction +here to be more precise. Let BCk +hpsq :“ bck +hps, aq ` Ps,aBCk +h`1 with a “ πk +hpsq and boundary condition +BCk +H`1psq :“ 0. We can prove by induction that +BCk +hpsq “ pV k +h ´ V πk +h qpsq P r0, 1s. +(6) +First, +BCk +Hpsq “ bck +Hps, aq “ V k +Hpsq ´ rps, aq “ V k +Hpsq ´ V πk +H psq. +Then assume that BCk +h`1 “ V k +h`1 ´ V πk +h`1, we have +BCk +hpsq “ bck +hps, aq ` Ps,apV k +h`1 ´ V πk +h`1q +“ V k +h psq ´ Ps,aV k +h`1 ´ rps, aq ` Ps,apV k +h`1 ´ V πk +h`1q +“ V k +h psq ´ prps, aq ` Ps,aV πk +h`1q +“ V k +h psq ´ V πk +h psq. +Define a series of random variables and their truncated values: for any k P rKs, +W k :“ +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V πk +h`1qq, +W k :“ mintW k, 50ιu. +Correspondingly, define the following event, which means there is no truncation: +EW :“ tW k “ W k, @k P rKsu. +We now calculate the probability of no truncation happens. For any fixed 1 ď k ď K, +W k +(i) +ď +H +ÿ +h“1 +rPsk +h,ak +hpV πk +h`1q2 ´ pV πk +h`1psk +h`1qq2s ` +H +ÿ +h“1 +rpV πk +h psk +hqq2 ´ pPsk +h,ak +hV πk +h`1q2s ` +H +ÿ +h“1 +rpsk +h, ak +hq ´ pV πk +1 +psk +1qq2 +(ii) +ď 2 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h, pV πk +h`1q2qι ` 6ι ` 2 +H +ÿ +h“1 +pV πk +h psk +hq ´ Psk +h,ak +hV πk +h`1q ` +H +ÿ +h“1 +rpsk +h, ak +hq +(iii) +ď 4 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h, V πk +h`1qι ` 3 +H +ÿ +h“1 +rpsk +h, ak +hq ` 6ι +ď 4 +? +2W kι ` 3 ` 6ι, +where (i) is by Lemma 20, VpRps, aqq ď ErRps, aqs; (ii) is by Lemma 17 with c “ ǫ “ 1, which happens with +probability at least 1 ´ 2ιδ, and a2 ´ b2 ď pa ` bq maxta ´ b, 0u when a, b ě 0; (iii) is by Lemma 19 with +C “ 1. Solving the inequality of W k, we have that +W k ď 50ι. +This means PrEWs ě 1 ´ 2Kιδ. +23 + +From now on, we suppose EW holds. We are ready to bound M4: +M4 “ +K +ÿ +k“1 +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V k +h`1qqIpk, h ` 1q +(i) +ď 2 +K +ÿ +k“1 +H +ÿ +h“1 +pVpRpsk +h, ak +hqq ` VpPsk +h,ak +h, V πk +h`1qqIpk, h ` 1q ` 2 +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, BCk +h`1qIpk, h ` 1q +looooooooooooooooooooooomooooooooooooooooooooooon +“:Z +(7) +ď 2 +K +ÿ +k“1 +W k ` 2Z +(ii) +“ 2 +K +ÿ +k“1 +W k ` 2Z +(iii) +ď 6 +K +ÿ +k“1 +ErW k | Fks ` 2Z ` 100ι +ď 6 +K +ÿ +k“1 +ErW k | Fks ` 2Z ` 100ι2 +“ 6 +K +ÿ +k“1 +Varπk +1 psk +1q ` 2Z ` 100ι2 +ď 6 +K +ÿ +k“1 +Varπk ` 2Z ` 100ι2 +ď 6Var‹K ` 2Z ` 100ι2, +where (i) is by Equation (6) and VpX ` Y q ď 2VpXq ` 2VpY q; (ii) is by EW ; (iii) is by Lemma 18 with +l “ 50ι, which happens with probability at least 1 ´ δ. +It remains to bound the quantity Z we encountered: +Z “ +K +ÿ +k“1 +H +ÿ +h“1 +rPsk +h,ak +hpBCk +h`1q2 ´ pBCk +h`1psk +h`1qq2sIpk, h ` 1q +` +K +ÿ +k“1 +H +ÿ +h“1 +rpBCk +hpsk +hqq2Ipk, hq ´ pPsk +h,ak +hBCk +h`1q2Ipk, h ` 1qs ´ +K +ÿ +k“1 +pBCk +1psk +1qq2 +ď +K +ÿ +k“1 +H +ÿ +h“1 +rPsk +h,ak +hpBCk +h`1q2 ´ pBCk +h`1psk +h`1qq2sIpk, h ` 1q +` +K +ÿ +k“1 +H +ÿ +h“1 +rpBCk +hpsk +hqq2 ´ pPsk +h,ak +hBCk +h`1q2sIpk, h ` 1q ` +ˇˇKCˇˇ +(i) +ď 2 +g +f +f +e2 +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, pBCk +h`1q2qIpk, h ` 1qι ` 6ι +` 2 +K +ÿ +k“1 +H +ÿ +h“1 +maxtBCk +hpsk +hq ´ Psk +h,ak +hBCk +h`1, 0uIpk, h ` 1q ` +ˇˇKCˇˇ +(ii) +ď 4 +g +f +f +e2 +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, BCk +h`1qIpk, h ` 1qι ` 6ι ` 2 +K +ÿ +k“1 +H +ÿ +h“1 +maxtbck +hpsk +h, ak +hq, 0uIpk, h ` 1q ` +ˇˇKCˇˇ +24 + +ď 4 +? +2Zι ` 6ι ` 2 +K +ÿ +k“1 +H +ÿ +h“1 +qβk +hpsk +h, ak +hq ` 2 +ˇˇKCˇˇ +ď 4 +? +2Zι ` 2M2 ` 8SAι, +where (i) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1´2ιδ; (ii) is by Lemma 19 +with C “ 1. Solving the inequality of Z, we have that +Z ď 4M2 ` 48SAι. +(8) +So plugging back into the bound of M4 gives the final result. +Lemma 27. Conditioned on the successful events of Lemma 26, with probability at least 1 ´ 2ιδ, we have +that +M3 ď Op +a +M4ι ` +a +M2ι ` SAιq. +Proof of Lemma 27. +M3 “ +K +ÿ +k“1 +˜ H +ÿ +h“1 +rpsk +h, ak +hqIpk, hq ´ V πk +1 +psk +1qIpk, 1q +¸ +“ +K +ÿ +k“1 +˜ H +ÿ +h“1 +pV πk +h psk +hq ´ Psk +h,ak +hV πk +h`1qIpk, hq ´ V πk +1 +psk +1qIpk, 1q +¸ +ď +K +ÿ +k“1 +H +ÿ +h“1 +pV πk +h`1psk +h`1q ´ Psk +h,ak +hV πk +h`1qIpk, h ` 1q ` +ˇˇKCˇˇ +(i) +ď 2 +g +f +f +e2 +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, V πk +h`1qIpk, h ` 1qι ` 6ι ` SAι +(ii) +ď 4 +a +M4ι ` 4 +? +Zι ` 7SAι +(iii) +ď 4 +a +M4ι ` 8 +a +M2ι ` 35SAι, +where (i) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1 ´ 2ιδ; (ii) is by +Equation (6), VpX ` Y q ď 2VpXq ` 2VpY q and definition of Z (Equation (7)); (iii) is by Equation (8). +Lemma 28. Conditioned on the successful events of Lemma 25, with probability at least 1 ´ 2ιδ, we have +that +M5 ď OpM2 ` SAιq. +Proof of Lemma 28. Define rV k +h “ V k +h ´ V ‹ +h . +M5 “ +K +ÿ +k“1 +H +ÿ +h“1 +rPsk +h,ak +hprV k +h`1q2 ´ prV k +h`1psk +h`1qq2sIpk, h ` 1q +` +K +ÿ +k“1 +H +ÿ +h“1 +rprV k +h psk +hqq2Ipk, hq ´ pPsk +h,ak +h rV k +h`1q2Ipk, h ` 1qs ´ +K +ÿ +k“1 +prV k +1 psk +1qq2 +ď +K +ÿ +k“1 +H +ÿ +h“1 +rPsk +h,ak +hprV k +h`1q2 ´ prV k +h`1psk +h`1qq2sIpk, h ` 1q +25 + +` +K +ÿ +k“1 +H +ÿ +h“1 +rprV k +h psk +hqq2 ´ pPsk +h,ak +h rV k +h`1q2sIpk, h ` 1q ` +ˇˇKCˇˇ +(i) +ď 2 +g +f +f +e2 +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h, prV k +h`1q2qIpk, h ` 1qι ` 6ι +` 2 +K +ÿ +k“1 +H +ÿ +h“1 +maxtrV k +h psk +hq ´ Psk +h,ak +h rV k +h`1, 0uIpk, h ` 1q ` +ˇˇKCˇˇ +(ii) +ď 4 +a +2M5ι ` 2 +K +ÿ +k“1 +H +ÿ +h“1 +qβk +hpsk +h, ak +hq ` 2 +ˇˇKCˇˇ +ď 4 +a +2M5ι ` 2M2 ` 8SAι, +where (i) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1´2ιδ; (ii) is by Lemma 19 +with C “ 1 and the following argument: by Lemma 25, +rV k +h psk +hq ´ Psk +h,ak +h rV k +h`1 ď rQk +hpsk +h, ak +hq ´ Psk +h,ak +h rV k +h`1 ď βk +hpsk +h, ak +hq. +Solving the inequality of M5, we have that +M5 ď 4M2 ` 48SAι. +This completes the proof. +Lemma 29. With probability at least 1 ´ 4Kιδ, we have that for any k P rKs, +VarΣ +pkq ď Opιq. +As a result, +M6 ď VarΣ +K ď OpKιq. +Proof of Lemma 29. For any k P rKs, +VarΣ +pkq ď +H +ÿ +h“1 +rPsk +h,ak +hpV ‹ +h`1q2 ´ pV ‹ +h`1psk +h`1qq2s ` +H +ÿ +h“1 +rpV ‹ +h psk +hqq2 ´ pPsk +h,ak +hV ‹ +h`1q2s ` +H +ÿ +h“1 +rpsk +h, ak +hq ´ pV ‹ +1 psk +1qq2 +(i) +ď 2 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h, pV ‹ +h`1q2qι ` 6ι ` 2 +H +ÿ +h“1 +maxt V ‹ +h psk +hq ´ Psk +h,ak +hV ‹ +h`1 +loooooooooooomoooooooooooon +ěQ‹ +hpsk +h,ak +hq´Psk +h,ak +hV ‹ +h`1ě0 +, 0u ` 1 +(ii) +ď 4 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h, V ‹ +h`1qι ` 7ι ` 2 +H +ÿ +h“1 +pV ‹ +h`1psk +h`1q ´ Psk +h,ak +hV ‹ +h`1q ` 2V ‹ +1 psk +1q +looomooon +ď2 +(iii) +ď 4 +b +2VarΣ +pkqι ` 9ι ` 4 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h, V ‹ +h`1qι ` 12ι +ď 8 +b +2VarΣ +pkqι ` 21ι, +where (i) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1´2ιδ; (ii) is by Lemma 19 +with C “ 1; (iii) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1 ´ 2ιδ. Solving +the inequality of VarΣ +pkq, we have that +VarΣ +pkq ď 170ι. +So taking a union bound over k we have the desired result. +26 + +B.3 +Model-free Algorithm: UCB-Advantage-V (Algorithm 2) +Summary of notations. +Let sk +h, ak +h and rk +h denote the state, action and reward at the h-th step of the +k-th episode. Let V k +h psq, Qk +hps, aq, V ref,k +h +, N k +hps, aq and q +N k +hps, aq denote Vhpsq, Qhps, aq, Nhps, aq and q +Nhps, aq +at the beginning of the k-th episode. Let V REF +h +:“ V ref,K`1 +h +denote the final reference value function. Let +N k +hpsq :“ ř +aPA N k +hps, aq. N K`1 +h +ps, aq denotes the total number of visits of ps, a, hq after all K episodes are +done. +Define e1 “ H and ei`1 “ tp1 ` 1{Hqeiu. The definition of stages is with respect to the triple ps, a, hq. +For any fixed pair of k and h, we say that pk, hq falls in the j-th stage of ps, a, hq if and only if ps, aq “ psk +h, ak +hq +and the total visit number of psk +h, ak +hq after the k-th episode is in přj´1 +i“1 ei, řj +i“1 eis. +Let qυk +h, qµk +h, qσk +h, µref,k +h +, σref,k +h +, z +VarR +k +h, ¯bk +h, νref,k +h +, qνk +h and bk +h denote qυ, qµ, qσ, µref, σref, z +VarR, ¯b, νref, qν and b +calculated for the value of Qk +hpsk +h, ak +hq. +For each k and h, let nk +h be the total number of visits to psk +h, ak +h, hq prior to the current stage with respect +to the same triple and let nk +h be the number of visits to the same triple during the stage immediately before +the current stage. Let lk +h,i and qlk +h,i denote the index of the i-th episode among the nk +h and qnk +h episodes defined +above, respectively. When h and k are clear from the context, we use li and qli for short. +Proof of Theorem 10. We first run UCB-Advantage-V (Algorithm 2) with ι “ 99plnpHSAK{δq ` 1q which is +large enough for all the probabilistic inequalities to hold. This choice will make the success probability be +1 ´ polypS, A, H, K, ιqδ. The lemmas are also proved assuming this choice of ι at first. +Define +ψk +h`1 :“ 1 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpV ref,li +h`1 ´ V REF +h`1q, +ξk +h`1 :“ 1 +qnk +h +qnk +h +ÿ +i“1 +rPsk +h,ak +h,hpV ref,qli +h`1 ´ V ‹ +h`1q ´ pV ref,qli +h`1 ps +qli +h`1q ´ V ‹ +h`1ps +qli +h`1qqs, +φk +h`1 :“ Psk +h,ak +h,hpV ‹ +h`1 ´ V πk +h`1q ´ pV ‹ +h`1psk +h`1q ´ V πk +h`1psk +h`1qq. +Combining Section 4.2 in Zhang et al. [2020] with Lemmas 30 to 33 and 36 to 41, we have that with probability +at least 1 ´ 35HSAKιδ, +RegretpKq ď +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +pψk +h`1 ` ξk +h`1 ` φk +h`1 ` 2bk +hq +ď Op +b +VarΣ +KHSAι ` +a +H5SAKι2{22i‹ ` H5S2A2i‹ι2q. +Taking i‹ “ +P +1{2 ¨ log2pK{H5S3Aι2q +T +, we have: +RegretpKq ď Op +b +VarΣ +KHSAι ` +4? +H15S5A3Kι6q. +Now we apply Lemma 42. With probability at least 1 ´ 46HSAKιδ, +RegretpKq ď O +¨ +˝ +g +f +f +eHSAKι +K +ÿ +k“1 +Varπk ` +4? +H15S5A3Kι6 +˛ +‚ď Op +? +Var‹HSAKι ` +4? +H15S5A3Kι6q. +The final result is established by scaling δ to make the success probability be 1 ´ δ1. We upper-bound ι +by 100pHSAK{ +? +δ ` 1q and solve the inequality: +4600HSAK +ˆHSAK +? +δ +` 1 +˙ +δ ď δ1. +Take δ “ pδ1{7000H2S2A2K2q2. By lnpHSAK{pδ{7000H2S2A2K2q2q ď Opιq, we conclude the proof. +27 + +Lemma 30. With probability at least 1 ´ 15HSAKιδ, we have that for any ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, +Q‹ +hps, aq ď Qk`1 +h +ps, aq ď Qk +hps, aq. +Proof of Lemma 30. Recall that the update rule is: +Qhpsk +h, ak +hq Ð min +" +prhpsk +h, ak +hq ` qυ +qn ` ¯b +loooooooooomoooooooooon +① +, prhpsk +h, ak +hq ` µref +n ` qµ +qn ` b +looooooooooooooomooooooooooooooon +② +, Qhpsk +h, ak +hq +* +. +(9) +We prove by induction on k. Clearly for k “ 1 the argument is true. +For case ① in Equation (9), we have that (omit the subscripts of h and superscripts of k for simplicity) +Qk`1 +h +ps, aq “ rhps, aq ` 1 +qn +qn +ÿ +i“1 +V li +h`1psli +h`1q ` pprhps, aq ´ rhps, aqq ` ¯b +(i) +ě rhps, aq ` 1 +qn +qn +ÿ +i“1 +V ‹ +h`1psli +h`1q ` pprhps, aq ´ rhps, aqq ` ¯b +(ii) +ě rhps, aq ` Ps,a,hV ‹ +h`1 ´ +c +H2ι +2qn ` pprhps, aq ´ rhps, aqq ` ¯b +(iii) +ě rhps, aq ` Ps,a,hV ‹ +h`1 ´ +c +H2ι +2qn ´ +c ι +2n ` ¯b +ě Q‹ +h`1ps, aq, +where (i) is by induction V u ě V ‹ for any 1 ď u ď k; (ii) is by Lemma 14 with b “ H, which holds with +probability at least 1 ´ δ; (iii) is by Lemma 14 with b “ 1, which holds with probability at least 1 ´ δ. +Define +χ1 :“ 1 +n +n +ÿ +i“1 +pV ref,li +h`1 psli +h`1q ´ Ps,a,hV ref,li +h`1 q, +χ2 :“ 1 +qn +n +ÿ +i“1 +rpV li +h`1 ´ V ref,li +h`1 qpsli +h`1q ´ Ps,a,hpV li +h`1 ´ V ref,li +h`1 qs. +For case ② in Equation (9), we have that +Qk`1 +h +ps, aq “ prhps, aq ` Ps,a,h +˜ +1 +n +n +ÿ +i“1 +V ref,li +h`1 +¸ +` Ps,a,h +˜ +1 +qn +qn +ÿ +i“1 +pV +qli +h`1 ´ V ref,qli +h`1 q +¸ +` χ1 ` χ2 ` b +(i) +ě rhps, aq ` Ps,a,h +˜ +1 +qn +qn +ÿ +i“1 +V +qli +h`1 +¸ +` χ1 ` χ2 ` prhps, aq ´ prhps, aqq ` b +(ii) +ě rhps, aq ` Ps,a,hV ‹ +h`1 ` χ1 ` χ2 ` prhps, aq ´ prhps, aqq ` b +“ Q‹ +hps, aq ` χ1 ` χ2 ` prhps, aq ´ prhps, aqq ` b, +where (i) is by that V ref,u +h`1 is non-increasing in u; (ii) is by the induction V u ě V ‹ for any 1 ď u ď k. +From Lemma 17 with c “ H, ǫ “ c2, we have that with probability at least 1 ´ 2ιδ, +|χ1| ď 1 +n +¨ +˚ +˚ +˚ +˚ +˝ +2 +g +f +f +f +f +e +2 +n +ÿ +i“1 +VpPs,a,h, V ref,li +h`1 q +looooooooooomooooooooooon +“:X +ι ` 6Hι +˛ +‹‹‹‹‚ +. +(10) +28 + +Define +χ3 :“ +nÿ +i“1 +rPs,a,hpV ref,li +h`1 q2 ´ pV ref,li +h`1 psli +h`1qq2s, +χ4 :“ 1 +n +˜ n +ÿ +i“1 +V ref,li +h`1 psli +h`1q +¸2 +´ 1 +n +˜ n +ÿ +i“1 +Ps,a,hV ref,li +h`1 +¸2 +, +χ5 :“ 1 +n +˜ n +ÿ +i“1 +Ps,a,hV ref,li +h`1 +¸2 +´ +nÿ +i“1 +pPs,a,hV ref,li +h`1 q2, +then it is easy to verify that +X “ nνref ` χ3 ` χ4 ` χ5. +(11) +By Lemma 17 with c “ H2, ǫ “ c2, and Lemma 19 with C “ H, we have that with probability at least +1 ´ 2ιδ, +χ3 ď 2 +g +f +f +e2 +nÿ +i“1 +VpPs,a,h, pV ref,li +h`1 q2qι ` 6H2ι ď 4H +? +2Xι ` 6H2ι. +(12) +By Lemma 17 with c “ H, ǫ “ c2, we have that with probability at least 1 ´ 2ιδ, +χ4 ď 1 +n +ˇˇˇˇˇ +nÿ +i“1 +pV ref,li +h`1 psli +h`1q ` Ps,a,hV ref,li +h`1 q +ˇˇˇˇˇ +ˇˇˇˇˇ +n +ÿ +i“1 +pV ref,li +h`1 psli +h`1q ´ Ps,a,hV ref,li +h`1 q +ˇˇˇˇˇ ď 2Hp2 +? +2Xι ` 6Hιq. +(13) +By Cauchy-Schwarz inequality, χ5 ď 0. Thus, X ď nνref ` 18H2ι ` 8H +? +2Xι. Solving the inequality, +X ď 2nνref ` 164H2ι. +Plugging back into Equation (10), we have +χ1 ď 4 +c +νrefι +n +` p4 +? +82 ` 6qHι +n +. +By a similar reasoning, we have that with probability at least 1 ´ 6ιδ, +χ2 ď 4 +c +qνι +qn ` p4 +? +82 ` 6qHι +qn +. +By Lemma 16 and +1 +x´1 ď 2 +x, we have that with probability at least 1 ´ δ, +|rhps, aq ´ prhps, aq| ď 2 +d +z +VarRhps, aqι +n +` 14ι +3n . +(14) +Therefore, we have b ě |χ1| ` |χ2| ` |rhps, aq ´ prhps, aq|, which means Qk`1 +h +ps, aq ě Q‹ +hps, aq. +Lemma 31 (Adapted from Lemma 5 and Corollary6 in Zhang et al. [2020], and Corollary6 in Chen et al. +[2021]). Conditioned on the successful events of Lemma 30, with probability at least 1 ´ HKδ, we have that +for any ǫ P p0, Hs and any h P rHs, +K +ÿ +k“1 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě ǫs ď 60000H5SAι +ǫ2 +“: N0pǫq. +As a result, for every state s we have that +N k +hpsq ě N0pǫq ùñ 0 ď V k +h psq ´ V ‹ +h psq ď ǫ. +29 + +Proof of Lemma 31. To derive the constant 60000, we only need to solve the inequality: +K +ÿ +k“1 +1rδk +h ě ǫs ď +řK +k“1 +1rδk +h ě ǫsδk +h +ǫ +ď +240H5{2 +b +}w}8 SAι řK +k“1 +1rδk +h ě ǫs ` 3SAH3 }w}8 +ǫ +which is below Equation (48) in Zhang et al. [2020], using x ď a?x ` b ùñ x ď a2 ` 2b. +The second part can be proven in a similar way as Corollary6 in Chen et al. [2021]. +Lemma 32. Conditioned on the successful events of Lemma 31, we have that +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq ď Op +? +H7SAKιq, +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq2 ď OpH6SAι2q. +Proof of Lemma 32. Let c be a fixed constant, then +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq “ +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq1rV k +h psk +hq ´ V ‹ +h psk +hq ă cs +` +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq1rV k +h psk +hq ´ V ‹ +h psk +hq ě cs +(i) +ď cHK ` +K +ÿ +k“1 +H +ÿ +h“1 +ż H +0 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs 1rpV k +h psk +hq ´ V ‹ +h psk +hqq ě cs dx +“ cHK ` +K +ÿ +k“1 +H +ÿ +h“1 +ż H +c +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs dx +“ cHK ` +ż H +c +˜ K +ÿ +k“1 +H +ÿ +h“1 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs +¸ +dx +(ii) +ď cHK ` +ż H +c +O +ˆH6SAι +x2 +˙ +dx +ď O +ˆ +cHK ` H6SAι +c +˙ +, +where (i) is by n “ +ş8 +0 +1rn ě xs dx for any non-negative real n and V ‹ +h ď V k +h ď H (Lemma 30); (ii) is by +Lemma 31. Taking c “ +a +H5SAι{K gives the first result. +Similarly, +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq2 +“ +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq2 +1rV k +h psk +hq ´ V ‹ +h psk +hq ă cs +` +K +ÿ +k“1 +H +ÿ +h“1 +pV k +h psk +hq ´ V ‹ +h psk +hqq2 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě cs +ď c2HK ` +K +ÿ +k“1 +H +ÿ +h“1 +˜ż H +0 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs 1rpV k +h psk +hq ´ V ‹ +h psk +hqq ě cs dx +¸2 +30 + +“ c2HK ` +K +ÿ +k“1 +H +ÿ +h“1 +˜ż H +c +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs dx +¸2 +“ c2HK ` +ż H +c +˜ż H +c +˜ K +ÿ +k“1 +H +ÿ +h“1 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs 1rV k +h psk +hq ´ V ‹ +h psk +hq ě ys +¸ +dx +¸ +dy +“ c2HK ` +ż H +c +˜ż y +c +˜ K +ÿ +k“1 +H +ÿ +h“1 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě ys +¸ +dx ` +ż H +y +˜ K +ÿ +k“1 +H +ÿ +h“1 +1rV k +h psk +hq ´ V ‹ +h psk +hq ě xs +¸ +dx +¸ +dy +ď c2HK ` +ż H +c +˜ +py ´ cqO +ˆH6SAι +y2 +˙ +` +ż H +y +O +ˆH6SAι +x2 +˙ +dx +¸ +dy +ď c2HK ` O +˜ +H6SAι +ż H +c +dy +y +¸ +“ O +ˆ +c2HK ` H6SAι ln H +c +˙ +, +and taking c “ 1{ +? +HK gives the second result. +Lemma 33. Define βi :“ H{2i for i P t0, 1, . . . , i‹u, N 0 +0 :“ 0 and N i +0 :“ N0pβiq “ 60000¨22iSAH3ι (defined +in Lemma 31) for i P ri‹s. Define +Bref,k +h +psq :“ +i‹ +ÿ +i“1 +βi´1 +1rN i´1 +0 +ď N k +hpsq ă N i +0s. +Conditioned on the successful events of Lemma 31, we have that +V ref,k +h +psq ´ V REF +h +psq ď Bref,k +h +psq, +V ref,k +h +psq ´ V ‹ +h psq ď Bref,k +h +psq ` βi‹, +and +K +ÿ +k“1 +H +ÿ +h“1 +Bref,k +h +psk +hq ď OpH5S2A2i‹ιq, +K +ÿ +k“1 +H +ÿ +h“1 +pBref,k +h +psk +hqq2 ď OpH6S2Ai‹ιq. +Proof of Lemma 33. For i ď i‹ ´ 1, by Lemma 31, if N k +hpsq ě N i +0 “ N0pβiq then V k +h psq ´ V ‹ +h psq ď βi. Let k0 +be the minimum k such that N k +hpsq ě N i +0. By the updating rule in Algorithm 2 and non-increasing property +of V k (Lemma 30), it must satisfy that V k +h psq ď V ref,k +h +psq ď V k0 +h psq. Since V ‹ +h psq ď V REF +h +psq (Lemma 30), we +have that +V ref,k +h +psq ´ V REF +h +psq ď V k0 +h psq ´ V ‹ +h psq ď βi. +If N k +hpsq ě N i‹ +0 +“ N0pβi‹q then V ref,k +h +psq “ V REF +h +psq and V ref,k +h +psq ´ V ‹ +h psq ď βi‹, which corresponds to +Bref,k +h +psq “ 0. Since the indicator functions are disjoint, we have the first part of results. +The remaining result is proven by: +K +ÿ +k“1 +H +ÿ +h“1 +Bref,k +h +psk +hq “ +ÿ +sPS +K +ÿ +k“1 +H +ÿ +h“1 +i‹ +ÿ +i“1 +H +2i´1 +1rs “ sk +h, N i´1 +0 +ď N k +hpsq ă N i +0s +31 + +ď +ÿ +sPS +H +ÿ +h“1 +i‹ +ÿ +i“1 +H +2i´1 N i +0 +ď O +˜ +SH +i‹ +ÿ +i“1 +H +2i ¨ 22iSAH3ι +¸ +ď OpH5S2A2i‹ιq; +K +ÿ +k“1 +H +ÿ +h“1 +pBref,k +h +psk +hqq2 “ +K +ÿ +k“1 +H +ÿ +h“1 +i‹ +ÿ +i“1 +β2 +i´1 +1rN i´1 +0 +ď N k +hpsq ă N i +0s +ď O +˜ +SH +i‹ +ÿ +i“1 +H2 +22i ¨ 22iSAH3ι +¸ +ď OpH6S2Ai‹ιq. +This completes the proof. +Lemma 34 (Lemma 11 in Zhang et al. [2020]). For any non-negative weights pwhps, aqqsPS,aPA,hPrHs and +α P p0, 1q, it holds that +K +ÿ +k“1 +H +ÿ +h“1 +whpsk +h, ak +hq +pnk +hqα +ď +2α +1 ´ α +ÿ +s,a,h +whps, aqpN K`1 +h +ps, aqq1´α, +K +ÿ +k“1 +H +ÿ +h“1 +whpsk +h, ak +hq +pqnk +hqα +ď 22αHα +1 ´ α +ÿ +s,a,h +whps, aqpN K`1 +h +ps, aqq1´α. +In the case α “ 1, it holds that +K +ÿ +k“1 +H +ÿ +h“1 +whpsk +h, ak +hq +nk +h +ď 2 +ÿ +s,a,h +whps, aq ln N K`1 +h +ps, aq, +K +ÿ +k“1 +H +ÿ +h“1 +whpsk +h, ak +hq +qnk +h +ď 4H +ÿ +s,a,h +whps, aq ln N K`1 +h +ps, aq. +Lemma 35. For any non-negative sequence pXk +hqkPrKs,hPrHs, we have that +K +ÿ +k“1 +H +ÿ +h“1 +1 +nk +h +nk +h +ÿ +i“1 +X +lk +h,i +h +ď 2ι +K +ÿ +k“1 +H +ÿ +h“1 +Xk +h, +K +ÿ +k“1 +H +ÿ +h“1 +1 +qnk +h +qnk +h +ÿ +i“1 +X +qlk +h,i +h +ď +ˆ +1 ` 1 +H +˙ K +ÿ +k“1 +H +ÿ +h“1 +Xk +h. +Proof of Lemma 35. Refer to Equation (58) in Zhang et al. [2020] for the first inequality. Refer to Equa- +tion (15) and the paragraph below it in Zhang et al. [2020] for the second inequality. +Lemma 36. Conditioned on the successful events of Lemma 33, with probability at least 1 ´ δ, we have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +ψk +h`1 ď OpH5S2A2i‹ιq. +32 + +Proof of Lemma 36. Since ψk +h is non-negative and p1 ` 1{Hqh´1 ď 3 when h ď H, we have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +ψk +h`1 ď O +˜ K +ÿ +k“1 +H +ÿ +h“1 +ψk +h`1 +¸ +“ O +¨ +˝ +K +ÿ +k“1 +H +ÿ +h“1 +1 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpV +ref,lk +h,i +h`1 +´ V REF +h`1q +˛ +‚ +(i) +ď O +¨ +˝ +K +ÿ +k“1 +H +ÿ +h“1 +1 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hB +ref,lk +h,i +h`1 +˛ +‚ +(ii) +ď O +˜ K +ÿ +k“1 +H +ÿ +h“1 +Psk +h,ak +h,hBref,k +h`1 +¸ +(iii) +ď O +˜ K +ÿ +k“1 +H +ÿ +h“1 +Bref,k +h +psk +hq ` Hι +¸ +(iv) +ď OpH5S2A2i‹ιq, +where (i) is by Lemma 33; (ii) is by Lemma 35; (iii) is by Lemma 18 with l “ H, which holds with probability +at least 1 ´ δ; (iv) is by Lemma 33. +Lemma 37. Conditioned on the successful events of Lemma 32, with probability at least 1 ´ 5HSAιδ, we +have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +ξk +h`1 ď OpH7{2SAι3{2q. +Proof of Lemma 37. We borrow the beginning part of proof of Lemma 15 in Zhang et al. [2020], and perform +more fine-grained analyses on the remaining part. Let xk +h be the number of elements in current stage with +respect to psk +h, ak +h, hq. Define +θj +h`1 :“ +ˆ +1 ` 1 +H +˙h´1 +K +ÿ +k“1 +1 +qnk +h +qnk +h +ÿ +i“1 +1rqlk +h,i “ js, +rθj +h`1 :“ +ˆ +1 ` 1 +H +˙h´1 +Y +p1 ` 1{Hqxj +h +] +xj +h +ď 3, +and +K :“ tpk, hq | θk +h`1 “ rθk +h`1u, +KK +h ps, aq :“ tk | psk +h, ak +hq “ ps, aq, k is in the second last stage of ps, a, hqu. +Let θh`1ps, aq and rθh`1ps, aq denote θk +h`1 and rθk +h`1 respectively for some k P KK +h ps, aq. By Equation (61) in +Zhang et al. [2020], +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +ξk +h`1 ď +K +ÿ +k“1 +H +ÿ +h“1 +rθk +h`1rPsk +h,ak +h,hpV k +h`1 ´ V ‹ +h`1q ´ pV k +h`1psk +h`1q ´ V ‹ +h`1psk +h`1qqs +looooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooon +“:① +` +ÿ +pk,hqPK +pθk +h`1 ´ rθk +h`1qrPsk +h,ak +h,hpV k +h`1 ´ V ‹ +h`1q ´ pV k +h`1psk +h`1q ´ V ‹ +h`1psk +h`1qqs +looooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooon +“:② +. +33 + +We now bound both terms: +① +(i) +ď O +¨ +˚ +˚ +˚ +˚ +˝ +g +f +f +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h,h, pV k +h`1 ´ V ‹ +h`1qq +loooooooooooooooooooooomoooooooooooooooooooooon +“:Z +ι ` Hι +˛ +‹‹‹‹‚ +, +② +(ii) +“ +ÿ +s,a,h +pθh`1ps, aq ´ rθh`1ps, aqq +ÿ +kPKK +h ps,aq +rPsk +h,ak +h,hpV k +h`1 ´ V ‹ +h`1q ´ pV k +h`1psk +h`1q ´ V ‹ +h`1psk +h`1qqs +ď +ÿ +s,a,h +ˇˇˇθh`1ps, aq ´ rθh`1ps, aq +ˇˇˇ +ˇˇˇˇˇˇ +ÿ +kPKK +h ps,aq +rPsk +h,ak +h,hpV k +h`1 ´ V ‹ +h`1q ´ pV k +h`1psk +h`1q ´ V ‹ +h`1psk +h`1qqs +ˇˇˇˇˇˇ +(iii) +ď O +¨ +˝ ÿ +s,a,h +¨ +˝ +d +ÿ +kPKK +h ps,aq +VpPsk +h,ak +h,h, V k +h`1 ´ V ‹ +h`1qι ` Hι +˛ +‚ +˛ +‚ +(iv) +ď O +¨ +˝ +d +HSAι +ÿ +s,a,h +ÿ +kPKK +h ps,aq +VpPsk +h,ak +h,h, V k +h`1 ´ V ‹ +h`1q ` H2SAι +˛ +‚ +(v) +ď Op +? +HSAZι ` H2SAιq, +where (i) is by Lemma 17 with c “ 3H, ǫ “ c2, which holds with probability at least 1 ´ 2ιδ; (ii) is by the +step above Equation (63) in Zhang et al. [2020]; (iii) is by +ˇˇˇθh`1ps, aq ´ rθh`1ps, aq +ˇˇˇ ď 3 and Lemma 17 with +c “ H, ǫ “ c2, which holds with probability at least 1 ´ 2HSAιδ; (iv) is by Cauchy-Schwarz inequality; (v) +is by the following argument: for any non-negative sequence pXk +hqkPrKs,hPrHs, +ÿ +s,a,h +ÿ +kPKK +h ps,aq +Xk +h “ +K +ÿ +k“1 +H +ÿ +h“1 +Xk +h +ÿ +s,a,h1 +ÿ +k1PKK +h1ps,aq +1rpk1, h1q “ pk, hqs +“ +K +ÿ +k“1 +H +ÿ +h“1 +Xk +h +ÿ +s,a +1rk P KK +h ps, aqs +ď +K +ÿ +k“1 +H +ÿ +h“1 +Xk +h +ÿ +s,a +1rpsk +h, ak +hq “ ps, aqs +ď +K +ÿ +k“1 +H +ÿ +h“1 +Xk +h. +It remains to bound Z: +Z ď +K +ÿ +k“1 +H +ÿ +h“1 +Psk +h,ak +h,hpV k +h`1 ´ V ‹ +h`1q2 +(i) +ď O +˜ K +ÿ +k“1 +H +ÿ +h“1 +pV k +h`1psk +h`1q ´ V ‹ +h`1psk +h`1qq2 ` H2ι +¸ +(ii) +ď OpH6SAι2q, +where (i) is by Lemma 18 with l “ H2, which holds with probability at least 1 ´ δ; (ii) is by Lemma 32. +34 + +Lemma 38. With probability at least 1 ´ 6ιδ, we have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +φk +h`1 ď OpHιq. +Proof of Lemma 38. Since p1 ` 1{Hqh´1 ď 3 when h ď H, we have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +φk +h`1 “ +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +rPsk +h,ak +h,hpV ‹ +h`1 ´ V πk +h`1q ´ pV ‹ +h`1psk +h`1q ´ V πk +h`1psk +h`1qqs +(i) +ď O +¨ +˚ +˚ +˚ +˚ +˝ +g +f +f +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h,h, V ‹ +h`1 ´ V πk +h`1q +loooooooooooooooooooomoooooooooooooooooooon +“:Y +ι ` Hι +˛ +‹‹‹‹‚ +, +where (i) is by Lemma 17 with c “ 3H, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ. Next we +bound Y : +Y “ +K +ÿ +k“1 +H +ÿ +h“1 +rPsk +h,ak +h,hpV ‹ +h`1 ´ V πk +h`1q2 ´ pV ‹ +h`1psk +h`1q ´ V πk +h`1psk +h`1qq2s +` +K +ÿ +k“1 +H +ÿ +h“1 +tpV ‹ +h psk +hq ´ V πk +h psk +hqq2 ´ rPsk +h,ak +h,hpV ‹ +h`1 ´ V πk +h`1qs2u ´ pV ‹ +1 psk +1q ´ V πk +1 +psk +1qq2 +(i) +ď 2 +g +f +f +e2 +K +ÿ +k“1 +H +ÿ +h“1 +VpPsk +h,ak +h,h, pV ‹ +h`1 ´ V πk +h`1q2qι ` 6H2ι +` 2H +K +ÿ +k“1 +H +ÿ +h“1 +maxtV ‹ +h psk +hq ´ V πk +h psk +hq ´ Psk +h,ak +h,hpV ‹ +h`1 ´ V πk +h`1q +loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon +ěQ‹ +hpsk +h,ak +hq´rhpsk +h,ak +hq´Psk +h,ak +h,hV ‹ +h`1“0 +, 0u +(ii) +ď 4H +? +2Y ι ` 6H2ι ` 2H +K +ÿ +k“1 +H +ÿ +h“1 +rV ‹ +h`1psk +h`1q ´ V πk +h`1psk +h`1q ´ Psk +h,ak +h,hpV ‹ +h`1 ´ V πk +h`1qs +` 2HpV ‹ +1 psk +1q ´ V πk +1 +psk +1qq +loooooooooooooomoooooooooooooon +ď2H2 +(iii) +ď 4H +? +2Y ι ` 8H2ι ` 4H +? +2Y ι ` 12H2ι +ď 8H +? +2Y ι ` 20H2ι, +where (i) is by Lemma 17 with c “ H2, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ, and +a2 ´ b2 ď pa ` bq maxta ´ b, 0u when a, b ě 0; (ii) is by Lemma 19 with C “ H; (iii) is by by Lemma 17 with +c “ H, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ. Solving the inequality of Y , we have that +Y ď 168H2ι. +Plugging Y back gives the desired result. +Lemma 39. Conditioned on the successful events of Lemma 33, with probability at least 1 ´ 4ιδ, we have +that for any pk, hq P rKs ˆ rHs +νref,k +h +ď O +¨ +˝VpPsk +h,ak +h,h, V ‹ +h`1q ` +H2ι ` řnk +h +i“1 Psk +h,ak +h,hpBref,li +h`1 q2 +nk +h +` β2 +i‹ +˛ +‚. +35 + +Proof of Lemma 39. Let pk, hq be fixed. We prove by first bounding νref,k +n +´ +1 +nk +h +řnk +h +i“1 VpPsk +h,ak +h,h, V ref,li +h`1 q. By +Equation (11), +νref,k +n +´ 1 +nk +h +nk +h +ÿ +i“1 +VpPsk +h,ak +h,h, V ref,li +h`1 q +looooooooooooomooooooooooooon +“X(abusing notation) +“ ´χ3 ` χ4 ` χ5 +nk +h +. +Since we can use Equations (12) and (13), we only need to bound ´χ5. +´χ5 “ +nk +h +ÿ +i“1 +pPsk +h,ak +h,hV ref,li +h`1 q2 ´ 1 +nk +h +¨ +˝ +nk +h +ÿ +i“1 +Psk +h,ak +h,hV ref,li +h`1 +˛ +‚ +2 +(i) +ď +nk +h +ÿ +i“1 +pPsk +h,ak +h,hV ref,li +h`1 q2 ´ 1 +nk +h +¨ +˝ +nk +h +ÿ +i“1 +Psk +h,ak +h,hV REF +h`1 +˛ +‚ +2 +“ +nk +h +ÿ +i“1 +pPsk +h,ak +h,hV ref,li +h`1 ` Psk +h,ak +h,hV REF +h`1 qpPsk +h,ak +h,hV ref,li +h`1 ´ Psk +h,ak +h,hV REF +h`1q +ď 2H +nk +h +ÿ +i“1 +pPsk +h,ak +h,hV ref,li +h`1 ´ Psk +h,ak +h,hV REF +h`1 q +(ii) +ď 2H +nk +h +ÿ +i“1 +Psk +h,ak +h,hBref,li +h`1 , +where (i) is by V ref,li +h`1 ě V REF +h`1 (Lemma 30); (ii) is by Lemma 33. Combining bounds of χ3 and χ4, we have: +νref,k +n +´ X +nk +h +ď +8H +? +2Xι ` 18H2ι ` 2H řnk +h +i“1 Psk +h,ak +h,hBref,li +h`1 +nk +h +. +Since 8H +? +2Xι ď X ` 32H2ι, we have: +νref,k +n +´ 2X +nk +h +ď O +¨ +˝H2ι ` H řnk +h +i“1 Psk +h,ak +h,hBref,li +h`1 +nk +h +˛ +‚. +For the desired result, we finally bound: +X +nk +h +´ 2VpPsk +h,ak +h,h, V ‹ +h`1q “ 1 +nk +h +nk +h +ÿ +i“1 +pVpPsk +h,ak +h,h, V ref,li +h`1 q ´ 2VpPsk +h,ak +h,h, V ‹ +h`1qq +(i) +ď 2 +nk +h +nk +h +ÿ +i“1 +VpPsk +h,ak +h,h, V ref,li +h`1 ´ V ‹ +h`1q +ď 2 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpV ref,li +h`1 ´ V ‹ +h`1q2 +(ii) +ď 2 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpBref,li +h`1 ` βi‹q2 +36 + +ď 4 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpBref,li +h`1 q2 ` 4β2 +i‹, +where (i) is by VpX ` Y q ď 2VpXq ` 2VpY q; (ii) is by Lemma 33. +So by HPsk +h,ak +h,hBref,li +h`1 ď OpH2 ` +Psk +h,ak +h,hpBref,li +h`1 q2q we have the result. +Lemma 40 (Analogous to Lemma 24). With probability at least 1 ´ 2HSAKιδ, we have that for any +ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, +z +VarR +k +hps, aq ď O +ˆ +VpRhps, aqq ` +ι +nk +hps, aq +˙ +. +Lemma 41. Conditioned on the successful events of Lemmas 39 and 40, with probability at least 1 ´ δ, we +have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +bk +h ď Op +b +VarΣ +KHSAι ` +a +H5SAKι2{22i‹ ` H4S3{2Ai‹1{2ι2q. +Proof of Lemma 41. Since bk +h is non-negative and p1 ` 1{Hqh´1 ď 3 when h ď H, we have that +K +ÿ +k“1 +H +ÿ +h“1 +ˆ +1 ` 1 +H +˙h´1 +bk +h ď O +¨ +˚ +˝ +K +ÿ +k“1 +H +ÿ +h“1 +¨ +˚ +˝ +d +νref,k +h +ι +nk +h +` +d +qνk +hι +qnk +h +` +g +f +f +e z +VarR +k +hι +nk +h +` Hι +qnk +h +˛ +‹‚ +˛ +‹‚ +ď O +¨ +˝ +K +ÿ +k“1 +H +ÿ +h“1 +¨ +˝ +d +νref,k +h +ι +nk +h +` +d +qνk +hι +qnk +h +` +d +VpRhpsk +h, ak +hqqι +nk +h +` Hι +qnk +h +˛ +‚ +˛ +‚, +where the last step is by Lemma 40. Using Lemma 34, we have that +K +ÿ +k“1 +H +ÿ +h“1 +Hι +qnk +h +ď OpH3SAι2q. +Next, we bound the terms of νref and qν separately. +Plugging in Lemma 39, we have +K +ÿ +k“1 +H +ÿ +h“1 +¨ +˝ +d +νref,k +h +ι +nk +h +` +d +VpRhpsk +h, ak +hqqι +nk +h +˛ +‚ +ď O +¨ +˚ +˝ +K +ÿ +k“1 +H +ÿ +h“1 +d +pVpRhpsk +h, ak +hqq ` VpPsk +h,ak +h,h, V ‹ +h`1qqι +nk +h +` +K +ÿ +k“1 +H +ÿ +h“1 +Hι +nk +h +` +K +ÿ +k“1 +H +ÿ +h“1 +1 +nk +h +g +f +f +e +nk +h +ÿ +i“1 +Psk +h,ak +h,hpB +ref,lk +h,i +h`1 +q2ι ` +K +ÿ +k“1 +H +ÿ +h“1 +d +β2 +i‹ι +nk +h +˛ +‹‚ +(i) +ď O +¨ +˚ +˝ +ÿ +s,a,h +b +N K`1 +h +ps, aqpVpRhpsk +h, ak +hqq ` VpPs,a,h, V ‹ +h`1qqι ` H2SAι2 +37 + +` +K +ÿ +k“1 +H +ÿ +h“1 +c ι +nk +h +g +f +f +e 1 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpB +ref,lk +h,i +h`1 +q2 ` +b +β2 +i‹ι +ÿ +s,a,h +b +N K`1 +h +ps, aq +˛ +‹‚ +(ii) +ď O +¨ +˚ +˝ +d +HSAι +ÿ +s,a,h +N K`1 +h +ps, aqpVpRhps, aqq ` VpPs,a,h, V ‹ +h`1qq ` H2SAι2` +` +g +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +ι +nk +h +g +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +1 +nk +h +nk +h +ÿ +i“1 +Psk +h,ak +h,hpB +ref,lk +h,i +h`1 +q2 ` +d +HSAβ2 +i‹ι +ÿ +s,a,h +N K`1 +h +ps, aq +˛ +‹‚ +(iii) +ď O +¨ +˚ +˝ +g +f +f +f +f +f +e +HSAι +K +ÿ +k“1 +H +ÿ +h“1 +pVpRhpsk +h, ak +hqq ` VpPsk +h,ak +h,h, V ‹ +h`1qq +loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon +“VarΣ +K +` H2SAι2 ` +b +H2SAβ2 +i‹Kι ` +? +HSAι2 +g +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +Psk +h,ak +h,hpBref,k +h`1 q2 +˛ +‹‚ +(iv) +ď O +¨ +˝ +b +VarΣ +KHSAι ` H2SAι2 ` +b +H2SAβ2 +i‹Kι ` +? +HSAι2 +g +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +pBref,k +h +psk +hqq2 ` Hι +˛ +‚ +(v) +ď Op +b +VarΣ +KHSAι ` +a +H4SAKι{22i‹ ` H7{2S3{2Ai‹1{2ι3{2q. +where (i) is by Lemma 34; (ii) is by Cauchy-Schwarz inequality; (iii) is by Lemmas 34 and 35; (iv) is by +Lemma 18 with l “ H, which happens with probability at least 1 ´ δ; (v) is by Lemma 33. +K +ÿ +k“1 +H +ÿ +h“1 +qνk +h ď +K +ÿ +k“1 +H +ÿ +h“1 +1 +qnk +h +qnk +h +ÿ +i“1 +pV +ref,qlk +h,i +h`1 +ps +qlk +h,i +h`1q ´ V +qlk +h,i +h`1ps +qlk +h,i +h`1qq2 +ď +K +ÿ +k“1 +H +ÿ +h“1 +1 +qnk +h +qnk +h +ÿ +i“1 +pV +ref,qlk +h,i +h`1 +ps +qlk +h,i +h`1q ´ V ‹ +h`1ps +qlk +h,i +h`1qq2 +(i) +ď +K +ÿ +k“1 +H +ÿ +h“1 +1 +qnk +h +qnk +h +ÿ +i“1 +pB +ref,qlk +h,i +h`1 +ps +qlk +h,i +h`1q ` βi‹q2 +(ii) +ď O +˜ K +ÿ +k“1 +H +ÿ +h“1 +pBref,k +h +psk +hqq2 ` β2 +i‹HK +¸ +(iii) +ď OpH6S2Ai‹ι ` H3K{22i‹q. +where (i) is by Lemma 33; (ii) is by Lemma 35; (iii) is by Lemma 33. So +K +ÿ +k“1 +H +ÿ +h“1 +d +qνk +hι +qnk +h +ď +g +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +ι +qnk +h +g +f +f +e +K +ÿ +k“1 +H +ÿ +h“1 +qνk +h +(i) +ď OpH4S3{2Ai‹1{2ι3{2 ` +a +H5SAKι2{22i‹q. +where (i) is by Lemma 34. +38 + +Lemma 42. With probability at least 1 ´ 11Kιδ, the following results hold: For any k P rKs, +VarΣ +pkq ď OpH2ιq, +hence +VarΣ +K ď OpH2Kιq. +Alternatively, +VarΣ +K ď O +˜ K +ÿ +k“1 +Varπk ` H2ι2 +¸ +ď OpVar‹K ` H2ι2q. +Proof of Lemma 42. We first prove the result depending on VarΣ +K similar to Lemma 29. For any k P rKs, +VarΣ +pkq +(i) +ď +H +ÿ +h“1 +rPsk +h,ak +h,hpV ‹ +h`1q2 ´ pV ‹ +h`1psk +h`1qq2s ` +H +ÿ +h“1 +rpV ‹ +h psk +hqq2 ´ pPsk +h,ak +h,hV ‹ +h`1q2s ` +H +ÿ +h“1 +rhpsk +h, ak +hq ´ pV ‹ +1 psk +1qq2 +(ii) +ď 2 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h,h, pV ‹ +h`1q2qι ` 6H2ι ` 2H +H +ÿ +h“1 +maxt V ‹ +h psk +hq ´ Psk +h,ak +h,hV ‹ +h`1 +looooooooooooomooooooooooooon +ěQ‹ +hpsk +h,ak +hq´Psk +h,ak +h,hV ‹ +h`1ě0 +, 0u ` H +(iii) +ď 4H +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h,h, V ‹ +h`1qι ` 7H2ι ` 2H +H +ÿ +h“1 +pV ‹ +h`1psk +h`1q ´ Psk +h,ak +h,hV ‹ +h`1q ` 2HV ‹ +1 psk +1q +loooomoooon +ď2H2 +(iv) +ď 4H +b +2VarΣ +pkqι ` 9H2ι ` 4H +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h,h, V ‹ +h`1qι ` 12Hι +ď 8H +b +2VarΣ +pkqι ` 21H2ι, +where (i) is by by Lemma 20, VpRhps, aqq ď ErRhps, aqs; (ii) is by Lemma 17 with c “ H2, ǫ “ c2, which +happens with probability at least 1 ´ 2ιδ; (iii) is by Lemma 19 with C “ H; (iv) is by Lemma 17 with +c “ H, ǫ “ c2, which happens with probability at least 1 ´2ιδ. Solving the inequality of VarΣ +pkq, we have that +VarΣ +pkq ď 170H2ι. +So taking a union bound over k we have the first result. +Next we prove the result depending on Var‹. This is similar to the proof of Lemma 26. Define a series of +random variables and their truncated values: for any k P rKs, +W k :“ +H +ÿ +h“1 +pVpRhpsk +h, ak +hqq ` VpPsk +h,ak +h,h, V πk +h`1qq, +W k :“ mintW k, 50H2ιu. +By VpPs,a,h, V ‹ +h`1q ď 2VpPs,a,h, V πk +h`1q ` 2VpPs,a,h, V ‹ +h`1 ´ V πk +h`1q, we know that +VarΣ +K ď 2 +K +ÿ +k“1 +W k ` 2Y, +where Y ď OpH2ιq (with probability at least 1 ´ 4ιδ) is defined in the proof of Lemma 38. Correspondingly, +define the following event, which means there is no truncation: +EW :“ tW k “ W k, @k P rKsu. +39 + +For any fixed 1 ď k ď K, +W k ď +H +ÿ +h“1 +rPsk +h,ak +h,hpV πk +h`1q2 ´ pV πk +h`1psk +h`1qq2s ` +H +ÿ +h“1 +rpV πk +h psk +hqq2 ´ pPsk +h,ak +h,hV πk +h`1q2s ` +H +ÿ +h“1 +rhpsk +h, ak +hq ´ pV πk +1 +psk +1qq2 +(i) +ď 2 +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h,h, pV πk +h`1q2qι ` 6H2ι ` 2H +H +ÿ +h“1 +pV πk +h psk +hq ´ Psk +h,ak +h,hV πk +h`1q ` H +(ii) +ď 4H +g +f +f +e2 +H +ÿ +h“1 +VpPsk +h,ak +h,h, V πk +h`1qι ` 7H2ι ` 2H +H +ÿ +h“1 +rhpsk +h, ak +hq +ď 4H +? +2W kι ` 9H2ι, +where (i) is by Lemma 17 with c “ H2, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ; (ii) is by +Lemma 19 with C “ H. Solving the inequality, W k ď 50H2ι. This means, PrEWs ě 1 ´ 2Kιδ. Now on +suppose EW holds, then +K +ÿ +k“1 +W k “ +K +ÿ +k“1 +W k +(i) +ď 3 +K +ÿ +k“1 +ErW k | Fks ` 50H2ι2 +ď 3 +K +ÿ +k“1 +ErW k | Fks ` 50H2ι2 +“ 3 +K +ÿ +k“1 +Varπk +1 psk +1q ` 50H2ι2 +ď 3 +K +ÿ +k“1 +Varπk ` 50H2ι2 +ď 3Var‹K ` 50H2ι2, +where (i) is by Lemma 18 with l “ 50H2ι, which happens with probability at least 1 ´ δ. +B.4 +Proof of Lower Bounds +We modify Theorem 9 in Domingues et al. [2021] for a bounded-reward, time-homogeneous lower bound +(Theorem 12). Theorem 13 is much more straightforward. To this end, we borrow necessary notations from +Domingues et al. [2021], adapted to time-homogeneous setting. +A policy π interacting with an MDP M defines a stochastic process denote by ppSk +h, Ak +h, Rk +hqhPrHsqkě1, +where Sk +h, Ak +h and Rk +h are the random variables representing the state, the action and the reward at time +h of episode k. As explained by Lattimore and Szepesv´ari [2020], the Ionescu-Tulcea theorem ensures the +existence of probability space pΩ, F, PMq such that +PMrSk +h`1 “ s|Ak +h, Ik +hs “ Pps|Sk +h, Ak +hq, +and +PMrAk +h “ a|Ik +hs “ πk +hpa|Ik +hq, +where π “ pπk +hqkPrKs,hPrHs and +Ik +h :“ pS1 +1, A1 +1, R1 +1, . . . , S1 +H, A1 +H, R1 +H, S2 +1, A2 +1, R2 +1, . . . , Sk´1 +H +, Ak´1 +H +, Rk´1 +H +, Sk +1 , Ak +1, Rk +1, . . . , Sk +hq +is the random vector containing all state-action pairs observed up to step h of episode k, but not including +Ak +h. Here we assume the rewards are deterministic as in Domingues et al. [2021]. Next, we denote by PIK +H +M +40 + +the pushforward measure of IK +H under PM, +PIK +H +M riK +Hs :“ PMrIK +H “ iK +Hs “ +K +ź +k“1 +H +ź +h“1 +πk +hpak +h|ik +hqPpsk +h`1|sk +t , ak +t q, +(15) +where iK +H is a realization of IK +H . +Definition 43. The Kullback-Leibler divergence between two distributions P1 and P2 on a measurable space +pΩ, Gq is defined as +KLpP1, P2q :“ +ż +Ω +ln +ˆdP1 +dP2 +pωq +˙ +dP1pωq, +if P1 ! P2 and `8 otherwise. For Bernoulli distributions, we define @pp, qq P r0, 1s2, +klpp, qq :“ KLpBppq, Bpqqq “ p ln +ˆp +q +˙ +` p1 ´ pq ln +ˆ1 ´ p +1 ´ q +˙ +. +Lemma 44 (Adapted from Lemma 5 in Domingues et al. [2021]). Let M and M1 be two MDPs that are +identical except for their transition probabilities, denoted by P and P 1, respectively. Assume that we have +@ps, aq, Pp¨|s, aq ! P 1p¨|s, aq. Then for any K, +KL +´ +PIK +H +M , PIK +H +M1 +¯ +“ +ÿ +ps,aqPSˆA +EM +“ +N K +s,a +‰ +KLpPp¨|s, aq, P 1p¨|s, aqq, +where N K +s,a :“ řK +k“1 +řH +h“1 +1rpSk +h, Ak +hq “ ps, aqs. +Lemma 45 (Lemma 1 in Garivier et al. [2016]). Consider a measurable space pΩ, Fq equipped with two +distributions P1 and P2. For any F-measurable function Z : Ω Ñ r0, 1s, we have +KLpP1, P2q ě klpE1rZs, E2rZsq, +where E1 and E2 are the expectations under P1 and P2 respectively. +Proof of Theorem 12. We retain most of the proof of Theorem 9 and Appendix C in Domingues et al. [2021], +while incorporating the hard instance design in Section 5.5.1 in Zhou et al. [2022]. Namely, we change the +A-ary tree in Domingues et al. [2021] with the binary tree in Zhou et al. [2022]. This change does not affect +the proof, while circumventing the requirement of S “ 3 ` pAd ´ 1q{pA ´ 1q where d is the tree height. We +can find S1 “ 2 ` 2tlog2pS´2qu “ ΩpSq and replace S with S1. We still use d “ tlog2pS ´ 2qu to denote the +tree height. +We change the transition at sg: Ppsb|sg, aq “ 1 for any a P A. This means for any trajectory, the agent +can only get reward once, then loops at sb. +Another important change in design is to scale the reward at sg by t ď 1, with t depending on V the +variance we desire. So rpsg, aq “ t. +To be precise, let E0 and Epℓ‹,a‹q be the expectation taken with respect to the reference MDP (with no +special leaf-action pair) and Mpℓ‹,a‹q. We have that +RKpπ, Mpℓ‹,a‹qq ě tKε +ˆ +1 ´ 1 +K Epℓ‹,a‹q +” +N K +pℓ‹,a‹q +ı˙ +, +where N K +pℓ‹,a‹q “ řK +k“1 +1rpSk +d`1, Ak +d`1q “ ps, aqs. Hence, +max +pℓ‹,a‹q RKpπ, Mpℓ‹,a‹qq ě tKε +¨ +˝1 ´ +1 +LAK +ÿ +pℓ‹,a‹q +Epℓ‹,a‹q +” +N K +pℓ‹,a‹q +ı +˛ +‚. +(16) +41 + +Since N K +pℓ‹,a‹q{K P r0, 1s, by Lemma 45, +kl +ˆ 1 +K E0 +” +N K +pℓ‹,a‹q +ı +, 1 +K Epℓ‹,a‹q +” +N K +pℓ‹,a‹q +ı˙ +ď KLpPIK +H +0 , PIK +H +pℓ‹,a‹qq. +By Lemma 44, +KL +´ +PIK +H +0 , PIK +H +pℓ‹,a‹q +¯ +“ E0 +” +N K +pℓ‹,a‹q +ı +kl +ˆ1 +2, 1 +2 ` ε +˙ +. +Assume that ε ď 1{4, then klp1{2, 1{2 ` εq ď 4ε2. By Pinsker’s inequality, pp ´ qq2 ď klpp, qq{2, it implies +1 +K Epℓ‹,a‹q +” +N K +pℓ‹,a‹q +ı +ď 1 +K E0 +” +N K +pℓ‹,a‹q +ı +` +? +2ε +c +E0 +” +N K +pℓ‹,a‹q +ı +. +Since ř +ph‹,a‹q N K +pℓ‹,a‹q “ K, by Cauchy-Schwarz inequality we have +1 +K +ÿ +pℓ‹,a‹q +Epℓ‹,a‹q +” +N K +pℓ‹,a‹q +ı +ď 1 ` +? +2ε +? +LAK. +Plugging this back to Equation (16), and taking ε “ p1 ´ 1{LAq +a +LA{8K, we have +max +pℓ‹,a‹q RKpπ, Mpℓ‹,a‹qq ě Ωpt +? +SAKq. +To ensure that ε ď 1{4, we need K ě SA. +Now we calculate the variances. We know that V ‹ +d`2psbq “ 0 and V ‹ +d`2psgq “ t. For any trajectory τ, we +look at step h “ d ` 1. If psh, ahq ‰ pℓ‹, a‹q, then +VarΣ +τ ě Vpp1{2, 1{2q, p0, tqq “ Ωpt2q. +If psh, ahq “ pℓ‹, a‹q, then +VarΣ +τ ě Vpp1{2 ´ ε, 1{2 ` εq, p0, tqq “ +ˆ1 +4 ´ ε2 +˙ +Ωpt2q. +Notice that ε ď 1{4, so VarΣ +τ ě Ωpt2q for any τ, and +Var‹ ě Varπ‹ ě min +τ +VarΣ +τ ě Ωpt2q. +Since the total reward in each episode is upper-bounded by t, we know that VarΣ +τ , Var‹ ď Opt2q. Thus, +VarΣ +τ , Var‹ “ Θpt2q. +For the desired result, we set t “ Θp +? +Vq. +Proof of Theorem 13. We retain most of the proof of Theorem 9 in Domingues et al. [2021], while incorpo- +rating the hard instance design in Section 5.5.1 in Zhou et al. [2022]. Namely, we change the A-ary tree in +Domingues et al. [2021] with the binary tree in Zhou et al. [2022]. This change does not affect the proof, +while circumventing the requirement of S “ 3 ` pAd ´ 1q{pA ´ 1q where d is the tree height. We can find +S1 “ 2 ` 2tlog2pS´2qu “ ΩpSq and replace S with S1. We still use d “ tlog2pS ´ 2qu to denote the tree height. +Another important change in design is to scale the reward at sg by t ď 1, with t depending on V the +variance we desire. So rhpsg, aq “ t1rh ě H ` d ` 1s. This modification does not affect the choice of ε and +H in Domingues et al. [2021], only scales the optimal value and regret linearly, so we have that +max +ph‹,ℓ‹,a‹q RKpπ, Mph‹,ℓ‹,a‹qq ě Ωpt +? +H3SAKq. +42 + +Now we calculate the variances. We know that V ‹ +H`d`1psbq “ 0 and V ‹ +H`d`1psgq “ tpH ´H ´dq “ ΩptHq. +For any trajectory τ, we look at step h “ H ` d. If psh, ahq ‰ pℓ‹, a‹q, then +VarΣ +τ ě Vpp1{2, 1{2q, p0, ΩptHqqq “ Ωpt2H2q. +If psh, ahq “ pℓ‹, a‹q, then +VarΣ +τ ě Vpp1{2 ´ ε, 1{2 ` εq, p0, ΩptHqqq “ +ˆ1 +4 ´ ε2 +˙ +Ωpt2H2q. +Notice that ε ď 1{4 in Domingues et al. [2021], so VarΣ +τ ě Ωpt2H2q for any τ, and +Var‹ ě Varπ‹ ě min +τ +VarΣ +τ ě Ωpt2H2q. +Since the total reward in each episode is upper-bounded by OptHq, we know that VarΣ +τ , Var‹ ď Opt2H2q. +Thus, +VarΣ +τ , Var‹ “ Θpt2H2q. +For the desired result, we set t “ Θp +? +V{Hq. +43 + diff --git a/_9FQT4oBgHgl3EQf8TZV/content/tmp_files/load_file.txt b/_9FQT4oBgHgl3EQf8TZV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eefecd8d8158a16150a776bab81777d14616c184 --- /dev/null +++ b/_9FQT4oBgHgl3EQf8TZV/content/tmp_files/load_file.txt @@ -0,0 +1,1834 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf,len=1833 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='13446v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='LG] 31 Jan 2023 Sharp Variance-Dependent Bounds in Reinforcement Learning: Best of Both Worlds in Stochastic and Deterministic Environments Runlong Zhou˚ Zihan Zhang: Simon S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Du;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' February 1, 2023 Abstract We study variance-dependent regret bounds for Markov decision processes (MDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Algorithms with variance-dependent regret guarantees can automatically exploit environments with low variance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', enjoying constant regret on deterministic MDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The existing algorithms are either variance- independent or suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We first propose two new environment norms to characterize the fine-grained variance properties of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For model-based methods, we design a variant of the MVP al- gorithm [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021a] and use new analysis techniques show to this algorithm enjoys variance- dependent bounds with respect to our proposed norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In particular, this bound is simultaneously min- imax optimal for both stochastic and deterministic MDPs, the first result of its kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We further initiate the study on model-free algorithms with variance-dependent regret bounds by designing a reference- function-based algorithm with a novel capped-doubling reference update schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lastly, we also provide lower bounds to complement our upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 1 Introduction We consider episodic reinforcement learning (RL) on tabular Markov Decision Processes (MDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Existing algorithms can be categorized into two classes: model-based methods whose space complexity scales quadrat- ically with the number of states [Auer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2008, Agrawal and Jia, 2017, Azar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2017, Dann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2017, 2019, Zanette and Brunskill, 2019, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021a] and model-free methods whose space complex- ity scales linearly with the number of states [Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2018, Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2019, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2020, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The MDPs in practice often enjoy benign structures, so problem-dependent regret bounds are of great interest [Zanette and Brunskill, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' RL algorithms often perform far better on these MDPs than what their worst-case guarantees would suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Motivated by this observation, we want to systematically study algorithms with regrets depending on quantities that characterizes the hardness of MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Ideally, such algorithms should automatically exploit the MDP instance without the prior knowledge of problem-dependent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' As a motivating example, for time-homogeneous MDPs with total reward bounde by 1, the minimax regret bound for deterministic MDPs is OpSAq where S and A are number of states and actions, respectively and the worst-case minimax optimal regret bound for stochastic MDPs is rO `?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAK ˘ where K is the number of episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Many problems can be formulated as deterministc MDPs, such as shortest path (maze, real world navigation), combinatorial optimization, Atari games [Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2013] and many games (mountain car, lunar lander, robotics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=') in OpenAI Gym [Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Deterministic systems can also approximate stochastic systems well (see Section 2 and 6 in Bertsekas [2012]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We want an algorithm designed for generic stochastic MDPs with worst-case minimax optimal regret bound while enjoying the OpSAq bound when the MDP is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ˚University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Email: vectorzh@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='edu :Email: zihan-zh17@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='cn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='University of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Email: ssdu@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='edu 1 Zanette and Brunskill [2019] is a pioneer work which provides a model-based algorithm whose regret scales with variance-depedent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' They defined a quantity, Q‹, named the maximum per-step con- ditional variance to characterize the randomness of the MDP instance, and showed a regret bound of rOp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='HQ‹ ¨ SAK ` H5{2S2Aq, where H is the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This bound is still not satisfactory because: ① There exist MDPs with Q‹ “ Ωp1q, so the regret reduces to rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' HSAKq which does not match the min- imax optimal bound rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ② For deterministic MDPs (Q‹ “ 0), the regret reduces to rOpH5{2S2Aq, which does not match the optimal OpSAq bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This paper makes the following contributions which significantly advance our understand- ing of problem-dependent bounds in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ First, We introduce the total multi-step conditional variance, VarΣ K and the maximum policy-value variance, Var‹, to provide fine-grained characterizations of the variance in the MDP (see Section 4 for the formal definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Importantly, regret bounds that depend on these quantities will reduce to the minimax optimal bound in the worst case whereas the existing notion HQ‹ cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ Second, for model-based methods, we identify the obstacles preventing the current state-of-the-art minimax optimal algorithm, MVP Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a], from being variance-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We make necessary improvements and introduce a truncation method to bound the total variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We show the regret bound of the improved algorithm, MVP-V, scales with Var‹ or VarΣ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In particular, these bounds imply that, MVP-V is minimax optimal for both the classes of stochastic and deterministic MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To our knowledge, this is first result of such kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' See Table 1 for comparions between model-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ Third, we initiate the study of model-free algorithms with variance-dependent regrets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We explain why existings model-free algorithms cannot be variance-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We futher propose a new model-free algorithm, UCB-Advantage-V, which relies on a a capped-doubling manner of updates for reference values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We further utilize a novel analysis technique which bounds value gaps directly from the existing uniform convergence bound to give the first variance-dependent bound for model-free algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Importantly, this bound reduces to the minimax optimal bound for the worst-case MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' See Table 2 for comparisons between model-free algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ Lastly, we prove minimax regret lower bounds for the class of MDPs with bounded variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We show that the main order terms of our regret upper bounds match these lower bounds, so our proposed algorithms are minimax optimal for the class of variance-bounded MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1 Technical Overview For model-based algorithms, existing state-of-the-art work [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021a] fails to be variance-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' It is hard to bound the total variance by its expectation using martingale concentration inequalities directly, while avoiding an H-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This is because the total variance within an episode can be as large as ΩpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We introduce a novel analysis technique which truncates the total variance of each episode to a con- stant and apply martingale concentration inequalities on this sequence, and show that with high probability there is no truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We also apply a more refined concentration inequality to the transition model to have a dependency on the maximum support instead of the size of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This step is crucial in obtaining the tight bound for deterministic MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For the model-free algorithm, existing work [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2020] upper-bounds all the four bias terms in their Equation (13) by variance-independent main order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We identify the problem incurred by the large bias in reference values, and replace the update with a capped-doubling manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since too frequent updates discard past data very often, this method balances between the summation of gaps of value functions and the waste of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We integrate directly over the error between the estimated value and the optimal value to bound the total squared gaps between them, whereas Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] bound them with a coarse binary gap of either H or the final gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Combined with many other finer-grained analyses throughout the proof, we can finally remove all the variance-independent main order terms except for the total variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2 Algorithm Regret Variance- Dependent Stochastic- Optimal Deterministic- Optimal Horizon- Free Euler Zanette and Brunskill [2019] rOp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='HQ‹ ¨ SAK ` H5{2S2Aq Yes No No No rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAK ` H5{2S2Aq No Yes No No MVP Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a] rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAK ` S2Aq No Yes No Yes MVP-V This work rOp b mintVarΣ K, Var‹KuSA ` ΓSAq Yes Yes Yes Yes Table 1: Comparisons between model-based algorithms for time-inhomogeneous MDPs with total reward bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' rO hides logarithm factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' S, A, Γ, H and K are number of states, actions, maximum support of the transition model, planning horizon and interaction episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Q‹, VarΣ K and Var‹ are variance notations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Q‹ and VarΣ K are upper bounded by 1 in the worst case and become 0 when the MDP is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' An “Yes” in each column means: Variance-Dependent: the regret has a main order term scaling with any variance notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Stochastic-Optimal: the regret has a main order term of rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAKq which matches the minimax lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Deterministic-Optimal: the regret is rOpSAq on deterministic MDPs (with variance equal to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Horizon-Free: the regret has only logarithmic dependency on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Paper Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We first list basic concepts of MDPs in Section 3, then define variance quantities in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Our main results then come in three sections: Sections 5 and 6 show the algorithms, theorems, corollaries and proof sketches of our model-based and model-free methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Section 7 shows our lower bounds for the class of variance-bounded MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2 Related Works Minimax optimal regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Algorithms for regret minimization can be categorized into two classes: model-based and model-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Being model-free means the space complexity is OpHSAq, prohibiting the estimation of the whole transition model Phps1|s, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For model-based methods, there are previous work [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021a, 2022] achieving a property called horizon-free, which allows only logarithmic dependency on H for regrets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' As explained in Jiang and Agarwal [2018], in many scenarios with a long planning horizon, the interesting regime is K !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This underscores the importance of being horizon-free, because for H-dependent bounds, only when K " H they become sub-linear in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Being horizon-free is challenging, because it requires utilizing transition data for the same state-action pair from different steps and handling a spike in rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' There are many works other than those we cite in Section 1 giving nearly minimax optimal bounds: Bartlett and Tewari [2012], Osband et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2013], Osband and Van Roy [2017], Fruit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2018a], Talebi and Maillard [2018], Simchowitz and Jamieson [2019], Russo [2019], Zhang and Ji [2019], Neu and Pike-Burke [2020], Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021], Pacchiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We compare our results with the state-of-the-art in Table 1 (model-based) and Table 2 (model-free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Other problem-dependent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Most problem-dependent results prior to Zanette and Brunskill [2019] focus on the infinite-horizon setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Some depend on the range of value functions [Bartlett and Tewari, 2012, Fruit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2018b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Maillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2014] introduces a hardness measure called distribution norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Talebi and Maillard [2018] provides a problem-dependent regret bound that scales as a function of the vari- ance of the next state distribution under strong assumptions on mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' There are gap-dependent results for multi-armed bandits and RL [Even-Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2006, Auer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2008, Simchowitz and Jamieson, 2019, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] shows that with a slight modification, the algorithm in Zanette and Brunskill [2019] can have a first-order regret, with the main order term depending on the optimal value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Wagenmaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2022] provides a first-order regret for linear MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When the total reward is bounded by 1 almost surely, for any policy its variance is not larger than this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This means a first-order dependency is weaker than a variance-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 3 Algorithm Regret Variance- Dependent Stochastic- Optimal Q-learning (UCB-B) Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2018] rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H4SAK ` H9{2S3{2A3{2q No No UCB-Advantage Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H3SAK ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H33S8A6Kq No Yes Q-EarlySettled- Advantage Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021] rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H3SAK ` H6SAq No Yes UCB-Advantage-V This work rOp b mintVarΣ K, Var‹KuHSA ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H15S5A3Kq Yes Yes Table 2: Comparison between model-free algorithms for time-inhomogeneous MDPs with every reward bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' An “Yes” in each column means: Variance-Dependent: the bound scales with the variance term that characterizes the randomness of the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Stochastic-Optimal: in the wortt-case, the regret’ dominating term becomes rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H3SAKq which matches the minimax lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 3 Preliminaries Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any event E, we use 1rEs to denote the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any random variable X, we use VpXq to denote its variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any set X, we use ∆pXq to denote the probability simplex over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any positive integer n, we denote rns :“ t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any probability distribution P, we use supppPq “ ř x 1rPpxq ą 0s to denote the size of its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Suppose x and y are n-dimensional vectors, we denote xy :“ řn i“1 xiyi and xq :“ pxq 1, xq 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , xq nq for any number q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If x P ∆prnsq, we use Vpx, yq “ ř i xipyi ´ xyq2 “ xy2 ´ pxyq2 to denote the variance of y under x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We use 1k to denote a vector with all 0 but an only 1 on the k-th position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Finite-horizon MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' A finite-horizon MDP can be described by a tuple M “ pS, A, P, R, Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' S is the finite state space with size S and A is the finite action space with size A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any h P rHs, Ph : SˆA Ñ ∆pSq is the transition function and Rh : SˆA Ñ ∆pr0, 1sq is the reward distribution with mean rh : SˆA Ñ r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H is the planning horizon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', episode length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We denote Γ :“ maxh,s,a supppPhp¨|s, aqq as the maximum support of the transition function, and Ps,a,h :“ Php¨|s, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditions for MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We have two conditions more general than the ordinary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' As explained below them, getting tight regret bounds are harder when they are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any policy π, the total reward in a single episode is upper-bounded by 1 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For those MDPs not satisfying Condition 1, we can normalize all the rewards by 1{H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Such a conversion usually multiplies a factor of 1{H to the regret, but cannot take into account a spike in rewards, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', some rhps, aq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The MDP is time-homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Namely, there exist P : SˆA Ñ ∆pSq, R : SˆA Ñ ∆pr0, 1sq and r : S ˆ A Ñ r0, 1s such that for any ps, aq P S ˆ A, Php¨|s, aq “ Pp¨|s, aq, Rhps, aq “ Rps, aq and rhps, aq “ rps, aq for any h P rHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For simplicity, we denote Ps,a :“ Pp¨|s, aq and Ps,a,s1 :“ Pps1|s, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Any time-inhomogeneous MDP can be instantiated by a time-homogeneous one to satisfy Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let a mega state space S “ YH h“1Sh, where each Sh corresponds to the state space of the time-inhomogeneous MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any ph, s, aq, we construct Pps1 h`1|sh, aq “ Phps1|s, aq and Rpsh, aq “ Rhps, aq, where sh is the corresponding state of s in Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' S is 4 multiplied by H while Γ remains the same, and the regret changes accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This condition underscores the algorithm’s ability to use information of the same state-action pair from different steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We will introduce quantities in Section 4 to quantify determinism, but a fully-deterministic MDP is very worth studying because the regret lower bound is the well-known ΩpSAq (under Conditions 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, we care about whether the algorithms can have a constant regret (up to logarithm factors) under the assumption of determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The MDP is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Namely, for any ph, s, aq P rHsˆS ˆA, Rhps, aq and Php¨|s, aq map to a single real number and a single state respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' A history-independent deterministic policy π chooses an action based on the current state and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Formally, π “ tπhuhPrHs where πh : S Ñ A maps a state to an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We use Π to denote the set of all such policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Episodic RL on MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Upon choosing action a at state s when it is the h-th step in an episode, the agent observes a reward r „ Rhps, aq and the next state s1 „ Php¨|s, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When h “ H, the episode ends after this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, a policy π induces a (random) trajectory ptsh, ah, rhuhPrHs, sH`1q where s1 is exogenously generated, ah “ πhpshq, rh „ Rhpsh, ahq and sh`1 „ Php¨|sh, ahq for h P rHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The episodic RL on MDPs proceeds in a total of K episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When one episode ends, a new initial state s1 is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The agent should (adaptively) choose a policy πk for the k-th episode, put it into action and cannot change it within an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Value functions and Q-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Given a policy π, we define its value function and Q-function as V π h psq :“ Eπ « H ÿ t“h rt ˇˇˇˇˇ sh “ s ff , Qπ hps, aq :“ Eπ « H ÿ t“h rt ˇˇˇˇˇ psh, ahq “ ps, aq ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' It is easy to verify that Qπ hps, aq “ rhps, aq ` Ps,a,hV π h`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Performance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The goal of episodic RL on MDPs is to find the policy which maximizes the total reward collected in an episode, regardless of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Given M, such a goal can be achieved using dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Given this, we denote V ‹ :“ V π‹ and Q‹ :“ Qπ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We use cumulative regret as a performance measure: RegretpKq :“ K ÿ k“1 pV ‹ 1 psk 1q ´ V πk 1 psk 1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 4 Variance Quantities for MDPs We use the notion of variance to quantify the degree of determinism of MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The first is called the maximum per-step conditional variance [Zanette and Brunskill, 2019], which is only relevant to the optimal value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The maximum per-step conditional variance for a particular MDP is defined as: Q‹ :“ max h,s,atVpRhps, aqq ` VpPs,a,h, V ‹ h`1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 5 Zanette and Brunskill [2019] directly use HQ‹ to upper-bound the total per-step conditional variances in an episode, a quantity which can be upper-bounded by a constant (see Lemmas 29 and 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So when Q‹ ě ΩpHq (or Ωp1{Hq under Condition 1), HQ‹ is not tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In light of this, we define the total multi-step conditional variance as a better notation in place of HQ‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The total multi-step conditional variance for a trajectory τ “ tsh, ahuhPrHs is defined as: VarΣ τ :“ H ÿ h“1 pVpRhpsh, ahqq ` VpPsh,ah,h, V ‹ h`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' During the learning process, let the trajectory of the k-th episode be τ k, then we denote VarΣ pkq :“ VarΣ τ k, and VarΣ K :“ řK k“1 VarΣ pkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We introduce another type of variance, called the maximum policy-value variance, which is novel in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any policy π P Π, its maximum value variance is defined as Varπ :“ maxsPS Varπ 1psq, where Varπ 1psq :“ Eπ « H ÿ h“1 ` VpRhpsh, ahqq ` VpPsh,ah,h, V π h`1q ˘ ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The maximum policy-value variance for a particular MDP is defined as: Var‹ :“ max πPΠ Varπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Varπ 1psq is the variance of total reward of π starting from s, and the justification can be found in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Under Condition 1, by Lemma 20 we know that Varπ 1psq ď V π 1 psq ď V ‹ 1 psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So Var‹ ď V ‹ 1 psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This means a variance-dependent regret is better than a first-order regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1 Comparing VarΣ pkq and Var‹ We use this subsection to demonstrate that a small VarΣ pkq does not imply a small Var‹, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Deterministic MDPs have VarΣ pkq “ Var‹ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Trivially, Var‹ “ 0 ùñ VarΣ pkq “ 0, while the reverse is not ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When VarΣ pkq “ 0 ă Var‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Consider the following time-homogeneous MDP with horizon H: For each state s there is a good action a1 with a deterministic reward rps, a1q “ 1{H, and all other actions a1 have a deterministic reward rps, a1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any state-action pair ps, aq, the transition is identically Ps,a,s1 “ 1{S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The optimal policy always chooses a1 at any state s, so for any s and h, V ‹ h psq “ pH ´ h ` 1q{H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any ph, s, aq, VpRps, aqq ` VpPs,a, V ‹ h`1q “ 0, which means VarΣ pkq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' However, let π be a policy with πHps1q “ a1 for a certain state s1, and πhpsq “ a1 for any other h or s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then π yields cumulative rewards of 1 and 1 ´ 1{H, each with non-zero probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So Var‹ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This example shows that deterministic MDPs are not the only MDPs satisfying VarΣ pkq “ 0, and that VarΣ pkq “ 0 does not imply Var‹ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 6 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' " # $ " !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='% !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content="& #'( #'( !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=') !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' * + , # !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' "#$%&\'$ ()*+$ ,-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='/*0#$ 1,23$*3*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='0$ 2,(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='24 Figure 1: Example of Var‹ being arbitrarily smaller than VarΣ pkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Dashed arrows represent probabilistic transitions and solid arrows represent deterministic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The only reward comes at state s4 and on choosing any action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When VarΣ pkq “ 1{4 ą Var‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Consider the time-homogeneous MDP in Figure 1: Ps1,a,s2 “ p for any a P A, and the rest probability is into an MDP with no reward at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' s2 is a state which we want to have a high VarΣ pkq: Ps2,a,s3 “ Ps2,a,s4 “ 1{2, where s3 and s4 are states with value 0 and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus at s2, a, h “ 3, VarΣ pkq ě VpRps2, aqq ` VpPs2,a, V ‹ 3 q “ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We also have that for any policy π, V π 1 ps1q “ p{2, so by Lemma 20, Var‹ ď p{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Taking p arbitrarily small gives an arbitrarily large gap between VarΣ pkq and Var‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This example shows that a small Var‹ does not imply a small VarΣ pkq, so using only VarΣ pkq is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 5 Results of the Model-Based Algorithm We propose MVP-V (Algorithm 1, where “V” stands for “Variance-dependent”), a model-based algorithm with a variance-dependent regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Based on MVP [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021a] which is minimax optimal under Conditions 1 and 2, we make necessary alterations to make the regret variance-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Common notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' These are notations shared with our model-free algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let sk h, ak h and rk h denote the state, action and reward at the h-th step of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let V k h and Qk h denote Vh and Qh at the beginning of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' rO hides polylogpH, S, A, K, 1{δq factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Algorithm description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' MVP-V re-plans whenever a state-action pair’s visitation is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The bonus function depends on the variance of the next-step value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' It achieves variance-dependent regret by using the empirical variances of rewards in the bonus, as opposed to using the empirical rewards themselves in MVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' MVP-V is capable of handling Conditions 1 and 2 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume that Conditions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ δ, the regret of MVP-V (Algorithm 1) run with ι “ 99 ˆ ln ˆ30002H5S7A5K5 δ2 ˙ ` 1 ˙ 7 Algorithm 1 MVP-V 1: Input and initialize: Logarithm term ι;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Trigger set L Ð t2i´1 | 2i ď KH, i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2: for ps, a, s1, hq P S ˆ A ˆ S ˆ rHs do 3: Nps, aq Ð 0, θps, aq Ð 0, φps, aq Ð 0, nps, aq Ð 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 4: Nps, a, s1q Ð 0, pPs,a,s1 Ð 0, Qhps, aq Ð 1, Vhpsq Ð 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 5: end for 6: \\\\ Main algorithm begins 7: for k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', K do 8: for h “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', H do 9: Observe sk h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 10: Take action ak h “ arg maxa Qhpsk h, aq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 11: Receive reward rk h and observe sk h`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 12: Set ps, a, s1, rq Ð psk h, ak h, sk h`1, rk hq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 13: Set Nps, aq Ð Nps, aq ` 1, θps, aq Ð θps, aq ` r, φps, aq Ð φps, aq ` r2, Nps, a, s1q Ð Nps, a, s1q ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 14: \\\\ Update empirical reward and transition probability 15: if Nps, aq P L then 16: Set prps, aq Ð θps, aq{Nps, aq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 17: Set z VarRps, aq Ð φps, aq{Nps, aq ´ prps, aq2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 18: Set pPs,a,rs Ð Nps, a, rsq{Nps, aq for all rs P S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 19: Set nps, aq Ð Nps, aq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 20: Set TRIGGERED = TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 21: end if 22: end for 23: \\\\ Update Q-function 24: if TRIGGERED then 25: for h “ H, H ´ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 1 do 26: for ps, aq P S ˆ A do 27: Set bhps, aq Ð 4 d Vp pPs,a, Vh`1qι maxtnps, aq, 1u ` 2 d z VarRps, aqι maxtnps, aq, 1u ` 21ι maxtnps, aq, 1u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Qhps, aq Ð mintprps, aq ` pPs,aVh`1 ` bhps, aq, 1u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Vhpsq Ð max a Qhps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 28: end for 29: end for 30: Set TRIGGERED = FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 31: end if 32: end for is bounded by RegretpKq ď rOp b mintVarΣ K, Var‹KuSA ` ΓSAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When Condition 1 holds, we have Var‹ ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, our results are better than the rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAK ` S2Aq regret of MVP, and achieve the horizon-free (only logarithm dependency on H) property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' They are also strictly better than the rOp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='HQ‹ ¨ SAK ` H5{2S2Aqq regret in Zanette and Brunskill [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 8 Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='2 for the rigorous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We follow the outline in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a], realizing that the total regret is upper-bounded by M1 ` M2 ` M3, where M1 « K ÿ k“1 H ÿ h“1 pPsk h,ak h ´ 1sk h`1qV k h`1, M2 « K ÿ k“1 H ÿ h“1 pV k h psk hq ´ rpsk h, ak hq ´ Psk h,ak hV k h`1q, M3 « K ÿ k“1 ˜ H ÿ h“1 rpsk h, ak hq ´ V πk 1 psk 1q ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We expand rpsk h, ak hq by Bellman equation to derive a tighter bound for M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This change is necessary to remove a variance-independent rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Kq term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' M1, M2, M3 can be then related to a crucial variance term M4 « K ÿ k“1 H ÿ h“1 pVpRpsk h, ak hqq ` VpPsk h,ak h, V k h`1qq so the regret is approximately rOp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='SAM4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The difference between VarΣ K and M4 is of a lower order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To upper bound M4 directly, we introduce bonus-correction terms bck hps, aq :“ V k h psq ´ Ps,aV k h`1 ´ rps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let BCk hpsq :“ bck hps, aq ` Ps,aBCk h`1 with a “ πk hpsq, then it can be proven that BCk hpsq “ V k h psq ´ V πk h psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, M4 can be bounded by the sum of Z « K ÿ k“1 H ÿ h“1 VpPsk h,ak h, BCk h`1q and W “ K ÿ k“1 H ÿ h“1 pVpRpsk h, ak hqq ` VpPsk h,ak h, V πk h`1qq, where Z is of a lower order and W ď rOpVar‹Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' However, the bound of W cannot be derived using martingale concentration inequalities directly, because the summation of variances within an episode can be of order ΩpHq, which will introduce a constant term of H, ruining the horizon-free property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We first prove that the total variance in an episode is bounded by rOp1q with high probability, then the martingale concentration inequality can be applied to terms truncated to rOp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To get the Γ-dependency in the lower order term, we observe that Ps,a “ 0 ùñ pPs,a “ 0 and put this into concentration bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We study deterministic MDPs first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume that Conditions 1 and 2 and Assumption 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ δ, the regret of MVP-V (Algorithm 1) run with ι “ 99plnp30002H5S7A5K5{δ2q ` 1q is bounded by RegretpKq ď rOpSAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This is because Var‹ “ 0 and Γ “ 1 when the MDP is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With a more refined analysis, we can totally eliminate the dependency on δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Up to logarithm factors, MVP-V matches the lower bound of ΩpSAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So MVP-V is minimax optimal for the class of deterministic MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Another corollary arises when we remove Conditions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For MVP-V to work properly, we need to apply the conversion methods written below the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 9 Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ δ, the regret of MVP-V (Algorithm 1) run with ι “ 99plnp30002H12S7A5K5{δ2q ` 1q is bounded by RegretpKq ď rOp b mintVarΣ K, Var‹KuHSA ` H2ΓSAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Readers may notice that the scaling in main order terms are not typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This is because when removing Condition 1, VarΣ K and Var‹ automatically scale by H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 6 Results of the Model-Free Algorithm We propose UCB-Advantage-V (Algorithm 2) to initiate the study of model-free algorithms with variance- dependent regrets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Algorithm description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In UCB-Advantage-V, the update of value functions is triggered by the beginning of stages for each ps, a, hq triple separately, and the stage design approximately makes use of the last 1{H fraction of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Besides, the algorithm maintains reference values by assigning value functions to them at another frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The update rule using the reference value decomposition can be illustrated as: Qhps, aq Ð { Ps,a,hV ref h`1 ` { Ps,a,hpVh`1 ´ V ref h`1q ` prhps, aq ` bk hps, aq, where bk hps, aq is the bonus, prhps, aq, { Ps,a,hV ref h`1 and { Ps,a,hpVh`1 ´ V ref h`1q are empirical estimates of rhps, aq, Ps,a,hV ref h`1 and Ps,a,hpVh`1 ´ V ref h`1q respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In addition, a very simple update rule Qhps, aq Ð { Ps,a,hVh`1 ` prhps, aq ` bk hps, aq is also in use to provide a guarantee of uniform convergence of estimated value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We make three major alterations to UCB-Advantage: ① We use empirical variances of rewards in bonuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ② Due to a more refined analysis, we remove the rOpHpn´3{4 ` qn´3{4qq term in bonuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ③ The reference value functions are updated in a capped-doubling manner (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Line ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Alteration ③ is crucial to make the main order term variance-dependent, because there exist constant gaps between reference values and optimal values, whose summation contributes to the regret as the main order term in UCB-Advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By choosing an appropriate number of updates, we can balance between the total constant gap and the deviation introduced by frequent updates, making the total variance the only factor in the main order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ δ, the regret of UCB-Advantage-V (Algorithm 2) run with ι “ 99 ˆ ln ˆ70002pHSAKq5 δ2 ˙ ` 1 ˙ and i‹ “ R1 2 log2 ˆ K H5S3Aι2 ˙V is bounded by RegretpKq ď rOp b mintVarΣ K, Var‹KuHSA ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H15S5A3Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We have Var‹ ď H2, so our result is strictly better than the rOp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H3SAK ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H33S8A6Kq regret of UCB-Advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 10 Algorithm 2 UCB-Advantage-V 1: Input and initialize: Logarithm term ι;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Stage lengths e1 “ H, ei`1 “ tp1 ` 1{Hqeiu and stage trigger set L Ð třj i“1 ei | j “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Reference trigger set R Ð t60000 ¨ 22iSAH3ι | i “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', i‹u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2: for ps, a, hq P S ˆ A ˆ rHs do 3: Nhps, aq Ð 0, q Nhps, aq Ð 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 4: θhps, aq Ð 0, φhps, aq Ð 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 5: Vhpsq Ð H ´ h ` 1, Qhps, aq Ð H ´ h ` 1, V ref h ps, aq Ð H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 6: qυhps, aq Ð 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 7: qµhps, aq Ð 0, qσhps, aq Ð 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 8: µref h ps, aq Ð 0, σref h ps, aq Ð 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 9: end for 10: \\\\ Main algorithm begins 11: for k “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', K do 12: for h “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', H do 13: Observe sk h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 14: Take action ak h “ arg maxa Qhpsk h, aq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 15: Receive reward rk h and observe sk h`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 16: Update accumulators: n :“ Nhpsk h, ak hq ` Ð 1, qn :“ q Nhpsk h, ak hq ` Ð 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' θ :“ θhpsk h, ak hq ` Ð rk h, φ :“ φhpsk h, ak hq ` Ð prk hq2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' qυ :“ qυhpsk h, ak hq ` Ð Vh`1psk h`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' qµ :“ qµhpsk h, ak hq ` Ð Vh`1psk h`1q ´ V ref h`1psk h`1q, qσ :“ qσhpsk h, ak hq ` Ð pVh`1psk h`1q ´ V ref h`1psk h`1qq2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' µref :“ µref h psk h, ak hq ` Ð V ref h`1psk h`1q, σref :“ σref h psk h, ak hq ` Ð pV ref h`1psk h`1qq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 17: \\\\ Reaching the end of a stage, update Q-function 18: if n P L then 19: Set prhpsk h, ak hq Ð θ n, z VarRpsk h, ak hq Ð φ n ´ ˆ θ n ˙2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ¯b Ð 2 c H2ι qn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' νref Ð σref n ´ ˆµref n ˙2 , qν “ qσ qn ´ ˆ qµ qn ˙2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' b Ð 4 c νrefι n ` 4 c qνι qn ` 2 d z VarRhι n ` 90Hι qn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Qhpsk h, ak hq Ð min " prhpsk h, ak hq ` qυ qn ` ¯b, prhpsk h, ak hq ` µref n ` qµ qn ` b, Qhpsk h, ak hq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Vhpsk hq Ð max a Qhpsk h, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 20: \\\\ Reset intra-stage accumulators 21: Set q Nhpsk h, ak hq Ð 0, qµhpsk h, ak hq Ð 0, qυhpsk h, ak hq Ð 0, qσhpsk h, ak hq Ð 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 22: end if 23: \\\\ Update reference value function 24: if ř aPA Nhpsk h, aq P R then V ref h psk hq Ð Vhpsk hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 25: end for 26: end for 11 Extra notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let V ref,k h denote V ref h at the beginning of the k-th episode, and V REF h :“ V ref,K`1 h denote the final reference value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let νref,k h , qνk h, bk h denote νref, qν, b for the value of Qk hpsk h, ak hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let N k hpsq denote ř a Nhps, aq at the beginning of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let nk h and qnk h be the total number of visits to psk h, ak h, hq prior to the current stage and during the stage immediately before the current stage with respect to the same triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='3 for the rigorous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' From Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020], the regret is roughly řK k“1 řH h“1pψk h`1 ` ξk h`1 ` φk h`1 ` bk hq, where ψk h`1 « V ref,k h`1 psk h`1q ´ V REF h`1psk h`1q, ξk h`1 « pPsk h,ak h,h ´ 1sk h`1qpV k h`1 ´ V ‹ h`1q, φk h`1 “ pPsk h,ak h,h ´ 1sk h`1qpV ‹ h`1 ´ V πk h`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' All these four terms are bounded loosely in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] such that they are all main order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To establish a variance-dependent regret, we prove that only the b term is the main order term after our aforementioned alterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The φ term is a martingale and shown to be rOpH2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For the rest terms, we need the following argument: N k hpsq ě N0pǫq “ rO ˆH5SA ǫ2 ˙ ùñ 0 ď V k h psq ´ V ‹ h psq ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Notice that the reference trigger set R in Algorithm 2 is composed of N0pβiq for i P ri‹s where βi :“ H{2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' There is a constant gap of at least βi‹ between V REF h psq and V ‹ h psq in the worst case, because the number of updates is capped by i‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This argument branches into two corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The first one is we can bound value gaps directly: K ÿ k“1 H ÿ h“1 pV k h psk hq ´ V ‹ h psk hqq2 ď rOpH6SAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This can be utilized to bound the ξ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The second one is that, we define Bref,k h psq :“ i‹ ÿ i“1 βi´1 1rN0pβi´1q ď N k hpsq ă N0pβiqs, then V ref,k h psq ´ V REF h psq ď Bref,k h psq, V ref,k h psq ´ V ‹ h psq ď Bref,k h psq ` βi‹ and ÿ k,h Bref,k h psk hq ď rOpH5S2A2i‹q, ÿ k,h pBref,k h psk hqq2 ď rOpH6S2Ai‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So we can directly bound the ψ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We show that νref,k h « rOpVpPsk h,ak h,h, V ‹ h`1q ` pBref,k h`1 psk h`1qq2 ` β2 i‹q, qνk h ď OppBref,k h`1 psk h`1qq2 ` β2 i‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The b term is hence bounded by rOp b VarΣ KHSA ` a H5SAK{22i‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Analogous to the proof of Theorem 7, the difference between VarΣ K and Var‹K is of a lower order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Finally, the lower order terms are rOp a H5SAK{22i‹ ` H5S2A2i‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We derive Theorem 10 by choosing the optimal i‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 12 Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We study deterministic MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume that Assumption 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let δ P p0, 1q be the confidence parameter and that H, S, A, K, δ be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ δ, the regret of UCB-Advantage-V (Algorithm 2) run with ι “ 99plnp70002pHSAKq5{δ2q ` 1q and i‹ “ P 1{2 ¨ log2pK{H5S3Aι2q T is bounded by RegretpKq ď rOp 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H15S5A3Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Notice that the regret under Assumption 3 is not constant which we desire, this may be due to some fundamental limit of model-free algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' However, since the research on model-free algorithms is still at its nascent stage and there lack thorough understanding, our result provides the first look into the potential of such algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Intuitively, for any algorithm to have a constant regret on deterministic MDPs, its value functions should also converge in a constant steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Previous model-free algorithms all use historical data to estimate value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' These data are biased because some of them are not up-to-date, making value functions hard to converge in a constant steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Here we identify difficulties for existing algorithms to be variance-dependent for all K-related terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Q-learning (UCB-B) [Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In their proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='3, when bounding |P3 ´ P4|, there is a variance-independent 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' t term in the gap between the estimations and true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Notice that their result is possible to be variance-dependent by not loosening a H7SAι{t ď H`H6SAι{t above their Equation(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='10) while introducing a variance-independent K1{4 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' UCB-Advantage [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' There are biases in the reference value functions, because they are updated for only finite times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If the update is not capped by a threshold, readers can easily verify that the ψ term will become a variance-independent main order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Q-EarlySettled-Advantage [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' There is a same issue about the constant gap between the reference value and the optimal value when bounding R3 defined in their Equation(39c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 7 Regret Lower Bounds We show that for any algorithm and any variance V, there always exists an MDP such that the regret main order terms of Theorem 7, Corollary 9 and Theorem 10 are tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This means that MVP-V and UCB-Advantage-V are minimax optimal for the class of variance-bounded MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The proofs for this section are deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume S ě 6, A ě 2, H ě 3 tlog2pS ´ 2qu and 0 ă V ď Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any algorithm π, there exists an MDP Mπ such that: ‚ It satisfies Conditions 1 and 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ VarΣ τ , Var‹ “ ΘpVq for any possible trajectory τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ For K ě SA, the expected regret of π in Mπ after K episodes satisfies E « K ÿ k“1 pV ‹ 1 psk 1q ´ V πk 1 psk 1qq ˇˇˇˇˇ Mπ, π ff “ Ωp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' VSAKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume S ě 6, A ě 2, H ě 3 tlog2pS ´ 2qu and 0 ă V ď OpH2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any algorithm π, there exists an MDP Mπ such that: ‚ VarΣ τ , Var‹ “ ΘpVq for any possible trajectory τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ‚ For K ě HSA, the expected regret of π in Mπ after K episodes satisfies E « K ÿ k“1 pV ‹ 1 psk 1q ´ V πk 1 psk 1qq ˇˇˇˇˇ Mπ, π ff “ Ωp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' VHSAKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 13 8 Conclusion We systematically study variance-dependent regret bounds for MDPs by introducing new notions of variances, proposing model-based and model-free algorithms respectively, and providing regret lower bounds for the class of variance-bounded MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Our results improve upon the previous algorithms and achieves minimax optimal regrets for the class of variance-bounded MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Our model-based algorithm is minimax optimal for deterministic MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Finally, we identify some possible limit of current model-free algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' One possible future direction is to find a new model-free algorithm with a constant regret for deterministic MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' References Shipra Agrawal and Randy Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Optimistic posterior sampling for reinforcement learning: worst-case regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Peter Auer, Thomas Jaksch, and Ronald Ortner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Near-optimal regret bounds for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in neural information processing systems, 21, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Mohammad Gheshlaghi Azar, Ian Osband, and R´emi Munos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Minimax regret bounds for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In ICML, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Yu Bai, Tengyang Xie, Nan Jiang, and Yu-Xiang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Provably efficient q-learning with low switching cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Peter L Bartlett and Ambuj Tewari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Regal: A regularization based algorithm for reinforcement learning in weakly communicating mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' arXiv preprint arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='2661, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Dimitri Bertsekas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Dynamic programming and optimal control: Volume I, volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Athena scientific, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Openai gym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' arXiv preprint arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='01540, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Liyu Chen, Mehdi Jafarnia-Jahromi, Rahul Jain, and Haipeng Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Implicit finite-horizon approximation and efficient optimal algorithms for stochastic shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In NeurIPS, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Christoph Dann, Tor Lattimore, and Emma Brunskill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Unifying pac and regret: Uniform pac bounds for episodic reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In NIPS, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Christoph Dann, Lihong Li, Wei Wei, and Emma Brunskill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Policy certificates: Towards accountable rein- forcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1507–1516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Omar Darwiche Domingues, Pierre M´enard, Emilie Kaufmann, and Michal Valko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Episodic reinforcement learning in finite mdps: Minimax lower bounds revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Vitaly Feldman, Katrina Ligett, and Sivan Sabato, editors, Proceedings of the 32nd International Conference on Algorithmic Learning Theory, vol- ume 132 of Proceedings of Machine Learning Research, pages 578–598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 16–19 Mar 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' URL https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='press/v132/domingues21a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Eyal Even-Dar, Shie Mannor, Yishay Mansour, and Sridhar Mahadevan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Action elimination and stopping conditions for the multi-armed bandit and reinforcement learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Journal of machine learning research, 7(6), 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Ronan Fruit, Matteo Pirotta, and Alessandro Lazaric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Near optimal exploration-exploitation in non- communicating markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 31, 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Ronan Fruit, Matteo Pirotta, Alessandro Lazaric, and Ronald Ortner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Efficient bias-span-constrained exploration-exploitation in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1578–1586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 14 Aur´elien Garivier, Pierre M´enard, and Gilles Stoltz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Explore first, exploit next: The true shape of regret in bandit problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Mathematics of Operations Research, 44, 02 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1287/moor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='0928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Nan Jiang and Alekh Agarwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Open problem: The dependence of sample complexity lower bounds on planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Conference On Learning Theory, pages 3395–3398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Is q-learning provably efficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in neural information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Chi Jin, Akshay Krishnamurthy, Max Simchowitz, and Tiancheng Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Reward-free exploration for reinforce- ment learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 4870–4879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Tor Lattimore and Csaba Szepesv´ari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Bandit Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Cambridge University Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1017/ 9781108571401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Gen Li, Laixi Shi, Yuxin Chen, Yuantao Gu, and Yuejie Chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Breaking the sample complexity barrier to regret-optimal model-free reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:17762–17776, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Odalric-Ambrym Maillard, Timothy A Mann, and Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' How hard is my mdp?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' the distribution- norm to the rescue”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 27, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Andreas Maurer and Massimiliano Pontil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Empirical bernstein bounds and sample-variance penalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In COLT, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Playing atari with deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='5602, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Gergely Neu and Ciara Pike-Burke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' A unifying view of optimism in episodic reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:1392–1403, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Ian Osband and Benjamin Van Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Why is posterior sampling better than optimism for reinforcement learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In International conference on machine learning, pages 2701–2710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Ian Osband, Daniel Russo, and Benjamin Van Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (more) efficient reinforcement learning via posterior sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 26, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Aldo Pacchiano, Philip Ball, Jack Parker-Holder, Krzysztof Choromanski, and Stephen Roberts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' On opti- mism in model-based reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='11911, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Daniel Russo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Worst-case regret bounds for exploration via randomized value functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Max Simchowitz and Kevin G Jamieson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Non-asymptotic gap-dependent regret bounds for tabular mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Mohammad Sadegh Talebi and Odalric-Ambrym Maillard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Variance-aware regret bounds for undiscounted reinforcement learning in mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Algorithmic Learning Theory, pages 770–805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Jean Tarbouriech, Runlong Zhou, Simon Shaolei Du, Matteo Pirotta, Michael Valko, and Alessandro Lazaric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Stochastic shortest path: Minimax, parameter-free and towards horizon-free regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Neural Information Processing Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Andrew J Wagenmaker, Yifang Chen, Max Simchowitz, Simon Du, and Kevin Jamieson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' First-order regret in reinforcement learning with linear function approximation: A robust estimation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 22384–22429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 15 Zhihan Xiong, Ruoqi Shen, and Simon S Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Randomized exploration is near-optimal for tabular mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='09703, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Haike Xu, Tengyu Ma, and Simon Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Fine-grained gap-dependent bounds for tabular mdps via adaptive multi-step bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Conference on Learning Theory, pages 4438–4472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Kunhe Yang, Lin Yang, and Simon Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Q-learning with logarithmic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In International Conference on Artificial Intelligence and Statistics, pages 1576–1584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Andrea Zanette and Emma Brunskill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In ICML, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Zihan Zhang and Xiangyang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Regret minimization for reinforcement learning by evaluating the optimal bias function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Zihan Zhang, Yuan Zhou, and Xiangyang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Almost optimal model-free reinforcement learning via reference- advantage decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:15198–15207, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Zihan Zhang, Xiangyang Ji, and Simon Shaolei Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Is reinforcement learning more difficult than bandits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' a near-optimal algorithm escaping the curse of horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In COLT, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Zihan Zhang, Yuan Zhou, and Xiangyang Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Model-free reinforcement learning: from clipped pseudo-regret to sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Proceedings of the 38th International Conference on Machine Learning, pages 12653–12662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PMLR, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Zihan Zhang, Xiangyang Ji, and Simon Shaolei Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Horizon-free reinforcement learning in polynomial time: the power of stationary policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In Annual Conference Computational Learning Theory, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Runlong Zhou, Ruosong Wang, and Simon S Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Horizon-free reinforcement learning for latent markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='11604, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 16 Appendix Table of Contents A Technical Lemmas 17 B Missing Proofs 18 A Technical Lemmas Lemma 14 (Hoeffding’s Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let Z, Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Zn be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' random variables with values in r0, bs and let δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then we have P «ˇˇˇˇˇErZs ´ 1 n nÿ i“1 Zi ˇˇˇˇˇ ą b c lnp2{δq 2n ff ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 15 (Bennett’s Inequality, Theorem 3 in Maurer and Pontil [2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let Z, Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Zn be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ran- dom variables with values in r0, bs and let δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define VrZs “ ErpZ ´ ErZsq2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then we have P «ˇˇˇˇˇErZs ´ 1 n n ÿ i“1 Zi ˇˇˇˇˇ ą c 2VrZs lnp2{δq n ` b lnp2{δq n ff ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 16 (Theorem 4 in Maurer and Pontil [2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let Z, Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Zn pn ě 2q be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' random variables with values in r0, bs and let δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define ¯Z “ 1 nZi and ˆVn “ 1 n řn i“1pZi ´ ¯Zq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then we have P » – ˇˇˇˇˇErZs ´ 1 n n ÿ i“1 Zi ˇˇˇˇˇ ą d 2 ˆVn lnp2{δq n ´ 1 ` 7b lnp2{δq 3pn ´ 1q fi fl ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 17 (Lemma 11 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let pMnqně0 be a martingale such that M0 “ 0 and |Mn ´ Mn´1| ď c for some c ą 0 and any n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let Varn “ řn k“1 ErpMk ´ Mk´1q2|Fk´1s for n ě 0, where Fk “ σpM1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Mkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then for any positive integer n and any ǫ, δ ą 0, we have that P ” |Mn| ě 2 a 2Varn lnp1{δq ` 2 a ǫ lnp1{δq ` 2c lnp1{δq ı ď 2 ˆ log2 ˆnc2 ǫ ˙ ` 1 ˙ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 18 (Lemma 10 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2022]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' be a sequence of random variables taking values in r0, ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define Fk “ σpX1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Xk´1q and Yk “ ErXk | Fks for k ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any δ ą 0, we have that P « Dn, nÿ k“1 Xk ě 3 n ÿ k“1 Yk ` l lnp1{δq ff ď δ, P « Dn, nÿ k“1 Yk ě 3 nÿ k“1 Xk ` l lnp1{δq ff ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 19 (Lemma 30 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any two random variables X, Y , we have VpXY q ď 2VpXqpsup |Y |q2 ` 2pErXsq2VpY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Consequently, sup |X| ď C implies VpX2q ď 4C2VpXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 20 (Bhatia–Davis Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any random variable X, VpXq ď psup X ´ErXsqpErXs´inf Xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 17 B Missing Proofs B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1 Justification for Definition 6 Let Xπ hpsq denote the random variable of cumulative reward starting from s as the h-th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Clearly, V π h psq “ ErXπ h psqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We denote Varπ hpsq :“ VpXπ h psqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since π P Π is deterministic, let a “ πhpsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Law of total variance states that VpY q “ ErVpY |Xqs ` VpErY |Xsq, so Varπ hpsq “ Er„Rhps,aq,s1„Ps,a,hrVpr ` Xπ h`1ps1qqs ` Vr„Rhps,aq,s1„Ps,a,hpErr ` Xπ h`1ps1qsq “ Er„Rhps,aq,s1„Ps,a,hrVpXπ h`1ps1qqs ` Vr„Rhps,aq,s1„Ps,a,hpr ` ErXπ h`1ps1qsq “ Es1„Ps,a,hrVarπ h`1ps1qs ` Vr„Rhps,aqprq ` Vs1„Ps,a,hpV π h`1ps1qq “ Ps,a,hVarπ h`1 ` VpRhps, aqq ` VpPs,a,h, V π h`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let dπ h P ∆pSq denote the state visitation distribution at the h-th step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', dπ hpsq :“ Pπrsh “ ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By induction, we can prove that (with ah “ πhpshq) Varπ 1psq “ H ÿ h“1 Esh„dπ hrVpRhpsh, ahqq ` VpPsh,ah,h, V π h`1qs “ Eπ « H ÿ h“1 ` VpRhpsh, ahqq ` VpPsh,ah,h, V π h`1q ˘ ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='2 Model-based Algorithm: MVP-V (Algorithm 1) Summary of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let sk h, ak h and rk h denote the state, action and reward at the h-th step of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let V k h psq, Qk hps, aq, nkps, aq and pP k s,a,s1 denote Vhpsq, Qhps, aq, nps, aq and pPs,a,s1 at the beginning of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let K be the set of indexes of episodes in which no update is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By the update rule, it is obvious that ˇˇKCˇˇ ď SAplog2pKHq ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let h0pkq be the first time an update is triggered in the k-th episode if there is an update in this episode and otherwise H ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define X0 :“ tpk, h0pkqq | k P KCu and X :“ tpk, hq | k P KC, h0pkq ` 1 ď h ď Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let Ipk, hq :“ 1rpk, hq R Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We use the “check” notation to denote the original value timed with Ipk, hq, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', qV k h :“ V k h Ipk, hq and qβk h :“ βk hIpk, hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We first run MVP-V (Algorithm 1) with ι “ 99plnpHSAK{δq ` 1q which is large enough for all the probabilistic inequalities to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This choice will make the success probability be 1´polypS, A, H, K, ιqδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The lemmas are also proved assuming this choice of ι at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Based on Lemma 7 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a], by Lemmas 23 and 25 we have that RegretpKq ď K ÿ k“1 H ÿ h“1 pPsk h,ak h qV k h`1 ´ qV k h`1psk h`1qq loooooooooooooooooooooomoooooooooooooooooooooon “:M1 ` K ÿ k“1 H ÿ h“1 qβk hpsk h, ak hq loooooooooomoooooooooon “:M2 ` K ÿ k“1 ˜ H ÿ h“1 rpsk h, ak hqIpk, hq ´ V πk 1 psk 1q ¸ loooooooooooooooooooooooomoooooooooooooooooooooooon “:M3 ` ˇˇKCˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We utilize Equation (39) in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a]: for any non-negative sequence pwk hqkPrKs,hPrHs, K ÿ k“1 H ÿ h“1 Ipk, hq nkpsk h, ak hq ď OpSAιq, K ÿ k“1 H ÿ h“1 d wk hIpk, hq nkpsk h, ak hq ď O ¨ ˝ g f f eSAι K ÿ k“1 H ÿ h“1 wk hIpk, hq ` SAι ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 18 Thus by Lemma 25 we have M2 ď O ¨ ˚ ˚ ˚ ˚ ˚ ˝ g f f f f e SA K ÿ k“1 H ÿ h“1 pVpRpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V k h`1qqIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' hq looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon «:M4 ι ` g f f f f e ΓSA K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V k h`1 ´ V ‹ h`1qIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' hq loooooooooooooooooooooooomoooooooooooooooooooooooon «:M5 ι ` g f f f f e SA K ÿ k“1 H ÿ h“1 pVpRpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qqIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' hq looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon «:M6 ι ` ΓSAι2 ˛ ‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We need to substitute Ipk, hq with Ipk, h`1q to get the precise definition of M4, M5 and M6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Such substitution only introduces an error of Op ˇˇKCˇˇq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By VpX ` Y q ď 2VpXq ` 2VpY q, M6 ď OpM4 ` M5q, and M4 ď OpM5 ` M6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If we use the former relation, M2 ď Op?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='SAM4ι ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ΓSAM5ι ` ΓSAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Plugging in Lemma 26 and Lemma 28, we have M2 ď O ¨ ˝a ΓSAM2ι ` g f f eSA K ÿ k“1 Varπkι ` ΓSAι2 ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality gives M2 ď O ¨ ˝ g f f eSA K ÿ k“1 Varπkι ` ΓSAι2 ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Lemma 8 of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a], M1 ď Op?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='M4ι ` ιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Further plugging in Lemma 27 gives RegretpKq ď M1 ` M2 ` M3 ` ˇˇKCˇˇ ď O ¨ ˝ g f f eSA K ÿ k“1 Varπkι ` ΓSAι2 ˛ ‚ď Op ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Var‹SAKι ` ΓSAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If we use the latter relation, M2 ď Op?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='SAM6ι ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ΓSAM5ι ` ΓSAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' First plug in Lemma 28, we have M2 ď Op?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ΓSAM2ι ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='SAM6ι ` ΓSAι2q, which implies M2 ď Op a SAM6ι ` ΓSAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For the regret, we need M1 ď Op?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='M4ι ` ιq, Lemmas 27 and 29 and M4 ď OpM5 ` M6q, so RegretpKq ď Op a SAM6ι ` ΓSAι2q ď Op b VarΣ KSAι ` ΓSAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The above results hold with probability at least 1 ´ 19HS2AKιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To establish the final result, we need to scale δ to make the success probability be 1 ´ δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We upper-bound ι by 100pHSAK{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' δ ` 1q and solve the inequality: 1900HS2AK ˆHSAK ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' δ ` 1 ˙ δ ď δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Take δ “ pδ1{3000H2S3A2K2q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By lnpHSAK{pδ{3000H2S3A2K2q2q ď Opιq, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 19 Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define the following events: E1 :“ $ & %@ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, ˇˇˇp pP k s,a,s1 ´ Ps,a,s1qV ‹ h`1 ˇˇˇ ď 2 d Vp pP k s,a, V ‹ h`1qι nkps, aq ` 14ι 3nkps, aq , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , (1) E2 :“ $ ’ & ’ % @ps, a, kq P S ˆ A ˆ rKs, ˇˇprkps, aq ´ rps, aq ˇˇ ď 2 g f f e z VarR kps, aqι nkps, aq ` 14ι 3nkps, aq , / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' / , (2) E3 :“ # @ps, a, s1, kq P S ˆ A ˆ S ˆ rKs, ˇˇˇ pP k s,a,s1 ´ Ps,a,s1 ˇˇˇ ď d 2Ps,a,s1ι nkps, aq ` 1rPs,a,s1 ą 0sι nkps, aq + , (3) E4 :“ # @ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, ˇˇˇp pP k s,a,s1 ´ Ps,a,s1qV ‹ h`1 ˇˇˇ ď d 2VpPs,a, V ‹ h`1qι nkps, aq ` ι nkps, aq + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (4) We have that PrE1s ě 1 ´ HSAKιδ, PrE2s ě 1 ´ SAKιδ, PrE3s ě 1 ´ S2AKιδ, PrE4s ě 1 ´ HSAKιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PrE1s and PrE2s are direct results by applying Lemma 16 and 1 x´1 ď 2 x, taking union bounds over the mentioned quantifiers and that nkps, aq P t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , tlog2pHKquu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' PrE3s and PrE4s are direct results by applying Lemma 15 and that Ps,a,s1 “ 0 ùñ pP k s,a,s1 “ 0, finally taking union bounds over the mentioned quantifiers and that nkps, aq P t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , tlog2pHKquu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 22 (Adapted from Lemma 14 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a] and Lemma 16 in Tarbouriech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any fixed dimension D, let Υ :“ tv P RD : v ě 0, }v}8 ď Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any two constants c1, c2 satisfying c2 1 ď c2, let f : ∆prDsq ˆ Υ ˆ R ˆ R Ñ R with fpp, v, n, ιq “ pv ` max " c1 b Vpp,vqι n , c2 Bι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then for all p P ∆prDsq, v P Υ and n, ι ą 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' fpp, v, n, ιq is non-decreasing in v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=', @pv, v1q P Υ2, v ď v1, it holds that fpp, v, n, ιq ď fpp, v1, n, ιq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' fpp, v, n, ιq ě pv ` c1 2 b Vpp,vqι n ` c2 2 Bι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 21, we have that for any ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, Qk hps, aq ě Q‹ hps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let k be fixed and omit it for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The proof is conducted by induction in the order of h “ H ` 1, H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' QH`1ps, aq “ 0 ě 0 “ Q‹ H`1ps, aq holds trivially for any ps, aq P S ˆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Now assume Qh`1ps, aq ě Q‹ h`1ps, aq for any ps, aq P S ˆ A, hence Vh`1psq “ maxaPA Qh`1ps, aq ě maxaPA Q‹ h`1ps, aq “ V ‹ h`1psq for any s P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' prps, aq ` pPs,aVh`1 ` bhps, aq “ ¨ ˝prps, aq ` 2 d z VarRps, aqι nps, aq ` 5ι nps, aq ˛ ‚` ¨ ˝ pPs,aVh`1 ` 4 d Vp pPs,a, Vh`1qι nps, aq ` 16ι nps, aq ˛ ‚ (i) ě rps, aq ` pPs,aVh`1 ` max $ & %4 d Vp pPs,a, Vh`1qι nps, aq , 16ι nps, aq , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon “fp pPs,a,Vh`1,ι,nps,aqq 20 (ii) ě rps, aq ` pPs,aV ‹ h`1 ` max $ & %4 d Vp pPs,a, V ‹ h`1qι nps, aq , 16ι nps, aq , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' looooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooon “fp pPs,a,V ‹ h`1,ι,nps,aqq (iii) ě rps, aq ` pPs,aV ‹ h`1 ` 2 d Vp pPs,a, V ‹ h`1qι nps, aq ` 8ι nps, aq (iv) ě rps, aq ` Ps,aV ‹ h`1 “ Q‹ hps, aq, where (i) is by E2 (Equation (2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by recognizing the last part as the function in Lemma 22, pc1, c2, Bq “ p4, 16, 1q satisfying that c2 1 ď c2 and using the first property based on the induction that Vh`1 ě V ‹ h`1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by the second property in Lemma 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iv) is E1 (Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So Qhps, aq ě Q‹ hps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 2SAKιδ, we have that for any ps, a, kq P S ˆ A ˆ rKs, z VarR kps, aq ď O ˆ VpRps, aqq ` ι nkps, aq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let s, a, k be fixed and omit k for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume that all the nps, aq realizations of Rps, aq are prpiq s,aqnps,aq i“1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We have that z VarRps, aq “ 1 nps, aq nps,aq ÿ i“1 prpiq s,aq2 ´ ¨ ˝ 1 nps, aq nps,aq ÿ i“1 rpiq s,a ˛ ‚ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' From Lemma 15, P » – ˇˇˇˇˇˇ 1 nps, aq nps,aq ÿ i“1 prpiq s,aq2 ´ ErRps, aq2s ˇˇˇˇˇˇ ą d 2VpRps, aq2qι nps, aq ` ι nps, aq fi fl ď δ, P » – ˇˇˇˇˇˇ 1 nps, aq nps,aq ÿ i“1 rpiq s,a ´ ErRps, aqs ˇˇˇˇˇˇ ą d 2VpRps, aqqι nps, aq ` ι nps, aq fi fl ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Lemma 19, VpRps, aq2q ď 4VpRps, aqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So ˇˇˇz VarRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ´ VpRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ˇˇˇ ď ˇˇˇˇˇˇ 1 nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq ÿ i“1 prpiq s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq2 ´ ErRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq2s ˇˇˇˇˇˇ ` ˇˇˇˇˇˇˇ ¨ ˝ 1 nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq ÿ i“1 rpiq s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a ˛ ‚ 2 ´ ErRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqs2 ˇˇˇˇˇˇˇ ď 2 d 2VpRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqqι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` ι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` ˇˇˇˇˇˇ 1 nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq ÿ i“1 rpiq s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a ` ErRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqs ˇˇˇˇˇˇ ˇˇˇˇˇˇ 1 nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq ÿ i“1 rpiq s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a ´ ErRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqs ˇˇˇˇˇˇ ď 2 d 2VpRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqqι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` ι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` 2 d 2VpRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqqι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` 2ι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq “ 4 d 2VpRps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqqι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` 3ι nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 21 Using 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='xy ď x ` y, we have z VarRps, aq ď VpRps, aqq ` 4 d 2VpRps, aqqι nps, aq ` 3ι nps, aq ď 2VpRps, aqq ` 11ι nps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemmas 21 and 24, we have that for any ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs Qk hps, aq ´ rps, aq ´ Ps,aV k h`1 ď βk hps, aq, where βk hps, aq “ O ¨ ˝ d VpPs,a, V k h`1qι nkps, aq ` d VpPs,a, V ‹ h`1qι nkps, aq ` d ΓVpPs,a, V k h`1 ´ V ‹ h`1qι nkps, aq ` d VpRps, aqqι nkps, aq ` Γι nkps, aq ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' kq P S ˆ A ˆ rHs ˆ rKs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' p pP k s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a ´ Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aqpV k h`1 ´ V ‹ h`1q “ ÿ s1PS p pP k s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1 ´ Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1qrV k h`1ps1q ´ V ‹ h`1ps1q ´ Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='apV k h`1 ´ V ‹ h`1qs (i) ď ÿ s1PS d 2Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1ι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ˇˇV k h`1ps1q ´ V ‹ h`1ps1q ´ Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='apV k h`1 ´ V ‹ h`1q ˇˇ ` ÿ s1PS 1rPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1 ą 0sι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ď d 2ι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ÿ s1PS b 1rPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1 ą 0sPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1rV k h`1ps1q ´ V ‹ h`1ps1q ´ Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='apV k h`1 ´ V ‹ h`1qs2 ` Γι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq (ii) ď d 2ι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq d ÿ s1PS 1rPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1 ą 0s d ÿ s1PS Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='s1rV k h`1ps1q ´ V ‹ h`1ps1q ´ Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='apV k h`1 ´ V ‹ h`1qs2 ` Γι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq “ d 2ΓVpPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V k h`1 ´ V ‹ h`1q nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` Γι nkps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by E3 (Equation (3));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Cauchy-Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' While retaining most other steps in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1 of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a] which require E4 (Equation (4)), we have βk hps, aq “ O ¨ ˚ ˝ d Vp pP k s,a, V k h`1qι nkps, aq ` d VpPs,a, V ‹ h`1qι nkps, aq ` d ΓVpPs,a, V k h`1 ´ V ‹ h`1qι nkps, aq ` g f f e z VarR kps, aqι nkps, aq ` Γι nkps, aq ˛ ‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Similar as the steps above Equation(36) in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021a] which require E3 (Equation (3)), we have that Vp pP k s,a, V k h`1q ď O ˆ VpPs,a, V k h`1q ` Γι nkps, aq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Combined with Lemma 24 we have the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 25, with probability at least 1 ´ 5Kιδ, we have that M4 ď O ˜ K ÿ k“1 Varπk ` M2 ` SAι2 ¸ ď OpVar‹K ` M2 ` SAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 22 Proof of Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define bck hps, aq :“ V k h psq ´ Ps,aV k h`1 ´ rps, aq P r´1, 1s, (5) which stands for bonus-correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Lemma 25, bck hps, aq ď βk hps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' However, we make the distinction here to be more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let BCk hpsq :“ bck hps, aq ` Ps,aBCk h`1 with a “ πk hpsq and boundary condition BCk H`1psq :“ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We can prove by induction that BCk hpsq “ pV k h ´ V πk h qpsq P r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (6) First, BCk Hpsq “ bck Hps, aq “ V k Hpsq ´ rps, aq “ V k Hpsq ´ V πk H psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then assume that BCk h`1 “ V k h`1 ´ V πk h`1, we have BCk hpsq “ bck hps, aq ` Ps,apV k h`1 ´ V πk h`1q “ V k h psq ´ Ps,aV k h`1 ´ rps, aq ` Ps,apV k h`1 ´ V πk h`1q “ V k h psq ´ prps, aq ` Ps,aV πk h`1q “ V k h psq ´ V πk h psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define a series of random variables and their truncated values: for any k P rKs, W k :“ H ÿ h“1 pVpRpsk h, ak hqq ` VpPsk h,ak h, V πk h`1qq, W k :“ mintW k, 50ιu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Correspondingly, define the following event, which means there is no truncation: EW :“ tW k “ W k, @k P rKsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We now calculate the probability of no truncation happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any fixed 1 ď k ď K, W k (i) ď H ÿ h“1 rPsk h,ak hpV πk h`1q2 ´ pV πk h`1psk h`1qq2s ` H ÿ h“1 rpV πk h psk hqq2 ´ pPsk h,ak hV πk h`1q2s ` H ÿ h“1 rpsk h, ak hq ´ pV πk 1 psk 1qq2 (ii) ď 2 g f f e2 H ÿ h“1 VpPsk h,ak h, pV πk h`1q2qι ` 6ι ` 2 H ÿ h“1 pV πk h psk hq ´ Psk h,ak hV πk h`1q ` H ÿ h“1 rpsk h, ak hq (iii) ď 4 g f f e2 H ÿ h“1 VpPsk h,ak h, V πk h`1qι ` 3 H ÿ h“1 rpsk h, ak hq ` 6ι ď 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2W kι ` 3 ` 6ι, where (i) is by Lemma 20, VpRps, aqq ď ErRps, aqs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1 ´ 2ιδ, and a2 ´ b2 ď pa ` bq maxta ´ b, 0u when a, b ě 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 19 with C “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality of W k, we have that W k ď 50ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This means PrEWs ě 1 ´ 2Kιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 23 From now on, we suppose EW holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We are ready to bound M4: M4 “ K ÿ k“1 H ÿ h“1 pVpRpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V k h`1qqIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q (i) ď 2 K ÿ k“1 H ÿ h“1 pVpRpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V πk h`1qqIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` 2 K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' BCk h`1qIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q looooooooooooooooooooooomooooooooooooooooooooooon “:Z (7) ď 2 K ÿ k“1 W k ` 2Z (ii) “ 2 K ÿ k“1 W k ` 2Z (iii) ď 6 K ÿ k“1 ErW k | Fks ` 2Z ` 100ι ď 6 K ÿ k“1 ErW k | Fks ` 2Z ` 100ι2 “ 6 K ÿ k“1 Varπk 1 psk 1q ` 2Z ` 100ι2 ď 6 K ÿ k“1 Varπk ` 2Z ` 100ι2 ď 6Var‹K ` 2Z ` 100ι2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Equation (6) and VpX ` Y q ď 2VpXq ` 2VpY q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by EW ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 18 with l “ 50ι, which happens with probability at least 1 ´ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' It remains to bound the quantity Z we encountered: Z “ K ÿ k“1 H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hpBCk h`1q2 ´ pBCk h`1psk h`1qq2sIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` K ÿ k“1 H ÿ h“1 rpBCk hpsk hqq2Ipk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' hq ´ pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hBCk h`1q2Ipk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1qs ´ K ÿ k“1 pBCk 1psk 1qq2 ď K ÿ k“1 H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hpBCk h`1q2 ´ pBCk h`1psk h`1qq2sIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` K ÿ k“1 H ÿ h“1 rpBCk hpsk hqq2 ´ pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hBCk h`1q2sIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` ˇˇKCˇˇ (i) ď 2 g f f e2 K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' pBCk h`1q2qIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1qι ` 6ι ` 2 K ÿ k“1 H ÿ h“1 maxtBCk hpsk hq ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hBCk h`1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 0uIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` ˇˇKCˇˇ (ii) ď 4 g f f e2 K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' BCk h`1qIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1qι ` 6ι ` 2 K ÿ k“1 H ÿ h“1 maxtbck hpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 0uIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` ˇˇKCˇˇ 24 ď 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Zι ` 6ι ` 2 K ÿ k“1 H ÿ h“1 qβk hpsk h, ak hq ` 2 ˇˇKCˇˇ ď 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Zι ` 2M2 ` 8SAι, where (i) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1´2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 19 with C “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality of Z, we have that Z ď 4M2 ` 48SAι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (8) So plugging back into the bound of M4 gives the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 26, with probability at least 1 ´ 2ιδ, we have that M3 ď Op a M4ι ` a M2ι ` SAιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' M3 “ K ÿ k“1 ˜ H ÿ h“1 rpsk h, ak hqIpk, hq ´ V πk 1 psk 1qIpk, 1q ¸ “ K ÿ k“1 ˜ H ÿ h“1 pV πk h psk hq ´ Psk h,ak hV πk h`1qIpk, hq ´ V πk 1 psk 1qIpk, 1q ¸ ď K ÿ k“1 H ÿ h“1 pV πk h`1psk h`1q ´ Psk h,ak hV πk h`1qIpk, h ` 1q ` ˇˇKCˇˇ (i) ď 2 g f f e2 K ÿ k“1 H ÿ h“1 VpPsk h,ak h, V πk h`1qIpk, h ` 1qι ` 6ι ` SAι (ii) ď 4 a M4ι ` 4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Zι ` 7SAι (iii) ď 4 a M4ι ` 8 a M2ι ` 35SAι, where (i) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1 ´ 2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Equation (6), VpX ` Y q ď 2VpXq ` 2VpY q and definition of Z (Equation (7));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 25, with probability at least 1 ´ 2ιδ, we have that M5 ď OpM2 ` SAιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define rV k h “ V k h ´ V ‹ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' M5 “ K ÿ k“1 H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hprV k h`1q2 ´ prV k h`1psk h`1qq2sIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` K ÿ k“1 H ÿ h“1 rprV k h psk hqq2Ipk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' hq ´ pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h rV k h`1q2Ipk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1qs ´ K ÿ k“1 prV k 1 psk 1qq2 ď K ÿ k“1 H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hprV k h`1q2 ´ prV k h`1psk h`1qq2sIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q 25 ` K ÿ k“1 H ÿ h“1 rprV k h psk hqq2 ´ pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h rV k h`1q2sIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` ˇˇKCˇˇ (i) ď 2 g f f e2 K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' prV k h`1q2qIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1qι ` 6ι ` 2 K ÿ k“1 H ÿ h“1 maxtrV k h psk hq ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h rV k h`1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 0uIpk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' h ` 1q ` ˇˇKCˇˇ (ii) ď 4 a 2M5ι ` 2 K ÿ k“1 H ÿ h“1 qβk hpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hq ` 2 ˇˇKCˇˇ ď 4 a 2M5ι ` 2M2 ` 8SAι,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Lemma 17 with c “ ǫ “ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' which happens with probability at least 1´2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 19 with C “ 1 and the following argument: by Lemma 25, rV k h psk hq ´ Psk h,ak h rV k h`1 ď rQk hpsk h, ak hq ´ Psk h,ak h rV k h`1 ď βk hpsk h, ak hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality of M5, we have that M5 ď 4M2 ` 48SAι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 4Kιδ, we have that for any k P rKs, VarΣ pkq ď Opιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' As a result, M6 ď VarΣ K ď OpKιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any k P rKs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' VarΣ pkq ď H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hpV ‹ h`1q2 ´ pV ‹ h`1psk h`1qq2s ` H ÿ h“1 rpV ‹ h psk hqq2 ´ pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hV ‹ h`1q2s ` H ÿ h“1 rpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hq ´ pV ‹ 1 psk 1qq2 (i) ď 2 g f f e2 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' pV ‹ h`1q2qι ` 6ι ` 2 H ÿ h“1 maxt V ‹ h psk hq ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hV ‹ h`1 loooooooooooomoooooooooooon ěQ‹ hpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hq´Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hV ‹ h`1ě0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 0u ` 1 (ii) ď 4 g f f e2 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qι ` 7ι ` 2 H ÿ h“1 pV ‹ h`1psk h`1q ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hV ‹ h`1q ` 2V ‹ 1 psk 1q looomooon ď2 (iii) ď 4 b 2VarΣ pkqι ` 9ι ` 4 g f f e2 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qι ` 12ι ď 8 b 2VarΣ pkqι ` 21ι,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Lemma 17 with c “ ǫ “ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' which happens with probability at least 1´2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 19 with C “ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 17 with c “ ǫ “ 1, which happens with probability at least 1 ´ 2ιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality of VarΣ pkq, we have that VarΣ pkq ď 170ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So taking a union bound over k we have the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 26 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='3 Model-free Algorithm: UCB-Advantage-V (Algorithm 2) Summary of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let sk h, ak h and rk h denote the state, action and reward at the h-th step of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let V k h psq, Qk hps, aq, V ref,k h , N k hps, aq and q N k hps, aq denote Vhpsq, Qhps, aq, Nhps, aq and q Nhps, aq at the beginning of the k-th episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let V REF h :“ V ref,K`1 h denote the final reference value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let N k hpsq :“ ř aPA N k hps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' N K`1 h ps, aq denotes the total number of visits of ps, a, hq after all K episodes are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define e1 “ H and ei`1 “ tp1 ` 1{Hqeiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The definition of stages is with respect to the triple ps, a, hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any fixed pair of k and h, we say that pk, hq falls in the j-th stage of ps, a, hq if and only if ps, aq “ psk h, ak hq and the total visit number of psk h, ak hq after the k-th episode is in přj´1 i“1 ei, řj i“1 eis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let qυk h, qµk h, qσk h, µref,k h , σref,k h , z VarR k h, ¯bk h, νref,k h , qνk h and bk h denote qυ, qµ, qσ, µref, σref, z VarR, ¯b, νref, qν and b calculated for the value of Qk hpsk h, ak hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For each k and h, let nk h be the total number of visits to psk h, ak h, hq prior to the current stage with respect to the same triple and let nk h be the number of visits to the same triple during the stage immediately before the current stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let lk h,i and qlk h,i denote the index of the i-th episode among the nk h and qnk h episodes defined above, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' When h and k are clear from the context, we use li and qli for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We first run UCB-Advantage-V (Algorithm 2) with ι “ 99plnpHSAK{δq ` 1q which is large enough for all the probabilistic inequalities to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This choice will make the success probability be 1 ´ polypS, A, H, K, ιqδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The lemmas are also proved assuming this choice of ι at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define ψk h`1 :“ 1 nk h nk h ÿ i“1 Psk h,ak h,hpV ref,li h`1 ´ V REF h`1q, ξk h`1 :“ 1 qnk h qnk h ÿ i“1 rPsk h,ak h,hpV ref,qli h`1 ´ V ‹ h`1q ´ pV ref,qli h`1 ps qli h`1q ´ V ‹ h`1ps qli h`1qqs, φk h`1 :“ Psk h,ak h,hpV ‹ h`1 ´ V πk h`1q ´ pV ‹ h`1psk h`1q ´ V πk h`1psk h`1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Combining Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='2 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] with Lemmas 30 to 33 and 36 to 41, we have that with probability at least 1 ´ 35HSAKιδ, RegretpKq ď K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 pψk h`1 ` ξk h`1 ` φk h`1 ` 2bk hq ď Op b VarΣ KHSAι ` a H5SAKι2{22i‹ ` H5S2A2i‹ι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Taking i‹ “ P 1{2 ¨ log2pK{H5S3Aι2q T , we have: RegretpKq ď Op b VarΣ KHSAι ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H15S5A3Kι6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Now we apply Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 46HSAKιδ, RegretpKq ď O ¨ ˝ g f f eHSAKι K ÿ k“1 Varπk ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H15S5A3Kι6 ˛ ‚ď Op ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Var‹HSAKι ` 4?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H15S5A3Kι6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The final result is established by scaling δ to make the success probability be 1 ´ δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We upper-bound ι by 100pHSAK{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' δ ` 1q and solve the inequality: 4600HSAK ˆHSAK ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' δ ` 1 ˙ δ ď δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Take δ “ pδ1{7000H2S2A2K2q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By lnpHSAK{pδ{7000H2S2A2K2q2q ď Opιq, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 27 Lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 15HSAKιδ, we have that for any ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, Q‹ hps, aq ď Qk`1 h ps, aq ď Qk hps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Recall that the update rule is: Qhpsk h, ak hq Ð min " prhpsk h, ak hq ` qυ qn ` ¯b loooooooooomoooooooooon ① , prhpsk h, ak hq ` µref n ` qµ qn ` b looooooooooooooomooooooooooooooon ② , Qhpsk h, ak hq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (9) We prove by induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Clearly for k “ 1 the argument is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For case ① in Equation (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' we have that (omit the subscripts of h and superscripts of k for simplicity) Qk`1 h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq “ rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` 1 qn qn ÿ i“1 V li h`1psli h`1q ` pprhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ´ rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ` ¯b (i) ě rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` 1 qn qn ÿ i“1 V ‹ h`1psli h`1q ` pprhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ´ rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ` ¯b (ii) ě rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1 ´ c H2ι 2qn ` pprhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ´ rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ` ¯b (iii) ě rhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ` Ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1 ´ c H2ι 2qn ´ c ι 2n ` ¯b ě Q‹ h`1ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by induction V u ě V ‹ for any 1 ď u ď k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 14 with b “ H, which holds with probability at least 1 ´ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 14 with b “ 1, which holds with probability at least 1 ´ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define χ1 :“ 1 n n ÿ i“1 pV ref,li h`1 psli h`1q ´ Ps,a,hV ref,li h`1 q, χ2 :“ 1 qn n ÿ i“1 rpV li h`1 ´ V ref,li h`1 qpsli h`1q ´ Ps,a,hpV li h`1 ´ V ref,li h`1 qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For case ② in Equation (9), we have that Qk`1 h ps, aq “ prhps, aq ` Ps,a,h ˜ 1 n n ÿ i“1 V ref,li h`1 ¸ ` Ps,a,h ˜ 1 qn qn ÿ i“1 pV qli h`1 ´ V ref,qli h`1 q ¸ ` χ1 ` χ2 ` b (i) ě rhps, aq ` Ps,a,h ˜ 1 qn qn ÿ i“1 V qli h`1 ¸ ` χ1 ` χ2 ` prhps, aq ´ prhps, aqq ` b (ii) ě rhps, aq ` Ps,a,hV ‹ h`1 ` χ1 ` χ2 ` prhps, aq ´ prhps, aqq ` b “ Q‹ hps, aq ` χ1 ` χ2 ` prhps, aq ´ prhps, aqq ` b, where (i) is by that V ref,u h`1 is non-increasing in u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by the induction V u ě V ‹ for any 1 ď u ď k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' From Lemma 17 with c “ H, ǫ “ c2, we have that with probability at least 1 ´ 2ιδ, |χ1| ď 1 n ¨ ˚ ˚ ˚ ˚ ˝ 2 g f f f f e 2 n ÿ i“1 VpPs,a,h, V ref,li h`1 q looooooooooomooooooooooon “:X ι ` 6Hι ˛ ‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (10) 28 Define χ3 :“ nÿ i“1 rPs,a,hpV ref,li h`1 q2 ´ pV ref,li h`1 psli h`1qq2s, χ4 :“ 1 n ˜ n ÿ i“1 V ref,li h`1 psli h`1q ¸2 ´ 1 n ˜ n ÿ i“1 Ps,a,hV ref,li h`1 ¸2 , χ5 :“ 1 n ˜ n ÿ i“1 Ps,a,hV ref,li h`1 ¸2 ´ nÿ i“1 pPs,a,hV ref,li h`1 q2, then it is easy to verify that X “ nνref ` χ3 ` χ4 ` χ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (11) By Lemma 17 with c “ H2, ǫ “ c2, and Lemma 19 with C “ H, we have that with probability at least 1 ´ 2ιδ, χ3 ď 2 g f f e2 nÿ i“1 VpPs,a,h, pV ref,li h`1 q2qι ` 6H2ι ď 4H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Xι ` 6H2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (12) By Lemma 17 with c “ H, ǫ “ c2, we have that with probability at least 1 ´ 2ιδ, χ4 ď 1 n ˇˇˇˇˇ nÿ i“1 pV ref,li h`1 psli h`1q ` Ps,a,hV ref,li h`1 q ˇˇˇˇˇ ˇˇˇˇˇ n ÿ i“1 pV ref,li h`1 psli h`1q ´ Ps,a,hV ref,li h`1 q ˇˇˇˇˇ ď 2Hp2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Xι ` 6Hιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (13) By Cauchy-Schwarz inequality, χ5 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, X ď nνref ` 18H2ι ` 8H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Xι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality, X ď 2nνref ` 164H2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Plugging back into Equation (10), we have χ1 ď 4 c νrefι n ` p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 82 ` 6qHι n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By a similar reasoning, we have that with probability at least 1 ´ 6ιδ, χ2 ď 4 c qνι qn ` p4 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 82 ` 6qHι qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Lemma 16 and 1 x´1 ď 2 x, we have that with probability at least 1 ´ δ, |rhps, aq ´ prhps, aq| ď 2 d z VarRhps, aqι n ` 14ι 3n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (14) Therefore, we have b ě |χ1| ` |χ2| ` |rhps, aq ´ prhps, aq|, which means Qk`1 h ps, aq ě Q‹ hps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 31 (Adapted from Lemma 5 and Corollary6 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020], and Corollary6 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 30, with probability at least 1 ´ HKδ, we have that for any ǫ P p0, Hs and any h P rHs, K ÿ k“1 1rV k h psk hq ´ V ‹ h psk hq ě ǫs ď 60000H5SAι ǫ2 “: N0pǫq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' As a result, for every state s we have that N k hpsq ě N0pǫq ùñ 0 ď V k h psq ´ V ‹ h psq ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 29 Proof of Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To derive the constant 60000, we only need to solve the inequality: K ÿ k“1 1rδk h ě ǫs ď řK k“1 1rδk h ě ǫsδk h ǫ ď 240H5{2 b }w}8 SAι řK k“1 1rδk h ě ǫs ` 3SAH3 }w}8 ǫ which is below Equation (48) in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020], using x ď a?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='x ` b ùñ x ď a2 ` 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The second part can be proven in a similar way as Corollary6 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 31, we have that K ÿ k“1 H ÿ h“1 pV k h psk hq ´ V ‹ h psk hqq ď Op ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H7SAKιq, K ÿ k“1 H ÿ h“1 pV k h psk hq ´ V ‹ h psk hqq2 ď OpH6SAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let c be a fixed constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h“1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='pV k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h psk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hq ´ V ‹ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ż H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˜ K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k“1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ÿ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='(ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ď cHK ` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ż H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ˆH6SAι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ď O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='cHK ` H6SAι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by n “ ş8 0 1rn ě xs dx for any non-negative real n and V ‹ h ď V k h ď H (Lemma 30);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Taking c “ a H5SAι{K gives the first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Similarly,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hq ´ V ‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h psk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hq ě xs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ď c2HK ` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ż H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='py ´ cqO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ˆH6SAι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='y2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='` ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ż H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ˆH6SAι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ď c2HK ` O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='H6SAι ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ż H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='¸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='“ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ˆ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c2HK ` H6SAι ln H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='˙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' and taking c “ 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' HK gives the second result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define βi :“ H{2i for i P t0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , i‹u, N 0 0 :“ 0 and N i 0 :“ N0pβiq “ 60000¨22iSAH3ι (defined in Lemma 31) for i P ri‹s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define Bref,k h psq :“ i‹ ÿ i“1 βi´1 1rN i´1 0 ď N k hpsq ă N i 0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 31, we have that V ref,k h psq ´ V REF h psq ď Bref,k h psq, V ref,k h psq ´ V ‹ h psq ď Bref,k h psq ` βi‹, and K ÿ k“1 H ÿ h“1 Bref,k h psk hq ď OpH5S2A2i‹ιq, K ÿ k“1 H ÿ h“1 pBref,k h psk hqq2 ď OpH6S2Ai‹ιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For i ď i‹ ´ 1, by Lemma 31, if N k hpsq ě N i 0 “ N0pβiq then V k h psq ´ V ‹ h psq ď βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let k0 be the minimum k such that N k hpsq ě N i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By the updating rule in Algorithm 2 and non-increasing property of V k (Lemma 30), it must satisfy that V k h psq ď V ref,k h psq ď V k0 h psq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since V ‹ h psq ď V REF h psq (Lemma 30), we have that V ref,k h psq ´ V REF h psq ď V k0 h psq ´ V ‹ h psq ď βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If N k hpsq ě N i‹ 0 “ N0pβi‹q then V ref,k h psq “ V REF h psq and V ref,k h psq ´ V ‹ h psq ď βi‹, which corresponds to Bref,k h psq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since the indicator functions are disjoint, we have the first part of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The remaining result is proven by: K ÿ k“1 H ÿ h“1 Bref,k h psk hq “ ÿ sPS K ÿ k“1 H ÿ h“1 i‹ ÿ i“1 H 2i´1 1rs “ sk h, N i´1 0 ď N k hpsq ă N i 0s 31 ď ÿ sPS H ÿ h“1 i‹ ÿ i“1 H 2i´1 N i 0 ď O ˜ SH i‹ ÿ i“1 H 2i ¨ 22iSAH3ι ¸ ď OpH5S2A2i‹ιq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' K ÿ k“1 H ÿ h“1 pBref,k h psk hqq2 “ K ÿ k“1 H ÿ h“1 i‹ ÿ i“1 β2 i´1 1rN i´1 0 ď N k hpsq ă N i 0s ď O ˜ SH i‹ ÿ i“1 H2 22i ¨ 22iSAH3ι ¸ ď OpH6S2Ai‹ιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 34 (Lemma 11 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any non-negative weights pwhps, aqqsPS,aPA,hPrHs and α P p0, 1q, it holds that K ÿ k“1 H ÿ h“1 whpsk h, ak hq pnk hqα ď 2α 1 ´ α ÿ s,a,h whps, aqpN K`1 h ps, aqq1´α, K ÿ k“1 H ÿ h“1 whpsk h, ak hq pqnk hqα ď 22αHα 1 ´ α ÿ s,a,h whps, aqpN K`1 h ps, aqq1´α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' In the case α “ 1, it holds that K ÿ k“1 H ÿ h“1 whpsk h, ak hq nk h ď 2 ÿ s,a,h whps, aq ln N K`1 h ps, aq, K ÿ k“1 H ÿ h“1 whpsk h, ak hq qnk h ď 4H ÿ s,a,h whps, aq ln N K`1 h ps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any non-negative sequence pXk hqkPrKs,hPrHs, we have that K ÿ k“1 H ÿ h“1 1 nk h nk h ÿ i“1 X lk h,i h ď 2ι K ÿ k“1 H ÿ h“1 Xk h, K ÿ k“1 H ÿ h“1 1 qnk h qnk h ÿ i“1 X qlk h,i h ď ˆ 1 ` 1 H ˙ K ÿ k“1 H ÿ h“1 Xk h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Refer to Equation (58) in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] for the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Refer to Equa- tion (15) and the paragraph below it in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020] for the second inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 33, with probability at least 1 ´ δ, we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 ψk h`1 ď OpH5S2A2i‹ιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 32 Proof of Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since ψk h is non-negative and p1 ` 1{Hqh´1 ď 3 when h ď H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 ψk h`1 ď O ˜ K ÿ k“1 H ÿ h“1 ψk h`1 ¸ “ O ¨ ˝ K ÿ k“1 H ÿ h“1 1 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='lk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 ´ V REF h`1q ˛ ‚ (i) ď O ¨ ˝ K ÿ k“1 H ÿ h“1 1 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hB ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='lk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 ˛ ‚ (ii) ď O ˜ K ÿ k“1 H ÿ h“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hBref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k h`1 ¸ (iii) ď O ˜ K ÿ k“1 H ÿ h“1 Bref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k h psk hq ` Hι ¸ (iv) ď OpH5S2A2i‹ιq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Lemma 33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 18 with l “ H, which holds with probability at least 1 ´ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iv) is by Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 32, with probability at least 1 ´ 5HSAιδ, we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 ξk h`1 ď OpH7{2SAι3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We borrow the beginning part of proof of Lemma 15 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020], and perform more fine-grained analyses on the remaining part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let xk h be the number of elements in current stage with respect to psk h, ak h, hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define θj h`1 :“ ˆ 1 ` 1 H ˙h´1 K ÿ k“1 1 qnk h qnk h ÿ i“1 1rqlk h,i “ js, rθj h`1 :“ ˆ 1 ` 1 H ˙h´1 Y p1 ` 1{Hqxj h ] xj h ď 3, and K :“ tpk, hq | θk h`1 “ rθk h`1u, KK h ps, aq :“ tk | psk h, ak hq “ ps, aq, k is in the second last stage of ps, a, hqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let θh`1ps, aq and rθh`1ps, aq denote θk h`1 and rθk h`1 respectively for some k P KK h ps, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Equation (61) in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 ξk h`1 ď K ÿ k“1 H ÿ h“1 rθk h`1rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV k h`1 ´ V ‹ h`1q ´ pV k h`1psk h`1q ´ V ‹ h`1psk h`1qqs looooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooon “:① ` ÿ pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hqPK pθk h`1 ´ rθk h`1qrPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV k h`1 ´ V ‹ h`1q ´ pV k h`1psk h`1q ´ V ‹ h`1psk h`1qqs looooooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooooon “:② .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 33 We now bound both terms: ① (i) ď O ¨ ˚ ˚ ˚ ˚ ˝ g f f f f e K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' pV k h`1 ´ V ‹ h`1qq loooooooooooooooooooooomoooooooooooooooooooooon “:Z ι ` Hι ˛ ‹‹‹‹‚ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ② (ii) “ ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h pθh`1ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ´ rθh`1ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ÿ kPKK h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV k h`1 ´ V ‹ h`1q ´ pV k h`1psk h`1q ´ V ‹ h`1psk h`1qqs ď ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h ˇˇˇθh`1ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ´ rθh`1ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ˇˇˇ ˇˇˇˇˇˇ ÿ kPKK h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV k h`1 ´ V ‹ h`1q ´ pV k h`1psk h`1q ´ V ‹ h`1psk h`1qqs ˇˇˇˇˇˇ (iii) ď O ¨ ˝ ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h ¨ ˝ d ÿ kPKK h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V k h`1 ´ V ‹ h`1qι ` Hι ˛ ‚ ˛ ‚ (iv) ď O ¨ ˝ d HSAι ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h ÿ kPKK h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='aq VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V k h`1 ´ V ‹ h`1q ` H2SAι ˛ ‚ (v) ď Op ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' HSAZι ` H2SAιq, where (i) is by Lemma 17 with c “ 3H, ǫ “ c2, which holds with probability at least 1 ´ 2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by the step above Equation (63) in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2020];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by ˇˇˇθh`1ps, aq ´ rθh`1ps, aq ˇˇˇ ď 3 and Lemma 17 with c “ H, ǫ “ c2, which holds with probability at least 1 ´ 2HSAιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iv) is by Cauchy-Schwarz inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (v) is by the following argument: for any non-negative sequence pXk hqkPrKs,hPrHs, ÿ s,a,h ÿ kPKK h ps,aq Xk h “ K ÿ k“1 H ÿ h“1 Xk h ÿ s,a,h1 ÿ k1PKK h1ps,aq 1rpk1, h1q “ pk, hqs “ K ÿ k“1 H ÿ h“1 Xk h ÿ s,a 1rk P KK h ps, aqs ď K ÿ k“1 H ÿ h“1 Xk h ÿ s,a 1rpsk h, ak hq “ ps, aqs ď K ÿ k“1 H ÿ h“1 Xk h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' It remains to bound Z: Z ď K ÿ k“1 H ÿ h“1 Psk h,ak h,hpV k h`1 ´ V ‹ h`1q2 (i) ď O ˜ K ÿ k“1 H ÿ h“1 pV k h`1psk h`1q ´ V ‹ h`1psk h`1qq2 ` H2ι ¸ (ii) ď OpH6SAι2q, where (i) is by Lemma 18 with l “ H2, which holds with probability at least 1 ´ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 34 Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 6ιδ, we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 φk h`1 ď OpHιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since p1 ` 1{Hqh´1 ď 3 when h ď H, we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 φk h`1 “ K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 rPsk h,ak h,hpV ‹ h`1 ´ V πk h`1q ´ pV ‹ h`1psk h`1q ´ V πk h`1psk h`1qqs (i) ď O ¨ ˚ ˚ ˚ ˚ ˝ g f f f f e K ÿ k“1 H ÿ h“1 VpPsk h,ak h,h, V ‹ h`1 ´ V πk h`1q loooooooooooooooooooomoooooooooooooooooooon “:Y ι ` Hι ˛ ‹‹‹‹‚ , where (i) is by Lemma 17 with c “ 3H, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Next we bound Y : Y “ K ÿ k“1 H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV ‹ h`1 ´ V πk h`1q2 ´ pV ‹ h`1psk h`1q ´ V πk h`1psk h`1qq2s ` K ÿ k“1 H ÿ h“1 tpV ‹ h psk hq ´ V πk h psk hqq2 ´ rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV ‹ h`1 ´ V πk h`1qs2u ´ pV ‹ 1 psk 1q ´ V πk 1 psk 1qq2 (i) ď 2 g f f e2 K ÿ k“1 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' pV ‹ h`1 ´ V πk h`1q2qι ` 6H2ι ` 2H K ÿ k“1 H ÿ h“1 maxtV ‹ h psk hq ´ V πk h psk hq ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV ‹ h`1 ´ V πk h`1q loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon ěQ‹ hpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hq´rhpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hq´Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1“0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 0u (ii) ď 4H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Y ι ` 6H2ι ` 2H K ÿ k“1 H ÿ h“1 rV ‹ h`1psk h`1q ´ V πk h`1psk h`1q ´ Psk h,ak h,hpV ‹ h`1 ´ V πk h`1qs ` 2HpV ‹ 1 psk 1q ´ V πk 1 psk 1qq loooooooooooooomoooooooooooooon ď2H2 (iii) ď 4H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Y ι ` 8H2ι ` 4H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Y ι ` 12H2ι ď 8H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Y ι ` 20H2ι, where (i) is by Lemma 17 with c “ H2, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ, and a2 ´ b2 ď pa ` bq maxta ´ b, 0u when a, b ě 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 19 with C “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by by Lemma 17 with c “ H, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality of Y , we have that Y ď 168H2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Plugging Y back gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemma 33, with probability at least 1 ´ 4ιδ, we have that for any pk, hq P rKs ˆ rHs νref,k h ď O ¨ ˝VpPsk h,ak h,h, V ‹ h`1q ` H2ι ` řnk h i“1 Psk h,ak h,hpBref,li h`1 q2 nk h ` β2 i‹ ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 35 Proof of Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let pk, hq be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We prove by first bounding νref,k n ´ 1 nk h řnk h i“1 VpPsk h,ak h,h, V ref,li h`1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Equation (11), νref,k n ´ 1 nk h nk h ÿ i“1 VpPsk h,ak h,h, V ref,li h`1 q looooooooooooomooooooooooooon “X(abusing notation) “ ´χ3 ` χ4 ` χ5 nk h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since we can use Equations (12) and (13), we only need to bound ´χ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ´χ5 “ nk h ÿ i“1 pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 q2 ´ 1 nk h ¨ ˝ nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ˛ ‚ 2 (i) ď nk h ÿ i“1 pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 q2 ´ 1 nk h ¨ ˝ nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV REF h`1 ˛ ‚ 2 “ nk h ÿ i“1 pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ` Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV REF h`1 qpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV REF h`1q ď 2H nk h ÿ i“1 pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV REF h`1 q (ii) ď 2H nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hBref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by V ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ě V REF h`1 (Lemma 30);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Combining bounds of χ3 and χ4, we have: νref,k n ´ X nk h ď 8H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Xι ` 18H2ι ` 2H řnk h i“1 Psk h,ak h,hBref,li h`1 nk h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since 8H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2Xι ď X ` 32H2ι, we have: νref,k n ´ 2X nk h ď O ¨ ˝H2ι ` H řnk h i“1 Psk h,ak h,hBref,li h`1 nk h ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For the desired result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' we finally bound: X nk h ´ 2VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1q “ 1 nk h nk h ÿ i“1 pVpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 q ´ 2VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qq (i) ď 2 nk h nk h ÿ i“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ´ V ‹ h`1q ď 2 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ´ V ‹ h`1q2 (ii) ď 2 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpBref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 ` βi‹q2 36 ď 4 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpBref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='li h`1 q2 ` 4β2 i‹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by VpX ` Y q ď 2VpXq ` 2VpY q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So by HPsk h,ak h,hBref,li h`1 ď OpH2 ` Psk h,ak h,hpBref,li h`1 q2q we have the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 40 (Analogous to Lemma 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 2HSAKιδ, we have that for any ps, a, h, kq P S ˆ A ˆ rHs ˆ rKs, z VarR k hps, aq ď O ˆ VpRhps, aqq ` ι nk hps, aq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Conditioned on the successful events of Lemmas 39 and 40, with probability at least 1 ´ δ, we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 bk h ď Op b VarΣ KHSAι ` a H5SAKι2{22i‹ ` H4S3{2Ai‹1{2ι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since bk h is non-negative and p1 ` 1{Hqh´1 ď 3 when h ď H, we have that K ÿ k“1 H ÿ h“1 ˆ 1 ` 1 H ˙h´1 bk h ď O ¨ ˚ ˝ K ÿ k“1 H ÿ h“1 ¨ ˚ ˝ d νref,k h ι nk h ` d qνk hι qnk h ` g f f e z VarR k hι nk h ` Hι qnk h ˛ ‹‚ ˛ ‹‚ ď O ¨ ˝ K ÿ k“1 H ÿ h“1 ¨ ˝ d νref,k h ι nk h ` d qνk hι qnk h ` d VpRhpsk h, ak hqqι nk h ` Hι qnk h ˛ ‚ ˛ ‚, where the last step is by Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Using Lemma 34, we have that K ÿ k“1 H ÿ h“1 Hι qnk h ď OpH3SAι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Next, we bound the terms of νref and qν separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Plugging in Lemma 39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' we have K ÿ k“1 H ÿ h“1 ¨ ˝ d νref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k h ι nk h ` d VpRhpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqqι nk h ˛ ‚ ď O ¨ ˚ ˝ K ÿ k“1 H ÿ h“1 d pVpRhpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qqι nk h ` K ÿ k“1 H ÿ h“1 Hι nk h ` K ÿ k“1 H ÿ h“1 1 nk h g f f e nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpB ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='lk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 q2ι ` K ÿ k“1 H ÿ h“1 d β2 i‹ι nk h ˛ ‹‚ (i) ď O ¨ ˚ ˝ ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h b N K`1 h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqpVpRhpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qqι ` H2SAι2 37 ` K ÿ k“1 H ÿ h“1 c ι nk h g f f e 1 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpB ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='lk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 q2 ` b β2 i‹ι ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h b N K`1 h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ˛ ‹‚ (ii) ď O ¨ ˚ ˝ d HSAι ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h N K`1 h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqpVpRhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ` VpPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qq ` H2SAι2` ` g f f e K ÿ k“1 H ÿ h“1 ι nk h g f f e K ÿ k“1 H ÿ h“1 1 nk h nk h ÿ i“1 Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpB ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='lk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 q2 ` d HSAβ2 i‹ι ÿ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h N K`1 h ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aq ˛ ‹‚ (iii) ď O ¨ ˚ ˝ g f f f f f e HSAι K ÿ k“1 H ÿ h“1 pVpRhpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hqq ` VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qq loooooooooooooooooooooooooooomoooooooooooooooooooooooooooon “VarΣ K ` H2SAι2 ` b H2SAβ2 i‹Kι ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' HSAι2 g f f e K ÿ k“1 H ÿ h“1 Psk h,ak h,hpBref,k h`1 q2 ˛ ‹‚ (iv) ď O ¨ ˝ b VarΣ KHSAι ` H2SAι2 ` b H2SAβ2 i‹Kι ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' HSAι2 g f f e K ÿ k“1 H ÿ h“1 pBref,k h psk hqq2 ` Hι ˛ ‚ (v) ď Op b VarΣ KHSAι ` a H4SAKι{22i‹ ` H7{2S3{2Ai‹1{2ι3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Lemma 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Cauchy-Schwarz inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemmas 34 and 35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iv) is by Lemma 18 with l “ H, which happens with probability at least 1 ´ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (v) is by Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' K ÿ k“1 H ÿ h“1 qνk h ď K ÿ k“1 H ÿ h“1 1 qnk h qnk h ÿ i“1 pV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 ps qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1q ´ V qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1ps qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1qq2 ď K ÿ k“1 H ÿ h“1 1 qnk h qnk h ÿ i“1 pV ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 ps qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1q ´ V ‹ h`1ps qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1qq2 (i) ď K ÿ k“1 H ÿ h“1 1 qnk h qnk h ÿ i“1 pB ref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1 ps qlk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='i h`1q ` βi‹q2 (ii) ď O ˜ K ÿ k“1 H ÿ h“1 pBref,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='k h psk hqq2 ` β2 i‹HK ¸ (iii) ď OpH6S2Ai‹ι ` H3K{22i‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Lemma 33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So K ÿ k“1 H ÿ h“1 d qνk hι qnk h ď g f f e K ÿ k“1 H ÿ h“1 ι qnk h g f f e K ÿ k“1 H ÿ h“1 qνk h (i) ď OpH4S3{2Ai‹1{2ι3{2 ` a H5SAKι2{22i‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by Lemma 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 38 Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' With probability at least 1 ´ 11Kιδ, the following results hold: For any k P rKs, VarΣ pkq ď OpH2ιq, hence VarΣ K ď OpH2Kιq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Alternatively, VarΣ K ď O ˜ K ÿ k“1 Varπk ` H2ι2 ¸ ď OpVar‹K ` H2ι2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We first prove the result depending on VarΣ K similar to Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any k P rKs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' VarΣ pkq (i) ď H ÿ h“1 rPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hpV ‹ h`1q2 ´ pV ‹ h`1psk h`1qq2s ` H ÿ h“1 rpV ‹ h psk hqq2 ´ pPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1q2s ` H ÿ h“1 rhpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' ak hq ´ pV ‹ 1 psk 1qq2 (ii) ď 2 g f f e2 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' pV ‹ h`1q2qι ` 6H2ι ` 2H H ÿ h“1 maxt V ‹ h psk hq ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1 looooooooooooomooooooooooooon ěQ‹ hpsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak hq´Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1ě0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 0u ` H (iii) ď 4H g f f e2 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qι ` 7H2ι ` 2H H ÿ h“1 pV ‹ h`1psk h`1q ´ Psk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='hV ‹ h`1q ` 2HV ‹ 1 psk 1q loooomoooon ď2H2 (iv) ď 4H b 2VarΣ pkqι ` 9H2ι ` 4H g f f e2 H ÿ h“1 VpPsk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='ak h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V ‹ h`1qι ` 12Hι ď 8H b 2VarΣ pkqι ` 21H2ι,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' where (i) is by by Lemma 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' VpRhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqq ď ErRhps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' aqs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 17 with c “ H2, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iii) is by Lemma 19 with C “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (iv) is by Lemma 17 with c “ H, ǫ “ c2, which happens with probability at least 1 ´2ιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality of VarΣ pkq, we have that VarΣ pkq ď 170H2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So taking a union bound over k we have the first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Next we prove the result depending on Var‹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This is similar to the proof of Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Define a series of random variables and their truncated values: for any k P rKs, W k :“ H ÿ h“1 pVpRhpsk h, ak hqq ` VpPsk h,ak h,h, V πk h`1qq, W k :“ mintW k, 50H2ιu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By VpPs,a,h, V ‹ h`1q ď 2VpPs,a,h, V πk h`1q ` 2VpPs,a,h, V ‹ h`1 ´ V πk h`1q, we know that VarΣ K ď 2 K ÿ k“1 W k ` 2Y, where Y ď OpH2ιq (with probability at least 1 ´ 4ιδ) is defined in the proof of Lemma 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Correspondingly, define the following event, which means there is no truncation: EW :“ tW k “ W k, @k P rKsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 39 For any fixed 1 ď k ď K, W k ď H ÿ h“1 rPsk h,ak h,hpV πk h`1q2 ´ pV πk h`1psk h`1qq2s ` H ÿ h“1 rpV πk h psk hqq2 ´ pPsk h,ak h,hV πk h`1q2s ` H ÿ h“1 rhpsk h, ak hq ´ pV πk 1 psk 1qq2 (i) ď 2 g f f e2 H ÿ h“1 VpPsk h,ak h,h, pV πk h`1q2qι ` 6H2ι ` 2H H ÿ h“1 pV πk h psk hq ´ Psk h,ak h,hV πk h`1q ` H (ii) ď 4H g f f e2 H ÿ h“1 VpPsk h,ak h,h, V πk h`1qι ` 7H2ι ` 2H H ÿ h“1 rhpsk h, ak hq ď 4H ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2W kι ` 9H2ι, where (i) is by Lemma 17 with c “ H2, ǫ “ c2, which happens with probability at least 1 ´ 2ιδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (ii) is by Lemma 19 with C “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Solving the inequality, W k ď 50H2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This means, PrEWs ě 1 ´ 2Kιδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Now on suppose EW holds, then K ÿ k“1 W k “ K ÿ k“1 W k (i) ď 3 K ÿ k“1 ErW k | Fks ` 50H2ι2 ď 3 K ÿ k“1 ErW k | Fks ` 50H2ι2 “ 3 K ÿ k“1 Varπk 1 psk 1q ` 50H2ι2 ď 3 K ÿ k“1 Varπk ` 50H2ι2 ď 3Var‹K ` 50H2ι2, where (i) is by Lemma 18 with l “ 50H2ι, which happens with probability at least 1 ´ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='4 Proof of Lower Bounds We modify Theorem 9 in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021] for a bounded-reward, time-homogeneous lower bound (Theorem 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Theorem 13 is much more straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To this end, we borrow necessary notations from Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021], adapted to time-homogeneous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' A policy π interacting with an MDP M defines a stochastic process denote by ppSk h, Ak h, Rk hqhPrHsqkě1, where Sk h, Ak h and Rk h are the random variables representing the state, the action and the reward at time h of episode k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' As explained by Lattimore and Szepesv´ari [2020], the Ionescu-Tulcea theorem ensures the existence of probability space pΩ, F, PMq such that PMrSk h`1 “ s|Ak h, Ik hs “ Pps|Sk h, Ak hq, and PMrAk h “ a|Ik hs “ πk hpa|Ik hq, where π “ pπk hqkPrKs,hPrHs and Ik h :“ pS1 1, A1 1, R1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , S1 H, A1 H, R1 H, S2 1, A2 1, R2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Sk´1 H , Ak´1 H , Rk´1 H , Sk 1 , Ak 1, Rk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' , Sk hq is the random vector containing all state-action pairs observed up to step h of episode k, but not including Ak h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Here we assume the rewards are deterministic as in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Next, we denote by PIK H M 40 the pushforward measure of IK H under PM, PIK H M riK Hs :“ PMrIK H “ iK Hs “ K ź k“1 H ź h“1 πk hpak h|ik hqPpsk h`1|sk t , ak t q, (15) where iK H is a realization of IK H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Definition 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' The Kullback-Leibler divergence between two distributions P1 and P2 on a measurable space pΩ, Gq is defined as KLpP1, P2q :“ ż Ω ln ˆdP1 dP2 pωq ˙ dP1pωq, if P1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' P2 and `8 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For Bernoulli distributions, we define @pp, qq P r0, 1s2, klpp, qq :“ KLpBppq, Bpqqq “ p ln ˆp q ˙ ` p1 ´ pq ln ˆ1 ´ p 1 ´ q ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 44 (Adapted from Lemma 5 in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Let M and M1 be two MDPs that are identical except for their transition probabilities, denoted by P and P 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume that we have @ps, aq, Pp¨|s, aq !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' P 1p¨|s, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Then for any K, KL ´ PIK H M , PIK H M1 ¯ “ ÿ ps,aqPSˆA EM “ N K s,a ‰ KLpPp¨|s, aq, P 1p¨|s, aqq, where N K s,a :“ řK k“1 řH h“1 1rpSk h, Ak hq “ ps, aqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Lemma 45 (Lemma 1 in Garivier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2016]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Consider a measurable space pΩ, Fq equipped with two distributions P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any F-measurable function Z : Ω Ñ r0, 1s, we have KLpP1, P2q ě klpE1rZs, E2rZsq, where E1 and E2 are the expectations under P1 and P2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We retain most of the proof of Theorem 9 and Appendix C in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021], while incorporating the hard instance design in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1 in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Namely, we change the A-ary tree in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021] with the binary tree in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This change does not affect the proof, while circumventing the requirement of S “ 3 ` pAd ´ 1q{pA ´ 1q where d is the tree height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We can find S1 “ 2 ` 2tlog2pS´2qu “ ΩpSq and replace S with S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We still use d “ tlog2pS ´ 2qu to denote the tree height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We change the transition at sg: Ppsb|sg, aq “ 1 for any a P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This means for any trajectory, the agent can only get reward once, then loops at sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Another important change in design is to scale the reward at sg by t ď 1, with t depending on V the variance we desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So rpsg, aq “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To be precise, let E0 and Epℓ‹,a‹q be the expectation taken with respect to the reference MDP (with no special leaf-action pair) and Mpℓ‹,a‹q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We have that RKpπ, Mpℓ‹,a‹qq ě tKε ˆ 1 ´ 1 K Epℓ‹,a‹q ” N K pℓ‹,a‹q ı˙ , where N K pℓ‹,a‹q “ řK k“1 1rpSk d`1, Ak d`1q “ ps, aqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Hence, max pℓ‹,a‹q RKpπ, Mpℓ‹,a‹qq ě tKε ¨ ˝1 ´ 1 LAK ÿ pℓ‹,a‹q Epℓ‹,a‹q ” N K pℓ‹,a‹q ı ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' (16) 41 Since N K pℓ‹,a‹q{K P r0, 1s, by Lemma 45, kl ˆ 1 K E0 ” N K pℓ‹,a‹q ı , 1 K Epℓ‹,a‹q ” N K pℓ‹,a‹q ı˙ ď KLpPIK H 0 , PIK H pℓ‹,a‹qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Lemma 44, KL ´ PIK H 0 , PIK H pℓ‹,a‹q ¯ “ E0 ” N K pℓ‹,a‹q ı kl ˆ1 2, 1 2 ` ε ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Assume that ε ď 1{4, then klp1{2, 1{2 ` εq ď 4ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' By Pinsker’s inequality, pp ´ qq2 ď klpp, qq{2, it implies 1 K Epℓ‹,a‹q ” N K pℓ‹,a‹q ı ď 1 K E0 ” N K pℓ‹,a‹q ı ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2ε c E0 ” N K pℓ‹,a‹q ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since ř ph‹,a‹q N K pℓ‹,a‹q “ K, by Cauchy-Schwarz inequality we have 1 K ÿ pℓ‹,a‹q Epℓ‹,a‹q ” N K pℓ‹,a‹q ı ď 1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 2ε ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' LAK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Plugging this back to Equation (16), and taking ε “ p1 ´ 1{LAq a LA{8K, we have max pℓ‹,a‹q RKpπ, Mpℓ‹,a‹qq ě Ωpt ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' SAKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' To ensure that ε ď 1{4, we need K ě SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Now we calculate the variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We know that V ‹ d`2psbq “ 0 and V ‹ d`2psgq “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any trajectory τ, we look at step h “ d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If psh, ahq ‰ pℓ‹, a‹q, then VarΣ τ ě Vpp1{2, 1{2q, p0, tqq “ Ωpt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If psh, ahq “ pℓ‹, a‹q, then VarΣ τ ě Vpp1{2 ´ ε, 1{2 ` εq, p0, tqq “ ˆ1 4 ´ ε2 ˙ Ωpt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Notice that ε ď 1{4, so VarΣ τ ě Ωpt2q for any τ, and Var‹ ě Varπ‹ ě min τ VarΣ τ ě Ωpt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since the total reward in each episode is upper-bounded by t, we know that VarΣ τ , Var‹ ď Opt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, VarΣ τ , Var‹ “ Θpt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For the desired result, we set t “ Θp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Proof of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We retain most of the proof of Theorem 9 in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021], while incorpo- rating the hard instance design in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content='1 in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Namely, we change the A-ary tree in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021] with the binary tree in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This change does not affect the proof, while circumventing the requirement of S “ 3 ` pAd ´ 1q{pA ´ 1q where d is the tree height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We can find S1 “ 2 ` 2tlog2pS´2qu “ ΩpSq and replace S with S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We still use d “ tlog2pS ´ 2qu to denote the tree height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Another important change in design is to scale the reward at sg by t ď 1, with t depending on V the variance we desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' So rhpsg, aq “ t1rh ě H ` d ` 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' This modification does not affect the choice of ε and H in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021], only scales the optimal value and regret linearly, so we have that max ph‹,ℓ‹,a‹q RKpπ, Mph‹,ℓ‹,a‹qq ě Ωpt ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' H3SAKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 42 Now we calculate the variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' We know that V ‹ H`d`1psbq “ 0 and V ‹ H`d`1psgq “ tpH ´H ´dq “ ΩptHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For any trajectory τ, we look at step h “ H ` d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If psh, ahq ‰ pℓ‹, a‹q, then VarΣ τ ě Vpp1{2, 1{2q, p0, ΩptHqqq “ Ωpt2H2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' If psh, ahq “ pℓ‹, a‹q, then VarΣ τ ě Vpp1{2 ´ ε, 1{2 ` εq, p0, ΩptHqqq “ ˆ1 4 ´ ε2 ˙ Ωpt2H2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Notice that ε ď 1{4 in Domingues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' [2021], so VarΣ τ ě Ωpt2H2q for any τ, and Var‹ ě Varπ‹ ě min τ VarΣ τ ě Ωpt2H2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Since the total reward in each episode is upper-bounded by OptHq, we know that VarΣ τ , Var‹ ď Opt2H2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' Thus, VarΣ τ , Var‹ “ Θpt2H2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' For the desired result, we set t “ Θp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' V{Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} +page_content=' 43' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9FQT4oBgHgl3EQf8TZV/content/2301.13446v1.pdf'} diff --git a/_NE1T4oBgHgl3EQf8gXW/content/tmp_files/load_file.txt b/_NE1T4oBgHgl3EQf8gXW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..30f2eab4ac7046716186dc5323013e1d66f9a108 --- /dev/null +++ b/_NE1T4oBgHgl3EQf8gXW/content/tmp_files/load_file.txt @@ -0,0 +1,227 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf,len=226 +page_content='ON PERTURBATIONS RETAINING CONSERVATION LAWS OF DIFFTRENTIAL EQUATIONS ALEXEY SAMOKHIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The paper deals with perturbations of the equation that have a number of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' When a small term is added to the equation its conserved quantities usually decay at in- dividual rates, a phenomenon known as a selective decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' These rates are described by the simple law using the conservation laws’ generating functions and the added term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' We intro- duce a procedure for finding such perturbations and demonstrate it by examples including the KdV-Burgers equation and a system from magnetodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Some interesting properties of solutions of such perturbed equations are revealed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Keywords: conservation laws, perturbed equations, selective de- cay, traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' MSC[2010]: 35Q53, 35B36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Introduction Many physical systems are modeled using equations that have a sig- nificant number of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Yet when an additional (usually dissipative) term is added to the equation its conserved quantities decay at individual rates, which are connected to their generating functions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The famous example is the KdV equation (it has infinitely many conservation laws) and the KdV-Burgers equation (with additional, with respect to KdV, dissipative term and only one conservation law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' To be precise let E(u) = 0 be a system of equations describing an ideal (unperturbed) media state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' A scalar H depending on u and its derivatives is a conservation law if for ⟨H⟩, the integral of H over some fixed spatial domain, ∂⟨H⟩ ∂t ���� E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' For the perturbed equation the quantity H is constant no more and ∂⟨H⟩ ∂t ̸= 0 is called the decay rate of H, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' A perturbed state usually satisfies the equation E(u) + LF(u) = 0, where L is a small parameter diagonal matrix diag(λi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' for L = 0 we get the ideal state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The decay rate depends on the additional term L F(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The connection between decay rate and LF(u) was called a ’balance law’ in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='03547v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='PS] 9 Jan 2023 2 ALEXEY SAMOKHIN This law expresses ∂t⟨H⟩ in terms of scalar product of LF(u) and the generating function g of the conserved quantity H, [1]: ∂⟨H⟩ ∂t = ⟨g · LF⟩ (1) Remarks The right-hand side of (1) is not unique: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='g, one can get a different but equivalent form integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' In the case of the integrand in the right-hand side of (1) is null or an exact form we get the situation when the conserved quantity ⟨H⟩ is conserved as well for the correspondent perturbed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Let us restrict considerations to R[u], the ring of differential polynoms of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Then all perturbations F retaining the conser- vation law with the generating function g must satisfy g · LFdx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' dxn ∈ Im(d), where d : Λn−1 → Λn and Λk are k-forms of spatial variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Of course, the intersection of the principal ideal g · R[u] with Im(d) is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' A considerable difference in decay rates leads to a simple method, first discovered by Taylor, [4], for finding quasi-stationary states of plasma which are of great practical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' He studied the model where the decay of energy E is monotonic but those of momentum M and helicity are not necessarily so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Such an inequality in decay rates leads to a distinct physical phenomenon of ’self–organization’ or quasi–stable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' There exist a very simple procedure for finding solutions of the above described behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' It was suggested in [4], and is known as ’Taylor trick’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Taking into consideration their comparative decay rates, minimize E with M as constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Put δ(E + λM = 0), M and presumed con- stant, λ being Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' This Euler–Lagrange equation is not necessarily compatible with the initial equation but nevertheless it gives a way for good approximations of self-organization phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' There is a considerable number of publication in the field, see a recent paper [5] for recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Another application of selective decay is given in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The problem is the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Using the relative decay of the KdV conserved quantities a simple algorithm to predict the number and amplitudes of resulting solitons was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' ON PERTURBATIONS RETAINING CONSERVATION LAWS 3 In [7] the selective decay approach was applied to some well-known equations of mathematical physics (KdV and KdV-Burgers equation, BBM and its dissipative generalization, two-dimensional generalized shallow water wave equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' It have showed that the Taylor trick extremals are associated with first-order PDEs and travelling wave so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' In this paper we search, for some popular equations, their low-order perturbations which retain a chosen conservation law (in a sense that the perturbed equation has the same conserved quantity as initial one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Examples include KdV and its conserved energy or momentum and the Kadomtsev-Pogutse system of equation from magnetohydrodynamics with its three known conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Some interesting properties of solutions of such perturbed equations are revealed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' KdV and KdV-Burgers The generalized KdV equation (KdV-Burgers equation) considered here is of the form ut = 2uux + uxxx + λuxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (2) The classical KdV equation corresponds to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The first three conserved quantities for KdV are m = � +∞ −∞ u(x, t) dx — mass, M = � +∞ −∞ u2(x, t) dx — momentum, E = � +∞ −∞ � 2u3(x, t) − 3(ux(x, t))2� dx — energy, and there are infinite number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The generating functions for the above conservation laws of the KdV are, up to multiplication constants, 1, u and u2 + uxx correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' As for the equation (2), it has a form of a conservation law, ut = Fx, the ”mass” � +∞ −∞ u dx is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' For a soliton this mass is equal to 12aγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' But the impulse ⟨u2⟩ = � +∞ −∞ u2 dx declines monotonically: Mt = 1 2⟨u2⟩t = ⟨uut⟩ = ⟨u(u2 + uxx + λux)x⟩ = 2 3u3��+∞ −∞ − u2 x|+∞ −∞ − λ⟨u2 x⟩ = (3) By analogy, for the energy 4 ALEXEY SAMOKHIN Et = ⟨ � 2u3(x, t) − 3(ux(x, t))2� ⟩t = 6λ⟨uxx(u2 + uxx)⟩ (4) Thus the energy does not necessary declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Transformations of KdV that retain momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Now let us find perturbations of the form F(u, ux, uxx) that retain momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Accordingly to the remark 2 above, the differential form λu · F(u, ux, uxx)dx must be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Thus u · F(u, ux, uxx) = Dx(A(u, ux)) (5) for some A(u, ux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Here Dx = ∂ ∂x + ∞ � n=0 uxn+1 ∂ ∂uxn is the operator of the full differentiation with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Below we restrict the search to polynomials of u and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Then in (5) the polynomial Dx(A(u, ux)) is divisible by u, so A(u, ux) = u2B(u, ux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' On the other hand Dx(u2B(u, ux)) = 2uuxB(u, ux) + u2(ux ∂B ∂u + uxx ∂B ∂ux ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Hence the second order retaining momentum perturbation is defined by F(u, ux, uxx) = 2uxB(u, ux) + u(ux ∂B ∂u + uxx ∂B ∂ux ) for an arbitrary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Note that F is linear in uxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' For instance, if B = ux the λ transformation of the KdV equation ut = 2uux + uxxx + λ(2u2 x + uuxx) (6) retains ⟨u2⟩ as its conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' This construction can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' If g is the gen- erating function for some conserved quantity Cl of an one-spational equation E, then F = g−1Dx(g2Φ) is the addendum to E which re- tains Cl, Φ being a arbitrary function of u and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The equation (6) has travelling wave solutions, in par- ticular shock waves of the form 3 2λ � a tanh �a3λ2 + 3a λ2 t + ax � + 1 λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (7) This shock moves to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' If require u|−∞ = 0 then (7) becomes the shock wave 3 2λ2 � 1 + tanh � 4 λ2t + 1 λx �� ON PERTURBATIONS RETAINING CONSERVATION LAWS 5 with the velocity 4/λ, see figure 1, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The travelling wave solution Left: for the equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Right: For the equation (9), a = 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The perturbed equation has only translations in x and t as its point symmetries, but a lot of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Transformations of KdV that retain energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Now for energy saving transformations of KdV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Since the generating function of energy is, up to a constant multiplier, u2 + uxx, one must solve (u2 + uxx) · F(u, ux, uxx, uxxx) = Dx(A(u, ux, uxx)) (8) for some A(u, ux, uxx), to find an low-order F(u, ux, uxx), the suitable transformation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' By analogy to the momentum case, the one pos- sibility is A = (u2 + uxx)2B F(u, ux, uxx) = 2Dx(u2+uxx)B+(u2+uxx)(ux ∂B ∂u +uxx ∂B ∂ux +uxxx ∂B ∂uxx ), for an arbitrary B = B(u, ux, uxx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' If B = u then F = 5u2ux+2uuxxx+ uxuxx The corresponding transformed equation is ut = 2uux + uxxx + λ(5u2ux + 2uuxxx + uxuxx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (9) Its point symmetries are only translations in x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The equation (9) has travelling wave solutions, in par- ticular — solutons of the form of a vertically shifted soliton u(x, t) = −6a2 tanh2(a(4a4·λt+x))+4a2 = 6a2 sech2(a(4a4·λt+x))−2a2 (10) found by Maple, with the velocity V = 4a4λ, see figure 1, right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 6 ALEXEY SAMOKHIN Yet it is not the whole answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Computer experiments demonstrate that an arbitrary initial datum for this equation scatters into a number of solitary peaks of different but constant height and velocity and a ’tail’ (see figures 2 and 3) — in a manner of the KdV itself, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Left: Initial profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='5 sech2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='5x) for the equation (9), λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Right: Resulting profile at t = 6: single soliton-like peak of a constant form and velocity and an oscillating tail moving in opposite direction Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Left: Initial profile sech2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1x) for the equa- tion (9), λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Right: Resulting profile at t = 40: multiple soliton-like peaks of a constant form and velocity and (seemingly) no tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The analytical description of these peaks is so far unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The reason is that the equation on travelling waves, u = u(x + V t), here V u′ = 2uu′ + u′′ + λ(5u2u′ + 2uu′′′ + u′u′′ can be readily integrated introducing the new dependent variable u′ = p(u) which leads to a linear first order ordinary differential equation on z(u) = p(u)p′(u), (2uλ + 1)z′ + λz = V − 5λu2 − 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' ON PERTURBATIONS RETAINING CONSERVATION LAWS 7 But the resulting general solution looks hopelessly implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The likes of (10) arise in the case of a very special combination of the arbitrary constants entering this general solution, and such combinations are hard to discover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Two-dimensional MHD System Consider the Kadomtsev-Pogutse sysnem of equations � ∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy = 0 vt + uxvy − uyvx = 0 (11) which describes quasi-stationary states of plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' It has three conser- vation laws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' that is there are three non–trivial conserved densities (two of them depending on arbitrary functions): the total energy E (mag- netic plus kinetic energy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' generalized ’cross helicity’ Hc and mean magnetic potential A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' E = 1 2⟨u2 x + u2 y + v2 x + v2 y⟩ H = ⟨f ′(v) · (uxvx + uyvy)⟩ A = ⟨Φ(v)⟩ (12) Their generating functions are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' respective order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' � u ∆v � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' � f(v) f ′(v)∆u � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' � 0 Φ′(v) � (13) where f and Φ are arbitrary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Let us seek transformations of (11) of the form � ∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy = νF(u, v) vt + uxvy − uyvx = ηG(u, v) (14) Here F, G are functions of u(x, y, t), v(x, y, t) and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Energy-retaining transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' In this instance ∂⟨E⟩/∂t = 0 implies (−νu · F − η∆v · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) = (DxA(u, v) + DyB(u, v))dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (15) There are a lot of solutions to (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' We restrict ourselves to some low-order examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Ortogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' One can always get zero right hand side in equation (15): just put F = η∆ and G = −νu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The vector (F, G) is orthogonal to the generating function so ∂⟨E⟩/∂t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' It works if the number of any system of equations is greater than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 8 ALEXEY SAMOKHIN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Splitted sum transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Another solution may be obtained assuming − νu · F(u, v) = DxA(u, v), η∆v · G(u, v) = DyB(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (16) Here again A, B are functions of u(x, y, t), v(x, y, t) and their deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' This equations may be solved by analogy to the KdV case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' One of numerous solutions here is A = νun, B = η(∆v)2, so F = −νnun−2ux, G = 2η∆vy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' {ν = η}—case transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Take A = Gvx, B = Gvy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Then uF = vxDxG + vyDyG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' For instance, choose G = u2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' it fol- lows that F = 2(uxvx + uyvy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Mean magnetic potential retaining transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Here ∂⟨A⟩/∂t = 0 implies (−ν0 · F − ηΦ′(v) · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) = (DxA(u, v) + DyB(u, v))dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (17) Thus F is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Then one possible solution is −ηΦ′(v) · Φ(v)(αvx + βvy) = DxαΦ2 + DyβΦ2, α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' That is, to retain the mean magnetic potential of (11), its first equation may be transformed in arbitrary way and the second one by ηG = −ηΦ(v)(αvx + βvy) for all α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Cross helicity retaining transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Here ∂⟨Hc⟩/∂t = 0 implies −νf(v)·F(u, v)−ηf ′(v)∆(u)·G(u, v) = DxA(u, v)+DyB(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' (18) In the case ηη = ν it is not hard to find some suitable transformations (F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Namely, take A = −ηf 2(v)f ′(v)ux, B = −ηf 2(v)f ′(v)uy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' It follows F = [2f ′2(v) + f(v)f ′′(v)](vxux + uyvy), G = f ′(v)f(v)∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' For f(v) = v it comes to F = −2η(vxux + uyvy) G = −ηv∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' ON PERTURBATIONS RETAINING CONSERVATION LAWS 9 Conclusion The paper deals with perturbations of the equation that have a num- ber of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenome- non known as a selective decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' These rates are described by the simple law using the conservation laws’ generating functions and the added term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' We introduced a procedure for finding such perturbations and demonstrated it by examples including the KdV-Burgers equation and a system from magnetodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Our worked out examples show that the perturbed equations retain- ing a specific conservation law frequently also retain additional alge- braic properties such as travelling wave solutions or a presence of other conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Thus the present paper as well as [5] and our previous research of the KdV solitons in nonhomogeneous media, [6], persuades that the selective decay approach is a valid and effective instrument to obtain qualitative approximations and estimates for behavior of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The figures in this paper were generated numerically using Maple PDETools package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' The mode of operation uses the default Euler method, which is a centered implicit scheme, and can be used to find solutions to PDEs that are first order in time, and arbitrary order in space, with no mixed partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Samokhin, Decay velocity of conservation laws for nonevolution equa- tions,Acta Applicanda Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=', v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 41 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 1, 1–11 (1995) [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Ting, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Matthaeus, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Montgomery, Turbulent relaxation processes in magnetohydrodynamics Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Fluids, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='29, 3261–3274 (1986) [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' van Groesen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Mainardi, Balance laws and centro velocity in dissipative systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 31 (11), 2136–2140 (1990) [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Relaxation of toroidal plasma and generation of reverse magnetic fields, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=', v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 33, 1139–1141 (1974) [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Brecht1, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Bauer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Bihlo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Gay-Balmaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' MacLachlan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Selective decay for the rotating shallow-water equations with a structure-preserving discretiza- tion Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Fluids, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='33, 116604 (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1063/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='0062573 [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Samokhin, The KdV soliton crosses a dissipative and dispersive border, Journal of Differential Geometry and its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='75, Part A, 11 pages(April 2021) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='difgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='101723 [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' Samokhin, Taylor Trick and Travelling Wave Solutions, Lobachevskii Journal of Mathematics, 2022, 43, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' 10, 2808—2815, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='1134/S1995080222130406 Institute of Control Sciences of Russian Academy of Sciences 65 Profsoyuznaya street, Moscow 117997, Russia Email address: samohinalexey@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'} diff --git a/atE3T4oBgHgl3EQfdQoZ/content/tmp_files/2301.04532v1.pdf.txt b/atE3T4oBgHgl3EQfdQoZ/content/tmp_files/2301.04532v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e1574576366098942cf64ed5745e577871639df --- /dev/null +++ b/atE3T4oBgHgl3EQfdQoZ/content/tmp_files/2301.04532v1.pdf.txt @@ -0,0 +1,2812 @@ +arXiv:2301.04532v1 [math.NT] 11 Jan 2023 +MODULARITY OF NAHM SUMS FOR THE TADPOLE +DIAGRAM +ANTUN MILAS AND LIUQUAN WANG +Abstract. We prove Rogers-Ramanujan type identities for the Nahm sums as- +sociated with the tadpole Cartan matrix of rank 3. These identities reveal the +modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu +and the first author in this case. We show that these Nahm sums together with +some shifted sums can be combined into a vector-valued modular function on the +full modular group. We also present some conjectures for a general rank. +1. Introduction and Main Results +The famous Rogers-Ramanujan identities state that +∞ +� +n=0 +qn2 +(q; q)n += +1 +(q, q4; q5)∞ +, +(1.1) +∞ +� +n=0 +qn2+n +(q; q)n += +1 +(q2, q3; q5)∞ +, +(1.2) +where (here and throughout this paper) we always assume |q| < 1 and use standard +q-series notations: +(a; q)0 := 1, +(a; q)n := +n−1 +� +k=0 +(1 − aqk), +(a; q)∞ := +∞ +� +k=0 +(1 − aqk), +(1.3) +(a1, . . . , am; q)n := (a1; q)n · · · (am; q)n, +n ∈ N ∪ {∞}. +(1.4) +When the base q is clear from the context, occasionally we omit it and simply write +(a; q)n as (a)n (n ∈ N ∪ {∞}). +The Rogers-Ramanujan identities first appeared in the 1894 paper of Rogers [17] +and were later rediscovered by Ramanujan before 1913. Besides (1.1) and (1.2), +Rogers [17, pp. 330-332] also proved some similar sum-product identities such as +∞ +� +n=0 +qn2 +(q; q)2n += (q2, q8, q10; q10)∞(q6, q14; q20)∞ +(q; q)∞ +, +(1.5) +∞ +� +n=0 +qn2+n +(q; q)2n += (q, q9, q10; q10)∞(q8, q12; q20)∞ +(q; q)∞ +, +(1.6) +2010 Mathematics Subject Classification. 11P84, 33D15, 33D45. +Key words and phrases. Rogers-Ramanujan identities; sum-product identities; Nahm sums; tad- +pole Cartan matrix; vector-valued modular forms. +1 + +2 +ANTUN MILAS AND LIUQUAN WANG +∞ +� +n=0 +qn2+n +(q; q)2n+1 += (q3, q7, q10; q10)∞(q4, q16; q20)∞ +(q; q)∞ +, +(1.7) +Later Rogers [18, p. 330 (3), 2nd Eq.]) proved another companion identity: +∞ +� +n=0 +qn2+2n +(q; q)2n+1 += (q4, q6, q10; q10)∞(q2, q18; q20)∞ +(q; q)∞ +. +(1.8) +The Rogers-Ramanujan identities also serve as important examples for close re- +lations between q-series and modular forms. The product sides are essentially re- +ciprocals of some generalized Dedekind eta functions (see (3.32)) and hence are +modular functions. +This is not observable from the sum sides, which is a ba- +sic q-hypergeometric series. +A natural question is to ask when does a basic q- +hypergeometric series become a modular form. In particular, a famous problem +of Nahm is to determine for which positive definite r × r rational matrix A, r- +dimensional rational vector B, and a rational scalar C such that +fA,B,C(q) := +� +n=(n1,...,nr)T∈(Z≥0)r +q +1 +2 nTAn+nTB+C +(q; q)n1 · · · (q; q)nr +is a modular form. Such (A, B, C) is called as a rank r modular triple. The series +fA,B,C(q) is therefore referred as Nahm sums. +Several important families of q-series identities (such as Andrews-Gordon iden- +tities) can be also studied using vertex operators and representation of infinite- +dimensional Lie algebras. This approach, pioneered by Lepowsky and Wilson in +1980s [10,11], was one of the starting points in the development of vertex operator +algebras and an important ingredient in the development of 2-dimensional confor- +mal field theory (CFT) in physics. In this setup, the graded dimension obtained +from a combinatorial bases of modules can be often interpreted as a Nahm sums. +The Nahm sums that are relevant for rational CFT almost always take form with +A = G ⊗ G′−1 where G and G′ are ADET type Cartan matrices, and all such Nahm +sums matrices are expected to give modular functions (with appropriate B and C). +Interestingly, any Nahm sum fA,0,0 can be interpreted as the graded dimension of a +special vertex algebra called principal subspace [15]. +Zagier [25] studied Nahm’s problem and made significant progress when the rank +r ≤ 3. In particular, he proved that there are exactly seven rank one modular triples. +In the rank two and three cases, Zagier provided a number of conjectural modular +triples. Most of the rank two examples have been confirmed in the literature. For +example, Vlasenko and Zwegers [20] confirmed one modular triple in Zagier’s list +and discovered some new examples. Recently, the second author [22] confirmed more +examples in Zagier’s list. As a consequence, among the eleven rank two examples +discovered by Zagier [25, Table 2], only one example is unproven. In the rank three +case, Zagier [25, Table 3] provided a list of twelve possible modular triples and +proved three of them. The remaining nine examples were confirmed by the second +author [23]. +In this paper, we are mainly concerned with the modularity of the Nahm sums +associated with the tadpole diagram T. The tadpole Nahm sums were considered by + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +3 +Penn, Calinescu, and the first author in their work on twisted modules of principal +subspace vertex algebras [6]. Given a positive integer r, let Tr be the tadpole Cartan +matrix, that is, Tr = (aij)r×r such that +arr = 1, aii = 2, 1 ≤ i ≤ r − 1, aij = −1 (|i − j| = 1), +and +aij = 0 +otherwise. +We let +χ0(x1, . . . , xr) = χ0(x1, . . . , xr; q) := +� +n=(n1,...,nr)∈Zr +≥0 +q +1 +2 nTTrnxn1 +1 · · · xnr +n +(q)n1 · · · (q)nr +be a generalized tadpole Nahm sum. It is easy to see that +qCχ0(qB1, . . . , qBr) = fA,B,C(q) +is the Nahm sum with A = Tr and B = (B1, . . . , Br). There is only one standard +A(2) +2n -module of level one (up to isomorphism) and thus only one principal subspace +of level 1, corresponding to the unique standard level one module, whose character +is χ0(1, 1, . . . , 1), i.e. +xi = 1 for all i. +In [6] the so-called shifted characters χi +were introduced by specializing xi = q and xj = 1 for j ̸= i. Penn, Calinescu, and +the first author [6] stated a conjecture concerning the modularity of the characters +χ0(x1, . . . , xr). +Conjecture 1.1. (Cf. [6, Conjecture 1].) The character qaχ0(1, . . . , 1) is modular +for some rational number a. +The rank two case (r = 2) was proved in [6]. The main goal of this paper is to +address the conjecture for r = 3 case. In this case, we write explicitly +χ0(x1, x2, x3) = χ0(x1, x2, x3; q) = +� +i,j,k≥0 +qi2+j2+ 1 +2k2−ij−jkxi +1xj +2xk +3 +(q)i(q)j(q)k +. +(1.9) +We record four shifted q-characters that are of interest here: +F1(q) := χ0(1, 1, 1), +F2(q) := χ0(1, 1, q +1 +2), +F3(q) := χ0(q, q−1, q +1 +2), +F4(q) := χ0(q−1, q, 1). +We are mainly concerned with modular properties of these sums. We first write +down the tadpole Cartan matrix T3 and its inverse: +T3 = + + +2 +−1 +0 +−1 +2 +−1 +0 +−1 +1 + + , +T −1 +3 += + + +1 +1 +1 +1 +2 +2 +1 +2 +3 + + . +Note that T −1 +3 +is the matrix part of the sixth example in Zagier’s list [25, Table +3] (see also [23, Example 6]) with the first row/column and the third row/column +interchanged. Zagier stated six possible modular triples with T −1 +3 +as the matrix +part. The vector parts are +B ∈ + + + + + +0 +0 +0 + + , + + +1/2 +1 +3/2 + + , + + +1/2 +0 +1/2 + + , + + +0 +1 +1 + + , + + +−1/2 +0 +−1/2 + + , + + +1/2 +1 +−1/2 + + + + + . +(1.10) + +4 +ANTUN MILAS AND LIUQUAN WANG +Here the first and the third entries in each vector have been interchanged and we +have reordered these vectors. Zagier [25, p. 50] conjectured that there are some +duality between modular triples. Namely, he mentioned that when (A, B, C) is a +rank r modular triple, then it is likely that +(A⋆, B⋆, C⋆) = (A−1, A−1B, 1 +2BTA−1B − r +24 − C) +is also a rank r modular triple. This motivates us to consider the dual cases to the +six modular triples related to T −1 +3 . To be specific, the dual vectors are +B⋆ ∈ + + + + + +0 +0 +0 + + , + + +0 +0 +1/2 + + , + + +1 +−1 +1/2 + + , + + +−1 +1 +0 + + , + + +−1 +1 +−1/2 + + , + + +−2 +2 +−1/2 + + + + + . +We find that they are indeed modular triples for suitable C. This is a consequence +of the following set of Rogers-Ramanujan type identities. Before we state them, we +introduce the compact notations +Jm := (qm; qm)∞, +Ja,m := (qa, qm−a, qm; qm)∞. +Theorem 1.2. We have +� +i,j,k≥0 +q2i2+2j2+k2−2ij−2jk +(q2; q2)i(q2; q2)j(q2; q2)k += +J5 +4J40 +J1J2 +2J2 +8J8,40 ++ 2qJ2 +8J6,20J8,40 +J1J2 +4J40 +, +(1.11) +� +i,j,k≥0 +qi2+j2+ 1 +2(k2+k)−ij−jk +(q; q)i(q; q)j(q; q)k += J6 +2J1,10J8,20 +J5 +1J2 +4J20 ++ 2qJ2 +4J4,10J2,20 +J3 +1J20 +, +(1.12) +� +i,j,k≥0 +qi2+j2+ 1 +2 (k2+k)−ij−jk+i−j +(q; q)i(q; q)j(q; q)k += 2J2J2 +4J20 +J3 +1J4,20 ++ J6 +2J3,10J4,20 +J5 +1J2 +4J20 +, +(1.13) +� +i,j,k≥0 +q2i2+2j2+k2−2ij−2jk−2i+2j +(q2; q2)i(q2; q2)j(q2; q2)k += 2J2 +8J2,20J16,40 +J1J2 +4J40 ++ qJ4 +4J8,20J4,40 +J1J2 +2J2 +8J40 +, +(1.14) +� +i,j,k≥0 +qi2+j2+ 1 +2 (k2−k)−ij−jk−i+j +(q; q)i(q; q)j(q; q)k += 4J2J2 +4J20 +J3 +1J4,20 ++ 2J6 +2J3,10J4,20 +J5 +1J2 +4J20 +, +(1.15) +� +i,j,k≥0 +qi2+j2+ 1 +2(k2−k)−ij−jk−2i+2j +(q; q)i(q; q)j(q; q)k += 2q−1J6 +2J1,10J8,20 +J5 +1J2 +4J20 ++ 4J2 +4J4,10J2,20 +J3 +1J20 +. +(1.16) +Note that the left sides of (1.11)–(1.16) are the Nahm sums +F1(q2), F2(q), F3(q), F4(q2), χ0(q−1, q, q− 1 +2; q) and χ0(q−2, q2, q− 1 +2; q), +respectively. In light of the modularity of the functions Jm and Ja,m, it is easy to +verify (e.g., using the algorithm in [8]) that the Nahm sums in (1.11)–(1.16) are all +modular functions after multiplying a factor qC with C being +− 7 +40, +1 +40, +9 +40, +17 +40, +9 +40, +41 +40, +respectively. In particular, we see that the identity (1.11) confirms Conjecture 1.1 in +the case r = 3. Interestingly, there are essentially four different modular functions + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +5 +for these six Nahm sums. In fact, comparing the right sides of (1.13) and (1.15), +and (1.12) and (1.16), we see that +χ0(q−1, q, q− 1 +2; q) = 2χ0(q, q−1, q +1 +2; q), +(1.17) +χ0(q−2, q2, q− 1 +2; q) = 2q−1χ0(1, 1, q +1 +2; q). +(1.18) +This is not obvious from their original definitions. We will explain this in the proof +of this theorem. +Next, we will study the tadpole Nahm sums from the point of vector-valued +modular forms. By doing so, we will be able to see their modular transformation +properties more clearly . +Let H denote the upper half complex plane. Throughout this paper we denote +q = e2πiτ, τ ∈ H. +Since the shifted characters χ0(1, 1, 1) and χ0(q−1, q, 1) have +q-powers in Z + 1 +2 it is convenient to consider +F5(q) := χ0(1, 1, 1)|τ→τ+1 = +� +i,j,k≥0 +(−1)k qi2+j2+ 1 +2k2−ij−jk +(q)i(q)j(q)k +, +(1.19) +F6(q) := χ0(q−1, q, 1)|τ→τ+1 = +� +i,j,k≥0 +(−1)k qi2+j2+ 1 +2k2−ij−jk−i+j +(q)i(q)j(q)k +. +(1.20) +For 1 ≤ i ≤ 6 we define ˜Fi(q) := qλiFi(q) where +λ1 = − 7 +80, λ2 = 1 +40, λ3 = 9 +40, λ4 = 17 +80, λ5 = − 7 +80, λ6 = 17 +80. +We define Weber’s modular functions [24]: +f(τ) := q−1/48(−q1/2; q)∞, +f1(τ) := q−1/48(q1/2; q)∞, +f2(τ) := q1/24(−q; q)∞. +(1.21) +For k > 0 and j ∈ Q we let +(∂Θ)j,k(τ) := +� +n∈Z +(2kn + j)q(2kn+j)2/(4k), +(1.22) +(∂G)j,k(τ) := +� +n∈Z +(−1)n(2kn + j)q(2kn+j)2/(4k). +(1.23) +When k, j ∈ Q, these are essentially Jacobi theta series of weight 3/2. +Theorem 1.3. We have the following q-identities: +˜F1(q) = f(τ)3 +η(τ)3(∂Θ)1, 5 +2(τ), +˜F2(q) = 2f2(τ)3 +η(τ)3 (∂Θ) 1 +2, 5 +2(τ), +˜F3(q) = 2f2(τ)3 +η(τ)3 (∂Θ) 3 +2, 5 +2(τ), +˜F4(q) = f(τ)3 +η(τ)3(∂Θ)2, 5 +2(τ), + +6 +ANTUN MILAS AND LIUQUAN WANG +˜F5(q) = f1(τ)3 +η(τ)3 (∂G)1, 5 +2(τ), +˜F6(q) = f1(τ)3 +η(τ)3 (∂G)2, 5 +2(τ). +In particular, ˜Fi(q) are modular functions on some congruence subgroups of SL(2, Z). +Moreover, ( ˜Fi(q))1≤i≤6 transforms as a vector valued modular function on SL(2, Z). +This in particular proves and extends Conjecture 1.1 for r = 3. +The rest of this paper is organized as follows. In Section 2 we present a proof +for Theorem 1.2. The idea is to use constant term method to reduce triple sums to +some single sums and use Rogers’ identities (1.5)–(1.8). In Section 3 we give two +different proofs for Theorem 1.3. Finally, in Section 4 we prove that some other +Nahm sums associated with the tadpole Cartan matrix are not modular, and we +give a general conjecure on modular Nahm sums. +Remark 1. It will be useful to observe another basis of V = Span{ ˜Fi(q) : 1 ≤ 1 ≤ 6}: +g1(q) := ˜F1(q) + ˜F5(q) = q−7/80(2 + 12q + 30q2 + · · · ) ∈ q−7/80C[[q]], +g2(q) := ˜F1(q) − ˜F5(q) = q33/80(6 + 18q + 54q2 + · · · ) ∈ q33/80C[[q]], +g3(q) := ˜F4(q) + ˜F6(q) = q17/80(4 + 6q + 30q2 + · · · ) ∈ q17/80C[[q]], +g4(q) := ˜F4(q) − ˜F6(q) = q57/80(6 + 16q + 42q2 + · · · ) ∈ q57/80C[[q]], +g5(q) := ˜F2(q) = q1/40(1 + 6q + 15q2 + · · · ) ∈ q1/40C[[q]], +g6(q) := ˜F3(q) = q9/40(3 + 11q + 30q2 + · · · ) ∈ q9/40C[[q]]. +Observe that now the leading q-series powers are non-congruent modulo Z. +2. Proof of Theorem 1.2 +Recall the q-binomial theorem [4, Theorem 1.3.1]: +∞ +� +n=0 +(a; q)n +(q; q)n +zn = (az; q)∞ +(z; q)∞ +, +|z| < 1. +(2.1) +As important corollaries of this theorem, Euler’s q-exponential identities state that +[4, Corollary 1.3.2] +∞ +� +n=0 +zn +(q; q)n += +1 +(z; q)∞ +, +∞ +� +n=0 +q(n +2)zn +(q; q)n += (−z; q)∞, +|z| < 1. +(2.2) +The Jacobi triple product identity [4, Theorem 1.3.3] is +(q, z, q/z; q)∞ = +∞ +� +n=−∞ +(−1)nq(n +2)zn. +(2.3) +It gives product representations for two important unary Jacobi theta functions: +θ2(τ) := +� +n∈Z +q(n+1/2)2 = 2q1/4J2 +4 +J2 +, +(2.4) + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +7 +θ3(τ) := +� +n∈Z +qn2 = +J5 +2 +J2 +1J2 +4 +. +(2.5) +We will use the constant term method. For any series f(z) = � +n∈Z a(n)zn, we +define the operator +CT[f(z)] = a(0), +which extracts the constant term of f(z). Obviously, for any complex number α +and integer β with αβ ̸= 0, we have +CT[f(αzβ)] = CT[f(z)]. +(2.6) +Proof of Theorem 1.2. After replacing q by q2 in (1.9), we have +χ0(x1, x2, x3; q2) = +� +i,j,k≥0 +q2i2+2j2+k2−2ij−2jkxi +1xj +2xk +3 +(q2; q2)i(q2; q2)j(q2; q2)k +. +(2.7) +We have by (2.2) and (2.3) that +χ0(x1, x2, x3; q2) = +� +i,j≥0 +q2i2+2j2−2ijxi +1xj +2 +(q2; q2)i(q2; q2)j +� +k≥0 +qk2−k · q(1−2j)kxk +3 +(q2; q2)k += +� +i,j≥0 +q2i2+2j2−2ijxi +1xj +2 +(q2; q2)i(q2; q2)j +(−x3q1−2j; q2)∞. +(2.8) +(1) We have by (2.8) that +χ0(1, 1, 1; q2) = +� +i≥0 +q2i2+2j2−2ij +(q2; q2)i(q2; q2)j +(−q1−2j; q2)∞ += (−q; q2)∞ +� +i,j≥0 +q2i2+j2−2ij(−q; q2)j +(q2; q2)i(q2; q2)j += (−q; q2)∞CT +�� +i≥0 +qi2zi +(q2; q2)i +� +j≥0 +(−q; q2)jz−j +(q2; q2)j +∞ +� +k=−∞ +z−kqk2 +� += (−q; q2)∞CT +�(−qz, −q/z, −qz, −q/z, q2; q2)∞ +(1/z; q2)∞ +� +(by (2.2) and (2.3)) += (−q; q2)∞ +(q2; q2)∞ +CT +�(−qz, −q/z, q2; q2)2 +∞ +(1/z; q2)∞ +� += (−q; q2)∞ +(q2; q2)∞ +CT +� ∞ +� +n=0 +z−n +(q2; q2)n +∞ +� +i=−∞ +ziqi2 +∞ +� +j=−∞ +zjqj2 +� +(by (2.2) and (2.3)) += (−q; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +1 +(q2; q2)n +� +i+j=n +qi2+j2 += (−q; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +qn2 +(q2; q2)n +∞ +� +i=−∞ +q2i2−2ni + +8 +ANTUN MILAS AND LIUQUAN WANG += (−q; q2)∞ +(q2; q2)∞ +(S0(q) + S1(q)). +(2.9) +Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively. +We have +S0(q) = +∞ +� +n=0 +q2n2 +(q2; q2)2n +∞ +� +i=−∞ +q2(i−n)2 = +∞ +� +n=0 +q2n2 +(q2; q2)2n +∞ +� +i=−∞ +q2i2 += +J6 +4J40 +J3 +2J2 +8J8,40 +. +(by (1.5) and (2.5)) +(2.10) +Similarly, we have +S1(q) = +∞ +� +n=0 +q4n2+4n+1 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2−4in−2i = +∞ +� +n=0 +q2n2+2n+1 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2(i−n)2−2(i−n) += +∞ +� +n=0 +q2n2+2n+1 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2−2i = 2qJ2 +8J6,20J8,40 +J2J4J40 +. +(by (1.7) and (2.4)) (2.11) +Substituting (2.10) and (2.11) into (2.9), we obtain (1.11). +(2) We have by (2.8) that +χ0(1, 1, q; q2) = +� +i,j≥0 +q2i2+2j2−2ij +(q2; q2)i(q2; q2)j +(−q2−2j; q2)∞ += (−q2; q2)∞ +� +i,j≥0 +q2i2+j2+j−2ij +(q2; q2)i(q2; q2)j +(−1; q2)j += (−q2; q2)∞CT +�� +i≥0 +qi2zi +(q2; q2)i +� +j≥0 +qjz−j(−1; q2)j +(q2; q2)j +∞ +� +k=−∞ +z−kqk2 +� += (−q2; q2)∞CT +�(−qz, −q/z, −q/z, −qz, q2; q2)∞ +(q/z; q2)∞ +� +(by (2.2) and (2.3)) (2.12) += (−q2; q2)∞ +(q2; q2)∞ +CT +� ∞ +� +n=0 +qn +(q2; q2)n +∞ +� +i=−∞ +ziqi2 +∞ +� +j=−∞ +zjqj2 +� +(by (2.2) and (2.3)) += (−q2; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +qn +(q2; q2)n +� +i+j=n +qi2+j2 += (−q2; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +qn2+n +(q2; q2)n +∞ +� +i=−∞ +q2i2−2ni += (−q2; q2)∞ +(q2; q2)∞ +(S0(q) + S1(q)). +(2.13) +Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively. + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +9 +We have +S0(q) = +∞ +� +n=0 +q4n2+2n +(q2; q2)2n +∞ +� +i=−∞ +q2i2−4ni = +∞ +� +n=0 +q2n2+2n +(q2; q2)2n +∞ +� +i=−∞ +q2(i−n)2 += +∞ +� +n=0 +q2n2+2n +(q2; q2)2n +∞ +� +i=−∞ +q2i2 = J5 +4J2,20J16,40 +J3 +2J2 +8J40 +. +(by (1.6) and (2.5)) +(2.14) +Similarly, +S1(q) = +∞ +� +n=0 +q4n2+6n+2 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2−4ni−2i = +∞ +� +n=0 +q2n2+4n+2 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2(n−i)2+2(n−i) += +∞ +� +n=0 +q2n2+4n+2 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2+2i = 2q2J2 +8J8,20J4,40 +J2J4J40 +. +(by (1.8) and (2.4)) +(2.15) +Substituting (2.14) and (2.15) into (2.13), we obtain (1.12). +(3) We have by (2.8) that +χ0(q2, q−2, q; q2) = +� +i,j≥0 +q2i2+2j2−2ij+2i−2j +(q2; q2)i(q2; q2)j +(−q2−2j; q2)∞ += (−q2; q2)∞ +� +i,j≥0 +q2i2+j2−j−2ij+2i +(q2; q2)i(q2; q2)j +(−1; q2)j += (−q2; q2)∞CT +�� +i≥0 +qi2+2izi +(q2; q2)i +� +j≥0 +q−jz−j(−1; q2)j +(q2; q2)j +∞ +� +k=−∞ +z−kqk2 +� += (−q2; q2)∞CT +�(−q3z, −1/(qz); q2)∞(−qz, −q/z, q2; q2)∞ +(1/(qz); q2)∞ +� +(by (2.2) and (2.3)) +(2.16) += (−q2; q2)∞ +(q2; q2)∞ +CT +�(−q3z, −1/(qz), q2; q2)∞(−qz, −q/z, q2; q2)∞ +(1/(qz); q2)∞ +� += (−q2; q2)∞ +(q2; q2)∞ +CT +� ∞ +� +n=0 +q−nz−n +(q2; q2)n +∞ +� +i=−∞ +qi2+2izi +∞ +� +j=−∞ +qj2zj +� +(by (2.2) and (2.3)) += (−q2; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +q−n +(q2; q2)n +� +i+j=n +qi2+2i+j2 += (−q2; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +qn2−n +(q2; q2)n +∞ +� +i=−∞ +q2i2−2ni+2i += (−q2; q2)∞ +(q2; q2)∞ +(S0(q) + S1(q)). +(2.17) +Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively. + +10 +ANTUN MILAS AND LIUQUAN WANG +We have +S0(q) = +∞ +� +n=0 +q4n2−2n +(q2; q2)2n +∞ +� +i=−∞ +q2i2−4ni+2i = +∞ +� +n=0 +q2n2 +(q2; q2)2n +∞ +� +i=−∞ +q2(i−n)2+2(i−n) += +∞ +� +n=0 +q2n2 +(q2; q2)2n +∞ +� +i=−∞ +q2i2+2i = 2 J2 +8J40 +J2J8,40 +. +(by (1.5) and (2.4)) +(2.18) +Similarly, +S1(q) = +∞ +� +n=0 +q4n2+2n +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2−4in = +∞ +� +n=0 +q2n2+2n +(q2; q2)2n+1 +∞ +� +i=−∞ +q2(i−n)2 += +∞ +� +n=0 +q2n2+2n +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2 = J5 +4J6,20J8,40 +J3 +2J2 +8J40 +. +(by (1.7) and (2.5)) +(2.19) +Substituting (2.18) and (2.19) into (2.17), we obtain (1.13). +(4) We have by (2.8) that +χ0(q−2, q2, 1; q2) = +� +i,j≥0 +q2i2+2j2−2ij−2i+2j +(q2; q2)i(q2; q2)j +(−q1−2j; q2)∞ += (−q; q2)∞ +� +i,j≥0 +q2i2+j2−2ij−2i+2j +(q2; q2)i(q2; q2)j +(−q; q2)j += (−q; q2)∞CT +�� +i≥0 +qi2−2izi +(q2; q2)i +� +j≥0 +q2jz−j(−q; q2)j +(q2; q2)j +∞ +� +k=−∞ +z−kqk2 +� += (−q; q2)∞CT +�(−z/q, −q3/z, −qz, −q/z, q2; q2)∞ +(q2/z; q2)∞ +� +(by (2.2) and (2.3)) += (−q; q2)∞ +(q2; q2)∞ +CT +�(−z/q, −q3/z, q2; q2)∞(−qz, −q/z, q2; q2)∞ +(q2/z; q2)∞ +� += (−q; q2)∞ +(q2; q2)∞ +CT +� ∞ +� +n=0 +q2nz−n +(q2; q2)n +∞ +� +i=−∞ +ziqi2−2i +∞ +� +j=−∞ +zjqj2 +� +(by (2.2) and (2.3)) += (−q; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +q2n +(q2; q2)n +� +i+j=n +qi2+j2−2i += (−q; q2)∞ +(q2; q2)∞ +∞ +� +n=0 +qn2+2n +(q2; q2)n +∞ +� +i=−∞ +q2i2−2ni−2i += (−q; q2)∞ +(q2; q2)∞ +(S0(q) + S1(q)). +(2.20) +Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively. +We have +S0(q) = +∞ +� +n=0 +q4n2+4n +(q2; q2)2n +∞ +� +i=−∞ +q2i2−4in−2i = +∞ +� +n=0 +q2n2+2n +(q2; q2)2n +∞ +� +i=−∞ +q2(n−i)2+2(n−i) + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +11 += +∞ +� +n=0 +q2n2+2n +(q2; q2)2n +∞ +� +i=−∞ +q2i2+2i = 2J2 +8J2,20J16,40 +J2J4J40 +. +(by (1.6) and (2.4)) +(2.21) +Similarly, +S1(q) = +∞ +� +n=0 +q4n2+8n+3 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2i2−4ni−4i = +∞ +� +n=0 +q2n2+4n+3 +(q2; q2)2n+1 +∞ +� +i=−∞ +q2(n−i)2+4(n−i) += q +∞ +� +n=0 +q2n2+4n +(q2; q2)2n+1 +∞ +� +i=−∞ +q2(i+1)2 = qJ5 +4J8,20J4,40 +J3 +2J2 +8J40 +. +(by (1.8) and (2.5)) +(2.22) +Substituting (2.21) and (2.22) into (2.20), we obtain (1.14). +(5) We have by (2.8) that +χ0(q−2, q2, q−1; q2) = +� +i,,j≥0 +q2i2+2j2−2ij−2i+2j +(q2; q2)i(q2; q2)j +(−q−2j; q2)∞ += (−1; q2)∞ +� +i,j≥0 +q2i2+j2+j−2ij−2i +(q2; q2)i(q2; q2)j +(−q2; q2)j += 2(−q2; q2)∞CT +�� +i≥0 +qi2−2izi +(q2; q2)i +� +j≥0 +(−q2; q2)jqjz−j +(q2; q2)j +∞ +� +k=−∞ +z−kqk2 +� += 2(−q2; q2)∞CT +�(−q3/z, −z/q, −qz, −q/z, q2; q2)∞ +(q/z; q2)∞ +� +. +(2.23) +Now if we replace z by q2z in (2.23), which does not change the result by (2.6), +and then compare with (2.16), after replacing q2 by q, we obtain (1.17). In view of +(1.13), we obtain (1.15). +(6) We have by (2.8) that +χ0(q−4, q4, q−1; q2) = +� +i,j≥0 +q2i2+2j2−2ij−4i+4j +(q2; q2)i(q2; q2)j +(−q−2j; q2)∞ += (−1; q2)∞ +� +i,j≥0 +q2i2+j2−2ij−4i+3j +(q2; q2)i(q2; q2)j +(−q2; q2)j += 2(−q2; q2)∞CT +�� +i≥0 +qi2−4izi +(q2; q2)i +� +j≥0 +q3jz−j(−q2; q2)j +(q2; q2)j +∞ +� +k=−∞ +z−kqk2 +� += 2(−q2; q2)∞CT +�(−q−3z, −q5/z, −qz, −q/z, q2; q2)∞ +(q3/z; q2)∞ +� +(by (2.2) and (2.3)) += 2(−q2; q2)∞CT +�(−q−1z, −q3/z, −q3z, −1/(qz), q2; q2)∞ +(q/z; q2)∞ +� +(replace z by q2z) += 2(−q2; q2)∞CT +�(1 + q−1z)(1 + q−1z−1) +(1 + qz)(1 + qz−1) +· (−qz, −q/z, −qz, −q/z, q2; q2)∞ +(q/z; q2)∞ +� + +12 +ANTUN MILAS AND LIUQUAN WANG += 2q−2(−q2; q2)∞CT +�(−qz, −q/z, −qz, −q/z, q2; q2)∞ +(q/z; q2)∞ +� +. +(2.24) +Note that (2.24) and (2.12) differ only by the factor 2q−2. After replacing q2 by q, +this proves (1.18). In view of (1.12), we obtain (1.16). +□ +3. Proofs of Theorem 1.3 +In this section, we provide two different proofs for Theorem 1.3. In the first proof, +we treat the functions ˜Fi(τ) (1 ≤ i ≤ 6) together by viewing them as vector-valued +modular form. In the second proof, we treat these identities one by one. +3.1. Theta functions. In this subsection, we shall make some preparations for the +proofs. Recall the full modular group +SL(2, Z) = +�� +a +b +c +d +� +: a, b, c, d ∈ Z, ad − bc = 1 +� +. +This group is also conveniently denoted as Γ(1). It is generated by the matrices +S = +� +0 +−1 +1 +0 +� +, +T = +� +1 +1 +0 +1 +� +. +For any congruence subgroup G of SL(2, Z) and Dirichlet character χ, we use +Mk(G, χ) (resp. Sk(G, χ)) to denote the space of modular forms (resp. cusp forms) +on G with weight k and multiplier χ. When χ is trivial, we omit it and write the +space as Mk(G) (resp. Sk(G)). Besides Γ(1) itself, we will mainly work with the +congruence subgroups +Γ1(N) := +�� +a +b +c +d +� +∈ SL(2, Z), +� +a +b +c +d +� +≡ +� +1 +∗ +0 +1 +� +(mod N) +� +, +(3.1) +Γ0(N) := +�� +a +b +c +d +� +∈ SL(2, Z), +� +a +b +c +d +� +≡ +� +∗ +∗ +0 +∗ +� +(mod N) +� +. +(3.2) +It is well-known that the Weber modular functions f(τ), f1(τ) and f2(τ) defined in +(1.21) transform as a vector-valued modular form of rank three under Γ(1): +f(−1/τ) = f(τ), +f2(−1/τ) = 1 +√ +2 +f1(τ), +f1(−1/τ) = +√ +2f2(τ) +f(τ + 1) = e−πi/24f1(τ), +f1(τ + 1) = e−πi/24f(τ), +f2(τ + 1) = eπi/12f2(τ). +We also require the Dedekind η-functions η(τ) and its transformation properties: +η(−1/τ) = +√ +−iτη(τ), +η(τ + 1) = eπi/12η(τ). +Recall the series (∂Θ)j,k(τ) and (∂G)j,k(τ) defined in (1.22) and (1.23). The follow- +ing lemma summarizes some useful properties and especially some transformation +formulas for them. +Lemma 3.1. (1) For any k > 0 and j we have +(∂Θ)0,k(τ) = 0, +(∂Θ)k,k(τ) = 0, +(3.3) +(∂Θ)j, k +2 (τ) = −(∂Θ)k−j, k +2 (τ), +(∂G)j, k +2 (τ) = (∂G)k−j, k +2 (τ), +(3.4) + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +13 +(∂Θ)j,k(τ) = 1 +2(∂Θ)2j,4k(τ) + 1 +2(∂Θ)2j+4k,4k(τ), +(3.5) +(∂G)j,k(τ) = 1 +2(∂Θ)2j,4k(τ) − 1 +2(∂Θ)2j+4k,4k(τ). +(3.6) +(2) For k ∈ N + 1 +2 and j ∈ N we have +(∂Θ)j,k(τ + 1) = eiπj2/2k(∂G)j,k(τ), +(3.7) +(∂G)j,k(τ + 1) = eiπj2/2k(∂Θ)j,k(τ), +(3.8) +(∂Θ)j,k(τ + 2) = eiπj2/k(∂Θ)j,k(τ), +(3.9) +(∂G)j,k(τ + 2) = eiπj2/k(∂G)j,k(τ). +(3.10) +For k ∈ N + 1 +2 and j ∈ N + 1 +2 we have +(∂Θ)j,k(τ + 1) = eiπj2/2k(∂Θ)j,k(τ), +(3.11) +(∂G)j,k(τ + 1) = eiπj2/2k(∂G)j,k(τ). +(3.12) +(3) For k ∈ 1 +2N and j ∈ N we have +(∂Θ)j,k(−1/τ) = (−τ) +� +−iτ/2k +2k−1 +� +j′=1 +eiπjj′/k(∂Θ)j′,k(τ), +(3.13) +(∂G)j,k(−1/τ) = (−τ) +� +−iτ/2k +2k +� +j′=1 +eiπj(2j′−1)/k(∂Θ) 2j′−1 +2 +,k(τ). +(3.14) +(4) For k ∈ 1 +2N and j ∈ N + 1 +2 we have +(∂Θ)j,k(−1/τ) = (−τ) +� +−iτ/2k +2k−1 +� +j′=1 +eiπjj′/k(∂G)j′,k(τ), +(3.15) +(∂G)j,k(−1/τ) = (−τ) +� +−iτ/2k +2k +� +j′=1 +eiπj(2j′−1)/k(∂G) 2j′−1 +2 +,k(τ). +(3.16) +Proof. Replacing n by −n in (1.22), we deduce that (∂Θ)0,k(τ) = 0. Replacing n by +−n − 1, we obtain (∂Θ)k,k(τ) = 0. This proves (3.3). In the same way we can prove +(3.4). +Splitting the sum according to the parity of n, we have +(∂Θ)j,k(τ) = +� +n∈Z +(4kn + j)q(4kn+j)2/4k + +� +n∈Z +(4kn + 2k + j)q(4kn+2k+j)2/(4k) += 1 +2 +� +n∈Z +(8kn + 2j)q(8kn+2j)2/(16k) + 1 +2 +� +n∈Z +(8kn + 4k + 2j)q(8kn+4k+2j)2/(16k) += 1 +2(∂Θ)2j,4k(τ) + 1 +2(∂Θ)2j+4k,4k(τ). +This proves (3.5). Similarly, we can prove (3.6) and hence finish the proof of part +(1). +Part (2) follows from definition. + +14 +ANTUN MILAS AND LIUQUAN WANG +It remains to prove parts (3) and (4). First, the formula (3.13) follows from [19, +Eq. (2.4)] and the fact (∂Θ)0,k(τ) = 0 (see (3.3)). It also appears as [1, Eq. (12.8)]. +Now assume that k ∈ 1 +2N and j ∈ N + 1 +2. Replacing τ by −1/τ in (3.5) and using +(3.13), we deduce that +(∂Θ)j,k(−1/τ) = 1 +2(−τ) +� +−iτ +8k +8k−1 +� +j′=1 +eiπjj′/2k(∂Θ)j′,4k(τ) ++ 1 +2(−τ) +� +−iτ +8k +8k−1 +� +j′=1 +eiπ(2k+j)j′/2k(∂Θ)j′,4k(τ) += −1 +4τ +� +−iτ +2k +8k−1 +� +j′=1 +(1 + eiπj′)eiπjj′/2k(∂Θ)j′,4k(τ) += −1 +2τ +� +−iτ +2k +4k−1 +� +j′=1 +eiπjj′/k(∂Θ)2j′,4k(τ) += −1 +2τ +� +−iτ +2k +2k−1 +� +j′=1 +� +eiπjj′/k(∂Θ)2j′,4k(τ) + eiπj(j′+2k)/k)(∂Θ)2j′+4k,4k(τ) +� += −1 +2τ +� +−iτ +2k +2k−1 +� +j′=1 +eiπjj′/k� +(∂Θ)2j′,4k(τ) − (∂Θ)2j′+4k,4k(τ) +� += −τ +� +−iτ +2k +2k−1 +� +j′=1 +eiπjj′/k(∂G)j,k(τ). +Here for the last third line we used the fact that (∂Θ)4k,4k(τ) = 0 (see (3.3)). This +proves (3.15). +For k ∈ 1 +2N and j ∈ 1 +2N, replacing τ by −1/τ in (3.6) and using (3.13), we deduce +that +(∂G)j,k(−1/τ) = 1 +2(−τ) +� +−iτ +8k +8k−1 +� +j′=1 +(eiπjj′/2k − eiπ(j+2k)j′/2k)(∂Θ)j′,4k(τ) += (−τ) +� +−iτ +8k +4k +� +j′=1 +eiπj(2j′−1)/2k(∂Θ)2j′−1,4k(τ) += (−τ) +� +−iτ +8k +2k +� +j′=1 +� +eiπj(2j′−1)/2k(∂Θ)2j′−1,4k(τ) + eiπj(2j′+4k−1)/2k(∂Θ)2j′+4k−1,4k(τ) +� += (−τ) +� +−iτ +8k +2k +� +j′=1 +eiπj(2j′−1)/2k� +(∂Θ)2j′−1,4k(τ) + e2iπj(∂Θ)2j′+4k−1,4k(τ) +� +. +Discussing according to j ∈ N or j ∈ N + 1 +2 and using (3.5)–(3.6), we obtain (3.14) +and (3.16), respectively. +□ +Combining these formulas give + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +15 +Proposition 3.2. Let k be a positive integer. Then the following functions: +f(τ)3 +η(τ)3(∂Θ)i, 2k+1 +2 (τ), +1 ≤ i ≤ k, +f1(τ)3 +η(τ)3 (∂G)i, 2k+1 +2 (τ), +1 ≤ i ≤ k, +f2(τ)3 +η(τ)3 (∂Θ) 2i−1 +2 +, 2k+1 +2 (τ), +1 ≤ i ≤ k, +combine into a vector-valued modular function under Γ(1). +Proof. First notice that all components are modular functions on an appropriate +congruence subgroup (see [5] for instance). Thus it suffices to prove that the span of +the functions (which are linearly independent) closes under τ → τ + 1 and τ → − 1 +τ . +For τ → τ + 1 this is clear and follows immediately from the formulas above. +Under τ → −1/τ we see using (3.13) and (3.4) that the span of +f3(τ) +η3(τ)(∂Θ)i, 2k+1 +2 (τ) +transforms to itself. Similarly, formulas f2(−1/τ) = +1 +√ +2f1(τ), (3.4), (3.14) and (3.15) +show that the span of f3 +2(τ) +η3(τ)(∂Θ) 2i−1 +2 +, 2k+1 +2 (τ) transforms under τ → −1/τ to the span +of f3 +1(τ) +η3(τ)(∂G)i, 2k+1 +2 (τ) and vice-versa. We proved the assertion. +□ +As a corollary we record +Corollary 3.3. (1) The vector-valued function +� f(τ)3 +η(τ)3(∂Θ)1, 5 +2(τ), f(τ)3 +η(τ)3(∂Θ)2, 5 +2(τ) +� +transforms as a vector-valued modular form (of weight zero) for Γθ = ⟨S, T 2⟩, the +sugbroup of Γ(1) generated by S and T 2. +(2) The vector-valued function +� f(τ)3 +η(τ)3(∂Θ)1, 5 +2(τ), f(τ)3 +η(τ)3(∂Θ)2, 5 +2(τ), f1(τ)3 +η(τ)3 (∂G)1, 5 +2(τ), +f1(τ)3 +η(τ)3 (∂G)2, 5 +2(τ), f2(τ)3 +η(τ)3 (∂Θ) 1 +2, 5 +2(τ), f2(τ)3 +η(τ)3 (∂Θ) 3 +2, 5 +2(τ) +� +transforms as a vector-valued modular function for Γ(1). +Part (2) of this corollary proves the second assertion in Theorem 1.3. Therefore, +to finish the proof of Theorem 1.3, it suffices to prove the six identities. Below we +present two different proofs. +3.2. First proof of Theorem 1.3. We first recall several known facts. Recall the +theta functions θ2(τ) and θ3(τ) defined in (2.4) and (2.5). The pair +(W1(τ), W2(τ)) := +�θ3(τ) +η(τ) , θ2(τ) +η(τ) +� +(3.17) + +16 +ANTUN MILAS AND LIUQUAN WANG +defines a two-dimensional vector-valued modular form ρ1 of weight zero whose S- +matrix is given by +� +W1(−1/τ) +W2(−1/τ) +� += 1 +√ +2 +� +1 +1 +1 +−1 +� � +W1(τ) +W2(τ) +� +. +These two series are precisely the level one characters of standard A(1) +1 -modules. +Let also +chr,s +3,5(τ) := +1 +η(τ) +� +n∈Z +� +q +(30n+5r−3s)2 +60 +− q +(30n+5r+3s)2 +60 +� +. +This notation indicates that ch1,1 +3,5(τ), ch1,2 +3,5(τ), ch2,1 +3,5(τ), ch2,2 +2,5(τ) are precisely char- +acters of four irreducible (3, 5) Virasoro minimal models. We need the following +known identities obtained using the quintuple product identity [4, Theorem 1.3.17] +that connect Rogers’ series (1.5)-(1.8) with these characters (see also [9, p. 170]): +Z1(τ) := q1/40 +∞ +� +n=0 +qn2+n +(q; q)2n += ch1,1 +3,5(τ), +(3.18) +Z2(τ) := q31/40 +∞ +� +n=0 +qn2+2n +(q; q)2n+1 += ch2,1 +3,5(τ), +(3.19) +Z3(τ) := q9/40 +∞ +� +n=0 +qn2+n +(q; q)2n+1 += ch2,2 +3,5(τ), +(3.20) +Z4(τ) := q−1/40 +∞ +� +n=0 +qn2 +(q; q)2n += ch1,2 +3,5(τ). +(3.21) +These expressions transforms under SL(2, Z), and they define a 4-dimensional +vector-valued modular form ρ2 of weight zero with a well-known S-matrix: + + + + +Z1(−1/τ) +Z2(−1/τ) +Z3(−1/τ) +Z4(−1/τ) + + + + = +� +2 +5 + + + + +sin(2π/5) +− sin(2π/5) +− sin(π/5) +sin(π/5) +− sin(2π/5) +− sin(2π/5) +sin(π/5) +sin(π/5) +− sin(π/5) +sin(π/5) +− sin(2π/5) +sin(2π/5) +sin(π/5) +sin(π/5) +sin(2π/5) +sin(2π/5) + + + + + + + + +Z1(τ) +Z2(τ) +Z3(τ) +Z4(τ) + + + + . +See for instance [9, p. 172, Eq. (15)] except for a typo in the (3, 3)-entry of this +matrix. One can easily write the diagonal T-matrices for ρ1 and ρ2 using the leading +terms. Observe that ρ1⊗ρ2, that is (WiZj)1≤i≤2,1≤j≤4, defines a 8-dimensional vector- +valued modular form on Γ(1). However, we also have a proper subspace in ρ1 ⊗ ρ2 +that also closes under the full modular group. +Proposition 3.4. Let Wi and Zi be as above. We have +� +˜F1(q), ˜F2(q), ˜F3(q), ˜F4(q), ˜F5(q), ˜F6(q) +� += (f(τ)(W1Z4 + W2Z3), f2(τ)(W1Z1 + W2Z2), +f2(τ)(W1Z3 + W2Z4), f(τ)(W1Z2 + W2Z1), f1(τ)(W1Z4 − W2Z3), f1(τ)(W2Z1 − W1Z2)), +(3.22) +and it defines a vector-valued modular form ˜ρ1 of weight zero on Γ(1). + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +17 +In fact, the equality (3.22) follows directly from Theorem 1.2. One can also see +this in a more straightforward way from the proof of Theorem 1.2. For instance, +from (2.9), (2.10) and (2.11) (after replacing q2 by q) we immediately conclude that +˜F1(q) = f(τ)(W1Z4 + W2Z3). +(3.23) +The second assertion of this proposition follows by direct computations using the S +and T-matrices of W1, W2 and Zi (1 ≤ i ≤ 4). +Denote by ˜ρ2 the vector-valued modular form of weight zero: +(h1(q), h2(q), h3(q), h4(q), h5(q), h6(q)) := +� f(τ)3 +η(τ)3(∂Θ)1, 5 +2(τ), 2f2(τ)3 +η(τ)3 (∂Θ) 1 +2, 5 +2(τ), +2f2(τ)3 +η(τ)3 (∂Θ) 3 +2 , 5 +2(τ), f(τ)3 +η(τ)3(∂Θ)2, 5 +2(τ), f1(τ)3 +η(τ)3 (∂G)1, 5 +2(τ), f1(τ)3 +η(τ)3 (∂G)2, 5 +2(τ) +� +. (3.24) +This is precisely the vector-valued modular form appearing in Corollary 3.3(2), +except that we added the factor 2 for the second and third entries. +So far we constructed two 6-dimensional vector valued modular forms: ˜ρ1 in (3.22) +and ˜ρ2 in (3.24). It is easy to see that (by direct computations) the S and T matrices +of them agree with each other. We claim that ˜ρ1 = ˜ρ2. +To prove the above claim, we switch to new bases of vector-valued modular forms. +Recall the basis gi(q) from Remark 1. Similarly we let +˜g1(q) := h1(q) + h5(q) = q−7/80(2 + 12q + 30q2 + · · · ) ∈ q−7/80C[[q]], +˜g2(q) := h1(q) − h5(q) = q33/80(6 + 18q + 54q2 + · · · ) ∈ q33/80C[[q]], +˜g3(q) := h4(q) + h6(q) = q17/80(4 + 6q + 30q2 + · · · ) ∈ q17/80C[[q]], +˜g4(q) := h4(q) − h6(q) = q57/80(6 + 16q + 42q2 + · · · ) ∈ q57/80C[[q]], +˜g5(q) := h2(q) = q1/40(1 + 6q + 15q2 + · · · ) ∈ q1/40C[[q]], +˜g6(q) := h3(q) = q9/40(3 + 11q + 30q2 + · · · ) ∈ q9/40C[[q]]. +Then (gi(q) − ˜gi(q))6 +i=1 also transforms as a vector-valued modular form. Using +their q-expansions it is easy to see (cf. Remark 3) that the Wronskian of gi(h)−˜gi(q), +1 ≤ i ≤ 6 has the order of vanishing that is strictly bigger than +5 +2. +But then, +according to Proposition 4.3, the Wronskian is identically zero and thus (gi(q) − +˜gi(q)) are linearly dependent. The last sentence in Remark 1 implies that the linear +dependence is equivalent to gj(q) = ˜gj(q) for some j. Thus we obtain a 5-dimensional +vector-valued modular form (gi(q) − ˜gi(q))i̸=j. Applying the same type of argument +to it yields gk(q) = ˜gk(q) for k ̸= j, etc. Thus we conclude gi(q) = ˜gi(q) for all i. +That clearly implies ˜Fi(q) = hi(q) and completes the proof of Theorem 1.3. +Remark 2. The claim ˜ρ1(τ) = ˜ρ2(τ) can also be proved by a Sturm-type criterion +of vector-valued modular forms. From [12, Proposition 1.1] we know that it suffices +to compare the first two terms in their q-expansions. Interestingly, in the above +proof using Wronskian, we also only need to compare the first two terms of gi(q) +and ˜gi(q), which guarantees that the Wronskian of gi(q) − ˜gi(q), 1 ≤ i ≤ 6 has the +order of vanishing > 5 +2. + +18 +ANTUN MILAS AND LIUQUAN WANG +3.3. Second proof of Theorem 1.3. We will need the order of a function f(τ) +with respect to a congruence subgroup G at the cusp p ∈ Q ∪ {∞} and denote it as +ord(f, p). +Recall the theta function θ3(τ) defined in (2.5). It is known that [16, Proposition +1.41] θ3(τ) ∈ M 1 +2(Γ0(4)). For any odd primitive Dirichlet character ψ with conductor +N, it is known that [16, Theorem 1.44] +θ(ψ, τ) := +∞ +� +n=1 +ψ(n)nqn2 ∈ S 3 +2(Γ0(4N2, ψχ−4), +where χ−4 is the nontrivial Dirichlet character modulo 4. +Lemma 3.5. For a ∈ {1, 2, 3, 4}, we have +� +n∈Z +n≡a (mod 5) +nqn2 ∈ M 3 +2(Γ1(100)). +(3.25) +Proof. Let ψk (k = 0, 1) be the primitive Dirichlet character with conductor 5 sat- +isfying ψk(2) = (−1)ki. Then +θ(ψ0, τ) = +� +n∈Z +ψ0(n)nqn2 = +� +n∈Z +(5n + 1)q(5n+1)2 − +� +n∈Z +(5n + 4)q(5n+4)2 ++ i +�� +n∈Z +(5n + 2)q(5n+2)2 − +� +n∈Z +(5n + 3)q(5n+3)2 +� += 2 +� +n∈Z +(5n + 1)q(5n+1)2 + 2i +� +n∈Z +(5n + 2)q(5n+2)2. +(3.26) +Here for the last equality we used the fact that for a ∈ {1, 2}, +� +n∈Z +(5n + a)q(5n+a)2 = − +� +n∈Z +(5n + 5 − a)q(5n+5−a)2, +(3.27) +which can be proved easily by changing n to −n − 1. +In the same way, we have +θ(ψ1, τ) = 2 +� +n∈Z +(5n + 1)q(5n+1)2 − 2i +� +n∈Z +(5n + 2)q(5n+2)2. +(3.28) +Since θ(ψk, τ) ∈ M 3 +2(Γ1(100)) (k = 0, 1), from (3.26) and (3.28) we deduce that +� +n∈Z +(5n + 1)q(5n+1)2 = 1 +4(θ(ψ0, τ) + θ(ψ1, τ)) ∈ M 3 +2(Γ1(100)), +(3.29) +� +n∈Z +(5n + 2)q(5n+2)2 = 1 +4i(θ(ψ0, τ) − θ(ψ1, τ)) ∈ M 3 +2(Γ1(100)). +(3.30) +This together with (3.27) proves the lemma. +□ +We also need the following identity +J2 +1 = J2J5 +8 +J2 +4J2 +16 +− 2qJ2J2 +16 +J8 +. +(3.31) + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +19 +This appeared frequently in the literature and is a direct consequence of [3, p. 40, +Entry 25(v),(vi)]. +Following the notion in [8], we define the generalized Dedekine eta function +ηδ;g(τ) := q +1 +2 δP2(g/δ) +� +m≡±g +(mod δ) +(1 − qm), +(3.32) +where P2(t) = {t}2 − {t} + 1 +6 is the second periodic Bernoulli polynomial, {t} is the +fractional part of t, g, δ, m ∈ Z+ and 0 < g < δ. +Now we are ready to give our second proof of Theorem 1.3. By the definition +given in (1.19) and (1.20), we know that the identities for �F5(q) and �F6(q) follow +from those of �F1(q) and �F4(q), respectively. Thus, it suffices to prove the first four +identities. By (1.11)–(1.14), the first four identities are equivalent to +T1(q) := +� +n∈Z +(5n + 1)q5n2+2n = J2 +1J8 +4J40 +J5 +2J2 +8J8,40 ++ 2qJ2 +1J4J2 +8J6,20J8,40 +J3 +2J40 +, +(3.33) +T2(q) := +� +n∈Z +(5n + 2)q5n2+4n = 2J2 +1J4J2 +8J2,20J16,40 +J3 +2J40 ++ qJ2 +1J7 +4J8,20J4,40 +J5 +2J2 +8J40 +, +(3.34) +T3(q) := +� +n∈Z +(10n + 1)q5n2+n = J2J3 +4J2,20J16,40 +J2 +8J40 ++ 2q2J3 +2J2 +8J8,20J4,40 +J3 +4J40 +, +(3.35) +T4(q) := +� +n∈Z +(10n + 3)q5n2+3n = 2J3 +2J2 +8J40 +J2 +4J8,40 ++ J2J3 +4J6,20J8,40 +J2 +8J40 +. +(3.36) +We will prove (3.33) first and then deduce the other identities from it. After replac- +ing q by q5 and multiplying both sides of (3.33) by qθ3(5τ), we see that (3.33) is +equivalent to +θ3(5τ) +� +n∈Z +(5n + 1)q(5n+1)2 = θ4 +3(5τ) (f1(τ) + f2(τ)) . +(3.37) +Here +f1(τ) := q J8 +5J14 +20J200 +J20 +10J2 +40J40,200 += +η8(5τ)η14(20τ) +η20(10τ)η2(40τ)η200,40(τ), +(3.38) +f2(τ) := 2q6J8 +5J7 +20J2 +40J30,100J40,200 +J18 +10J200 += η8(5τ)η7(20τ)η2(40τ)η(100τ)η100,30(τ)η200,40(τ) +η18(10τ) +. +(3.39) +With the help of Maple and the algorithm in [8], it is easy to check that both +f1(τ) and f2(τ) are modular functions on Γ1(200). Their poles and corresponding +orders of them are listed in Table 1. +On the other hand, it is easy to see that θ4 +3(5τ) ∈ M2(Γ0(20)). Hence θ4 +3(5τ) ∈ +M2(Γ1(200)). We can evaluate the orders of zeros of θ4 +3(5τ) at any cusp for Γ1(200) +(using Theorem 2.3 and the equation (2.12) in [8]). In Table 1 we only list the orders +of zeros of θ4 +3(5τ) at the poles of fi(τ) (i = 1, 2). It turns out that after multiplying + +20 +ANTUN MILAS AND LIUQUAN WANG +cusp p +ord(fi(τ), p) +ord(θ4 +3(5τ), p) +1 +2, 1 +6, 1 +14, 1 +18, 1 +22, 1 +26, 1 +34, 1 +38, 1 +42, 1 +46, +1 +54, 1 +58, 1 +62, 1 +66, 1 +74, 1 +78, 1 +82, 1 +86, 1 +94, 1 +98 +−16 +20 +1 +10, 1 +30, 1 +70, 1 +90, 3 +10, 3 +70, 7 +10, 7 +30, +7 +90, 9 +10, 9 +70, 23 +30, 23 +90, 29 +30, 29 +90, 67 +70 +−80 +100 +1 +50, 9 +50, 11 +50, 19 +50, 21 +50, 29 +50, 31 +50, 39 +50, 41 +50, 49 +50 +−16 +20 +3 +50, 7 +50, 13 +50, 17 +50, 23 +50, 27 +50, 33 +50, 37 +50, 43 +50, 47 +50 +−14 +20 +Table 1. Orders of poles and zeros at cusps for Γ1(200) +by θ4 +3(5τ), all the poles of fi(τ) will be eliminated. Hence θ4 +3(5τ)fi(τ) ∈ M2(Γ1(200)) +(i = 1, 2). +So far we have proved that both sides of (3.37) belong to M2(Γ1(200)). By Sturm’s +criterion (see [7, p. 185, Corollary 5.6.14], for example), to prove (3.37), it suffices +to verify that both sides agree for the first +1 + 2 +12 · 1 +2[SL(2, Z) : Γ1(200)] = 2401 +terms. We have checked this with Maple. Hence (3.37) holds and we finish the proof +of (3.33). +Next, we are going to prove (3.34)–(3.36) based on (3.33). +We aim to find a +2-dissection formula for T1(q): +T1(q) = +� +n∈Z +(5n + 1)q5n2+2n = L0(q2) + qL1(q2). +(3.40) +On the one hand, since 5n2 + 2n has the same parity with n, we have +L0(q2) = +� +n even +(5n + 1)q5n2+2n = +� +n∈Z +(10n + 1)q20n2+4n = T3(q4), +(3.41) +qL1(q2) = +� +n odd +(5n + 1)q5n2+2n = +� +n∈Z +(5(−2n − 1) + 1)q5(−2n−1)2+2(−2n−1) += −2q3 � +n∈Z +(5n + 2)q20n2+16n = −2q3T2(q4). +(3.42) +On the other hand, substituting (3.31) into (3.33), we obtain +� +n∈Z +(5n + 1)q5n2+2n = +� J2J5 +8 +J2 +4J2 +16 +− 2qJ2J2 +16 +J8 +� +· +� +J8 +4J40 +J5 +2J2 +8J8,40 ++ 2qJ4J2 +8J6,20J8,40 +J3 +2J40 +� += J6 +4J3 +8J40 +J4 +2J2 +16J8,40 +− 4q2J4J8J2 +16J6,20J2,40 +J2 +2J40 ++ 2q +�J7 +8J6,20J8,40 +J2 +2J4J2 +16J40 +− J8 +4J2 +16J40 +J4 +2J3 +8J8,40 +� +. +(3.43) + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +21 +Combining (3.40)–(3.43), we deduce that +L0(q) = T3(q2) = J6 +2J3 +4J20 +J4 +1J2 +8J4,20 +− 4qJ2J4J2 +8J3,10J4,20 +J2 +1J20 +, +(3.44) +L1(q) = −2qT2(q2) = J7 +4J3,10J4,20 +J2 +1J2J2 +8J20 +− J8 +2J2 +8J20 +J4 +1J3 +4J4,20 +. +(3.45) +Using (3.44) and (3.45) and the method in [8], it is easy to verify that (3.34) and +(3.35) hold. +It remains to prove (3.36). For this we make a 2-dissection for T2(q): +T2(q) = +� +n∈Z +(5n + 2)q5n2+4n = H0(q2) + qH1(q2). +(3.46) +In the same way as we did for T1(q), we can prove that +H0(q) = 2T1(q2) = 2J7 +4J1,10J8,20 +J2 +1J2J2 +8J20 +− 2qJ7 +2J2 +8J4,10J2,20 +J4 +1J3 +4J20 +, +(3.47) +H1(q) = −T4(q2) = J5 +2J3 +4J4,10J2,20 +J4 +1J2 +8J20 +− 4J2J4J2 +8J1,10J8,20 +J2 +1J20 +. +(3.48) +Now from (3.48) and using the method in [8], it is easy to verify that (3.36) holds. +4. General conjecture and concluding remarks +4.1. Non-modularity of some Nahm sums. In [6, Section 5] in addition to +χ0(1, 1, 1) and χ0(1, 1, q +1 +2) two additional specializations are considered: χ0(q, 1, 1) +and χ0(1, q, 1). +Thus it seems natural to ask whether these two series are also +modular after addition of a suitable multiplicative factor. We next show that this +is not the case. +Recall that we denote by fA,B,C(q) the Nahm sum associated to T3 matrix with +B = (B1, B2, B3) as in Section 1. Using this notation, we can write fA,(1,0,0),C(q) = +qCχ0(q, 1, 1) and fA,(0,1,1),C(q) = qCχ0(1, q, 1). Denote by +Q1 = 1 +2(3 − +√ +5), +Q2 = −2 + +√ +5, +Q3 = 1 +4(3 − +√ +5) +the unique solution inside the interval (0, 1) of the TBA system [20, Lemma 2.1]: +1 − Q1 = Q2 +1Q−1 +2 , +1 − Q2 = Q−1 +1 Q2 +2Q−1 +3 , +1 − Q3 = Q3Q−1 +2 . +Proposition 4.1. The Nahm sums fA,(1,0,0),C(q) and fA,(0,1,0),C(q) are not modular +for any rational number C. +Proof. To prove this, it suffices to argue that the two Nahm sums do not have +expected asymptotic expansion around zero. Letting q = e−ǫ, then according to [20, +Theorem 2.1] we have the following asymptotic behavior (as ǫ → 0+): +fA,B,C(e−ǫ)e− α +ǫ ∼ βe−γǫ(1 + +� +p≥1 +cpǫp) +where α is a positive constant, γ = C + 1 +24 +�r +i=1 +1+Qi +1−Qi, cp are some hard-to-compute +coefficients expressed using generalized 3-fold Gaussian integrals and β is a nonzero + +22 +ANTUN MILAS AND LIUQUAN WANG +constant not needed here. As a necessary condition for modularity we notice rela- +tions [20, Corollary 3.1]: +γp +p! − cp = 0, p ≥ 1. +In our situation, the condition c1 − γ = 0 is equivalent to +C = 9B1 +2 +4 +√ +5 − 3B1 +2 +4 ++ 3B1B2 +√ +5 +− B1B2 − B1B3 +2 +√ +5 + B1B3 +2 ++ 2B1 +√ +5 − 9B1 +10 ++ B2 +2 +√ +5 + B2B3 +2 +√ +5 + B2B3 +2 ++ 7B2 +4 +√ +5 − 17B2 +20 ++ B3 +2 +√ +5 + B3 +2 +4 +− B3 +2 +√ +5 + B3 +10 − 7 +80. +Plugging in B = (1, 0, 0) gives the value C = +1 +80(−139 + 68 +√ +5) which is irrational +and similarly for B = (0, 1, 0). Therefore the two Nahm sums in question cannot be +modular. +□ +Although χ0(q, 1, 1) and χ0(1, q, 1) cannot be made modular their sum satisfies +χ0(q, 1, 1) + χ0(1, q, 1) = χ0(q−1, q, 1) +(4.1) +which is modular after multiplying with q +17 +80. Relation (4.1) follows from a slightly +more general statement: +� +i,j,k≥0 +qi2+j2+k2/2−ij−jk+j−ixi +1xj +2xk +3 +(q)i(q)j(q)k += +� +i,j,k≥0 +qi2+j2+k2/2−ij−jkxi +1xj +2xk +3(x1qi + qj) +(q)i(q)j(q)k +. (4.2) +Comparing the coefficients of xi +1xj +2xk +3 of both sides of (4.2), we see that (4.2) is +equivalent to +qi2+j2+k2/2−ij−jk+j−i +(q)i(q)j(q)k += qi2+j2+k2/2−ij−jk+j +(q)i(q)j(q)k ++ q(i−1)2+j2+k2/2−(i−1)j−jk+(i−1) +(q)i−1(q)j(q)k +. +The last formula is trivial to check. +4.2. General conjecture. In this part we discuss the general conjecture on the +modularity of the rank n Nahm sum χ0(1) associated to Tn as in the introduction, +where for brevity we let 1 := (1, 1, ..., 1). +It is possible to formulate a slightly +stronger conjecture result analogous to Theorem 1.3 but we omit discussing it here. +Let f1,...,fℓ be any holomorphic functions in the upper half-plane. Denote by +D = +� +q d +dq +� += +1 +2πi +∂ +∂τ Ramanujan’s derivative and by +∂k := D − k +12E2 +where E2(q) = 1−24 � +n≥1 +nqn +1−qn is the second Eisenstein series. This map is known to +send the space of modular forms of weight k (on some congruence subgroup) into the +space of modular forms of weight k+2 for the same congruence subgroup. Denote by +WD(f1, ..., fℓ) the Wronskian determinant with respect to the D-derivation which is +again a holomorphic function. Additionally, denote by W∂k(f1, ..., fℓ) the Wronskian +with respect to the ∂k derivation, where the r-th derivative is defined as ∂r +k := +∂k+2r−2 ◦ · · · ◦ ∂k. Suppose that each fi admits a q-expansion. Then WD also has +a q-expansion so we can denote by � +WD the Wronskian normalized such that the + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +23 +leading coefficient in the q-expansion is 1. Then we have a known result (see for +instance [5,14]). +Theorem 4.2. Let f1,. . . ,fℓ be modular forms of weight k with respect to a congru- +ence subgroup, then the Wronskian +WD(f1, ..., fℓ) = W∂k(f1, ..., fℓ) +is a modular form of weight ℓ(ℓ + k − 1) on the same congruence subgroup. +In [5] a more precise information about the automorphy factor and congruence +subgroups for the Wronskian modular form WD is given. +Now we specialize Theorem 4.2 to a situation where V = Span(f1, ..., fℓ) define +an ℓ-dimensional modular invariant space under Γ(1). Then we have the following +result. +Proposition 4.3. (1) Let f1,..,fℓ be a basis of the modular invariant space V . If +ord(WD(f1, ..., fℓ), i∞) = λ +then +WD(f1, ..., fℓ) = η(τ)24λG(τ), +where G(τ) is a nonzero holomorphic modular form of weight ℓ(ℓ−1)−12λ on Γ(1). +In particular, if λ = ℓ(ℓ−1) +12 , then +� +WD(f1, ..., fℓ) = η(τ)2ℓ(ℓ−1). +(2) If +ord(WD(f1, ..., fℓ), i∞) > ℓ(ℓ − 1) +12 +then the Wronskian is identically zero and fi are linearly dependent. +Part (1) of this proposition can be found in [13, Theorem 3.7]; see also [14, The- +orem 2.2 and Proposition 2.4] and [2, Theorem 1]. Part (2) follows from part (1) +and the fact that Mk(Γ(1)) = 0 for k < 0. See also [13, Lemma 3.6]. +Remark 3. In the situation when fi(q) = qri(a(i) +0 + a(i) +1 q + · · ·), a(i) +0 +̸= 0 for every i +and ri ̸= rj for i ̸= j, it is easy to see that the order of vanishing of WD(f1, ..., fℓ) +is �ℓ +i=1 ri. Using this fact and Remark 1 we immediately see that the order of +vanishing of WD( ˜F1, ˜F2, ˜F3, ˜F4, ˜F5, ˜F6) is 3 +2, so in our case the Wronskian is not an +η-power and instead we have +� +WD( ˜F1, ˜F2, ˜F3, ˜F4, ˜F5, ˜F6) = (5892480)−1·η(τ)36(70027513E4(τ)3−64135033E6(τ)2). +Now we are ready to state a general conjecture which is based on some numerical +evidence. +Conjecture 4.4. Let χ0(1) be the Nahm sum associated to Tn, n ≥ 2. Then we +have +(1) For n = 2k ≥ 2 even: +qakχ0(1) = f(τ)2k � +WD(R2k,1, ...., R2k,k) +η(τ)k(2k−1) +, + +24 +ANTUN MILAS AND LIUQUAN WANG +where ak = −k(1+4k) +48(1+k) and +R2k,i(τ) = +� +n∈Z +(−1)nq(k+1)(n− (2i−1) +4(k+1) )2. +(2) For n = 2k − 1 ≥ 3 odd: +qakχ0(1) = +f(τ)2k−1 � +WD((∂Θ)1, 2k+1 +2 , ...., (∂Θ)k−1, 2k+1 +2 ) +η(τ)(k−1)(2k−1) +, +where ak = −1+6k−8k2 +96k+48 +. +More precisely, qakχ0(1) is a modular function which is a component of a 3k- +dimensional vector valued modular function under Γ(1). +Part (1) of the conjecture is known to hold for n = 2 [6] and we also verified n = 4 +numerically for high powers of q. Part (2) for n = 3 was proven in this paper. +Acknowledgements. The second author was supported by the National Natural +Science Foundation of China (12171375). +References +[1] D. Adamovi´c and A. Milas, The N = 1 triplet vertex operator superalgebras: twisted sec- +tor. SIGMA Symmetry, Integrability and Geometry: Methods and Applications. 2008 Dec +13;4:087. +[2] Y. Arike, M. Kaneko, K. Nagatomo, and Y. Sakai, Affine vertex operator algebras and +modular linear differential equations, Letters in Mathematical Physics 106, no. 5 (2016): +693-718. +[3] B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991. +[4] B.C. Berndt, Number Theory in the Spirit of Ramanujan, the American Mathematical So- +ciety, 2006. +[5] K. Bringmann, C. Calinescu, A. Folsom, and S. Kimport. Graded dimensions of princi- +pal subspaces and modular Andrews–Gordon-type series. Communications in Contemporary +Mathematics 16, no. 04 (2014): 1350050. +[6] C. Calinescu, A. Milas and M. Penn, Vertex algebraic structure of principal subspaces of +basic A(2) +2n -modules. Journal of Pure and Applied Algebra, 220 (2016), pp.1752-1784. +[7] H. Cohen and F. Str¨omberg, Modular Forms: a Classical Approach, Graduate Studies in +Mathematics 179, the American Mathematical Society, Providence, RI, 2017. +[8] F.G. Garvan and J. Liang, Automatic proof of theta-function identities, arXiv:1807.08051. +[9] K. Kawasetsu, The intermediate vertex subalgebras of the lattice vertex operator algebras, +Letters in Mathematical Physics 104, no. 2 (2014): 157-178. +[10] J. Lepowsky and R.L. Wilson, The structure of standard modules, I: Universal algebras and +the Rogers-Ramanujan identities. Invent. Math. 77 (1984), 199-290. +[11] J. Lepowsky and R.L. Wilson, The structure of standard modules II. the case A(1) +1 , principal +gradation, Invent. Math. 79 (1985), 417–442. +[12] T. Magnusson and M. Raum, On the Computation of General Vector-valued Modular Forms, +arXiv:2202.06676. +[13] G. Mason, Vector-valued modular forms and linear differential operators, International Jour- +nal of Number Theory 3, no. 03 (2007): 377-390. +[14] A. Milas, On certain automorphic forms associated to rational vertex operator algebras. +Moonshine - The First Quarter Century and Beyond: Proceedings of a Workshop, Edinburgh +2004, (2010): 330. + +MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM +25 +[15] A. Milas and M. Penn, Lattice vertex algebras and combinatorial bases: general case and +W-algebras, New York Journal of Mathematics 18 (2012): 621-650. +[16] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q- +series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Confer- +ence Board of the Mathematical Sciences, Washington, DC; by the American Mathematical +Society, Providence, RI, 2004. +[17] L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London +Math. Soc. 25 (1894), 318–343. +[18] L.J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. +London Math. Soc. 16 (1917), 315–336. +[19] G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. +[20] M. Vlasenko and S. Zwegers, Nahm’s conjecture: asymptotic computations and counterex- +amples. Communications in Number Theory and Physics. 2011;5(3):617-42. +[21] M. Wakimoto, Infinite-dimensional Lie algebras. Vol. 195. American Mathematical Soc., +2001. +[22] L. +Wang, +Identities +on +Zagier’s +rank +two +examples +for +Nahm’s +conjecture, +arXiv:2210.10748v2. +[23] L. Wang, Explict forms and proofs of Zagier’s rank three examples for Nahm’s problem, +arXiv:2211.04375v2. +[24] H. Weber, Lehrbuch der Algebra, Bd.3, Elliptische Funktionen and Algebraische Zahlen, +Braunschweig, 1908. +[25] D. Zagier, The dilogarithm function, in Frontiers in Number Theory, Physics and Geometry, +II, Springer, 2007, 3–65. +Department of Mathematics and Statistics, University at Albany (SUNY), Al- +bany, NY 12222, United States +Email address: amilas@math.albany.edu +School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, +People’s Republic of China +Email address: wanglq@whu.edu.cn;mathlqwang@163.com + diff --git a/atE3T4oBgHgl3EQfdQoZ/content/tmp_files/load_file.txt b/atE3T4oBgHgl3EQfdQoZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0ed788d443248749a2171de1f84837c5d6ecaab2 --- /dev/null +++ b/atE3T4oBgHgl3EQfdQoZ/content/tmp_files/load_file.txt @@ -0,0 +1,1192 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf,len=1191 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='04532v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='NT] 11 Jan 2023 MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM ANTUN MILAS AND LIUQUAN WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We prove Rogers-Ramanujan type identities for the Nahm sums as- sociated with the tadpole Cartan matrix of rank 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first author in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We also present some conjectures for a general rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Introduction and Main Results The famous Rogers-Ramanujan identities state that ∞ � n=0 qn2 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n = 1 (q, q4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q5)∞ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1) ∞ � n=0 qn2+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n = 1 (q2, q3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q5)∞ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) where (here and throughout this paper) we always assume |q| < 1 and use standard q-series notations: (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)0 := 1, (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n := n−1 � k=0 (1 − aqk), (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ := ∞ � k=0 (1 − aqk), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3) (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , am;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n := (a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n · · · (am;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n, n ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4) When the base q is clear from the context, occasionally we omit it and simply write (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n as (a)n (n ∈ N ∪ {∞}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The Rogers-Ramanujan identities first appeared in the 1894 paper of Rogers [17] and were later rediscovered by Ramanujan before 1913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Besides (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2), Rogers [17, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 330-332] also proved some similar sum-product identities such as ∞ � n=0 qn2 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n = (q2, q8, q10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q10)∞(q6, q14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q20)∞ (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5) ∞ � n=0 qn2+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n = (q, q9, q10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q10)∞(q8, q12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q20)∞ (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 11P84, 33D15, 33D45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Rogers-Ramanujan identities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' sum-product identities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Nahm sums;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' tad- pole Cartan matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' vector-valued modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 2 ANTUN MILAS AND LIUQUAN WANG ∞ � n=0 qn2+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n+1 = (q3, q7, q10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q10)∞(q4, q16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q20)∞ (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='7) Later Rogers [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 330 (3), 2nd Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=']) proved another companion identity: ∞ � n=0 qn2+2n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n+1 = (q4, q6, q10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q10)∞(q2, q18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q20)∞ (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) The Rogers-Ramanujan identities also serve as important examples for close re- lations between q-series and modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The product sides are essentially re- ciprocals of some generalized Dedekind eta functions (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='32)) and hence are modular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This is not observable from the sum sides, which is a ba- sic q-hypergeometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' A natural question is to ask when does a basic q- hypergeometric series become a modular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In particular, a famous problem of Nahm is to determine for which positive definite r × r rational matrix A, r- dimensional rational vector B, and a rational scalar C such that fA,B,C(q) := � n=(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=',nr)T∈(Z≥0)r q 1 2 nTAn+nTB+C (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n1 · · · (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)nr is a modular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Such (A, B, C) is called as a rank r modular triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The series fA,B,C(q) is therefore referred as Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Several important families of q-series identities (such as Andrews-Gordon iden- tities) can be also studied using vertex operators and representation of infinite- dimensional Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This approach, pioneered by Lepowsky and Wilson in 1980s [10,11], was one of the starting points in the development of vertex operator algebras and an important ingredient in the development of 2-dimensional confor- mal field theory (CFT) in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In this setup, the graded dimension obtained from a combinatorial bases of modules can be often interpreted as a Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The Nahm sums that are relevant for rational CFT almost always take form with A = G ⊗ G′−1 where G and G′ are ADET type Cartan matrices, and all such Nahm sums matrices are expected to give modular functions (with appropriate B and C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Interestingly, any Nahm sum fA,0,0 can be interpreted as the graded dimension of a special vertex algebra called principal subspace [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Zagier [25] studied Nahm’s problem and made significant progress when the rank r ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In particular, he proved that there are exactly seven rank one modular triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the rank two and three cases, Zagier provided a number of conjectural modular triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Most of the rank two examples have been confirmed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For example, Vlasenko and Zwegers [20] confirmed one modular triple in Zagier’s list and discovered some new examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recently, the second author [22] confirmed more examples in Zagier’s list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' As a consequence, among the eleven rank two examples discovered by Zagier [25, Table 2], only one example is unproven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the rank three case, Zagier [25, Table 3] provided a list of twelve possible modular triples and proved three of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The remaining nine examples were confirmed by the second author [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In this paper, we are mainly concerned with the modularity of the Nahm sums associated with the tadpole diagram T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The tadpole Nahm sums were considered by MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 3 Penn, Calinescu, and the first author in their work on twisted modules of principal subspace vertex algebras [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Given a positive integer r, let Tr be the tadpole Cartan matrix, that is, Tr = (aij)r×r such that arr = 1, aii = 2, 1 ≤ i ≤ r − 1, aij = −1 (|i − j| = 1), and aij = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We let χ0(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , xr) = χ0(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , xr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q) := � n=(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=',nr)∈Zr ≥0 q 1 2 nTTrnxn1 1 · · · xnr n (q)n1 · · · (q)nr be a generalized tadpole Nahm sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It is easy to see that qCχ0(qB1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , qBr) = fA,B,C(q) is the Nahm sum with A = Tr and B = (B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , Br).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' There is only one standard A(2) 2n -module of level one (up to isomorphism) and thus only one principal subspace of level 1, corresponding to the unique standard level one module, whose character is χ0(1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' xi = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In [6] the so-called shifted characters χi were introduced by specializing xi = q and xj = 1 for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Penn, Calinescu, and the first author [6] stated a conjecture concerning the modularity of the characters χ0(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [6, Conjecture 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=') The character qaχ0(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' , 1) is modular for some rational number a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The rank two case (r = 2) was proved in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The main goal of this paper is to address the conjecture for r = 3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In this case, we write explicitly χ0(x1, x2, x3) = χ0(x1, x2, x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q) = � i,j,k≥0 qi2+j2+ 1 2k2−ij−jkxi 1xj 2xk 3 (q)i(q)j(q)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='9) We record four shifted q-characters that are of interest here: F1(q) := χ0(1, 1, 1), F2(q) := χ0(1, 1, q 1 2), F3(q) := χ0(q, q−1, q 1 2), F4(q) := χ0(q−1, q, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We are mainly concerned with modular properties of these sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We first write down the tadpole Cartan matrix T3 and its inverse: T3 = \uf8eb \uf8ed 2 −1 0 −1 2 −1 0 −1 1 \uf8f6 \uf8f8 , T −1 3 = \uf8eb \uf8ed 1 1 1 1 2 2 1 2 3 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Note that T −1 3 is the matrix part of the sixth example in Zagier’s list [25, Table 3] (see also [23, Example 6]) with the first row/column and the third row/column interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Zagier stated six possible modular triples with T −1 3 as the matrix part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The vector parts are B ∈ \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed 0 0 0 \uf8f6 \uf8f8 , \uf8eb \uf8ed 1/2 1 3/2 \uf8f6 \uf8f8 , \uf8eb \uf8ed 1/2 0 1/2 \uf8f6 \uf8f8 , \uf8eb \uf8ed 0 1 1 \uf8f6 \uf8f8 , \uf8eb \uf8ed −1/2 0 −1/2 \uf8f6 \uf8f8 , \uf8eb \uf8ed 1/2 1 −1/2 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='10) 4 ANTUN MILAS AND LIUQUAN WANG Here the first and the third entries in each vector have been interchanged and we have reordered these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Zagier [25, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 50] conjectured that there are some duality between modular triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Namely, he mentioned that when (A, B, C) is a rank r modular triple, then it is likely that (A⋆, B⋆, C⋆) = (A−1, A−1B, 1 2BTA−1B − r 24 − C) is also a rank r modular triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This motivates us to consider the dual cases to the six modular triples related to T −1 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' To be specific, the dual vectors are B⋆ ∈ \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8ed 0 0 0 \uf8f6 \uf8f8 , \uf8eb \uf8ed 0 0 1/2 \uf8f6 \uf8f8 , \uf8eb \uf8ed 1 −1 1/2 \uf8f6 \uf8f8 , \uf8eb \uf8ed −1 1 0 \uf8f6 \uf8f8 , \uf8eb \uf8ed −1 1 −1/2 \uf8f6 \uf8f8 , \uf8eb \uf8ed −2 2 −1/2 \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We find that they are indeed modular triples for suitable C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This is a consequence of the following set of Rogers-Ramanujan type identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Before we state them, we introduce the compact notations Jm := (qm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' qm)∞, Ja,m := (qa, qm−a, qm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' qm)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We have � i,j,k≥0 q2i2+2j2+k2−2ij−2jk (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)k = J5 4J40 J1J2 2J2 8J8,40 + 2qJ2 8J6,20J8,40 J1J2 4J40 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11) � i,j,k≥0 qi2+j2+ 1 2(k2+k)−ij−jk (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)i(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)j(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)k = J6 2J1,10J8,20 J5 1J2 4J20 + 2qJ2 4J4,10J2,20 J3 1J20 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12) � i,j,k≥0 qi2+j2+ 1 2 (k2+k)−ij−jk+i−j (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)i(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)j(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)k = 2J2J2 4J20 J3 1J4,20 + J6 2J3,10J4,20 J5 1J2 4J20 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13) � i,j,k≥0 q2i2+2j2+k2−2ij−2jk−2i+2j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)k = 2J2 8J2,20J16,40 J1J2 4J40 + qJ4 4J8,20J4,40 J1J2 2J2 8J40 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14) � i,j,k≥0 qi2+j2+ 1 2 (k2−k)−ij−jk−i+j (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)i(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)j(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)k = 4J2J2 4J20 J3 1J4,20 + 2J6 2J3,10J4,20 J5 1J2 4J20 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15) � i,j,k≥0 qi2+j2+ 1 2(k2−k)−ij−jk−2i+2j (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)i(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)j(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)k = 2q−1J6 2J1,10J8,20 J5 1J2 4J20 + 4J2 4J4,10J2,20 J3 1J20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16) Note that the left sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16) are the Nahm sums F1(q2), F2(q), F3(q), F4(q2), χ0(q−1, q, q− 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q) and χ0(q−2, q2, q− 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In light of the modularity of the functions Jm and Ja,m, it is easy to verify (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', using the algorithm in [8]) that the Nahm sums in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16) are all modular functions after multiplying a factor qC with C being − 7 40, 1 40, 9 40, 17 40, 9 40, 41 40, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In particular, we see that the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11) confirms Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1 in the case r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Interestingly, there are essentially four different modular functions MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 5 for these six Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In fact, comparing the right sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16), we see that χ0(q−1, q, q− 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q) = 2χ0(q, q−1, q 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='17) χ0(q−2, q2, q− 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q) = 2q−1χ0(1, 1, q 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='18) This is not obvious from their original definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We will explain this in the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Next, we will study the tadpole Nahm sums from the point of vector-valued modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' By doing so, we will be able to see their modular transformation properties more clearly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let H denote the upper half complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Throughout this paper we denote q = e2πiτ, τ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Since the shifted characters χ0(1, 1, 1) and χ0(q−1, q, 1) have q-powers in Z + 1 2 it is convenient to consider F5(q) := χ0(1, 1, 1)|τ→τ+1 = � i,j,k≥0 (−1)k qi2+j2+ 1 2k2−ij−jk (q)i(q)j(q)k , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='19) F6(q) := χ0(q−1, q, 1)|τ→τ+1 = � i,j,k≥0 (−1)k qi2+j2+ 1 2k2−ij−jk−i+j (q)i(q)j(q)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='20) For 1 ≤ i ≤ 6 we define ˜Fi(q) := qλiFi(q) where λ1 = − 7 80, λ2 = 1 40, λ3 = 9 40, λ4 = 17 80, λ5 = − 7 80, λ6 = 17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We define Weber’s modular functions [24]: f(τ) := q−1/48(−q1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞, f1(τ) := q−1/48(q1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞, f2(τ) := q1/24(−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='21) For k > 0 and j ∈ Q we let (∂Θ)j,k(τ) := � n∈Z (2kn + j)q(2kn+j)2/(4k), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) (∂G)j,k(τ) := � n∈Z (−1)n(2kn + j)q(2kn+j)2/(4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='23) When k, j ∈ Q, these are essentially Jacobi theta series of weight 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We have the following q-identities: ˜F1(q) = f(τ)3 η(τ)3(∂Θ)1, 5 2(τ), ˜F2(q) = 2f2(τ)3 η(τ)3 (∂Θ) 1 2, 5 2(τ), ˜F3(q) = 2f2(τ)3 η(τ)3 (∂Θ) 3 2, 5 2(τ), ˜F4(q) = f(τ)3 η(τ)3(∂Θ)2, 5 2(τ), 6 ANTUN MILAS AND LIUQUAN WANG ˜F5(q) = f1(τ)3 η(τ)3 (∂G)1, 5 2(τ), ˜F6(q) = f1(τ)3 η(τ)3 (∂G)2, 5 2(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In particular, ˜Fi(q) are modular functions on some congruence subgroups of SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Moreover, ( ˜Fi(q))1≤i≤6 transforms as a vector valued modular function on SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This in particular proves and extends Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1 for r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In Section 2 we present a proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The idea is to use constant term method to reduce triple sums to some single sums and use Rogers’ identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In Section 3 we give two different proofs for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Finally, in Section 4 we prove that some other Nahm sums associated with the tadpole Cartan matrix are not modular, and we give a general conjecure on modular Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It will be useful to observe another basis of V = Span{ ˜Fi(q) : 1 ≤ 1 ≤ 6}: g1(q) := ˜F1(q) + ˜F5(q) = q−7/80(2 + 12q + 30q2 + · · · ) ∈ q−7/80C[[q]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' g2(q) := ˜F1(q) − ˜F5(q) = q33/80(6 + 18q + 54q2 + · · · ) ∈ q33/80C[[q]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' g3(q) := ˜F4(q) + ˜F6(q) = q17/80(4 + 6q + 30q2 + · · · ) ∈ q17/80C[[q]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' g4(q) := ˜F4(q) − ˜F6(q) = q57/80(6 + 16q + 42q2 + · · · ) ∈ q57/80C[[q]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' g5(q) := ˜F2(q) = q1/40(1 + 6q + 15q2 + · · · ) ∈ q1/40C[[q]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' g6(q) := ˜F3(q) = q9/40(3 + 11q + 30q2 + · · · ) ∈ q9/40C[[q]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Observe that now the leading q-series powers are non-congruent modulo Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2 Recall the q-binomial theorem [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1]: ∞ � n=0 (a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n zn = (az;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ , |z| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1) As important corollaries of this theorem, Euler’s q-exponential identities state that [4, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2] ∞ � n=0 zn (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n = 1 (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ , ∞ � n=0 q(n 2)zn (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)n = (−z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞, |z| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) The Jacobi triple product identity [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3] is (q, z, q/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)∞ = ∞ � n=−∞ (−1)nq(n 2)zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3) It gives product representations for two important unary Jacobi theta functions: θ2(τ) := � n∈Z q(n+1/2)2 = 2q1/4J2 4 J2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4) MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 7 θ3(τ) := � n∈Z qn2 = J5 2 J2 1J2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5) We will use the constant term method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For any series f(z) = � n∈Z a(n)zn, we define the operator CT[f(z)] = a(0), which extracts the constant term of f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Obviously, for any complex number α and integer β with αβ ̸= 0, we have CT[f(αzβ)] = CT[f(z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' After replacing q by q2 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='9), we have χ0(x1, x2, x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,j,k≥0 q2i2+2j2+k2−2ij−2jkxi 1xj 2xk 3 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='7) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3) that χ0(x1, x2, x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,j≥0 q2i2+2j2−2ijxi 1xj 2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j � k≥0 qk2−k · q(1−2j)kxk 3 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)k = � i,j≥0 q2i2+2j2−2ijxi 1xj 2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−x3q1−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) (1) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) that χ0(1, 1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i≥0 q2i2+2j2−2ij (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q1−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � i,j≥0 q2i2+j2−2ij(−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �� i≥0 qi2zi (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i � j≥0 (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)jz−j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j ∞ � k=−∞ z−kqk2 � = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−qz, −q/z, −qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (1/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT �(−qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2 ∞ (1/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT � ∞ � n=0 z−n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ ziqi2 ∞ � j=−∞ zjqj2 � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 1 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n � i+j=n qi2+j2 = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 qn2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ q2i2−2ni 8 ANTUN MILAS AND LIUQUAN WANG = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (S0(q) + S1(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='9) Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We have S0(q) = ∞ � n=0 q2n2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2(i−n)2 = ∞ � n=0 q2n2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2 = J6 4J40 J3 2J2 8J8,40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='10) Similarly, we have S1(q) = ∞ � n=0 q4n2+4n+1 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2−4in−2i = ∞ � n=0 q2n2+2n+1 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2(i−n)2−2(i−n) = ∞ � n=0 q2n2+2n+1 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2−2i = 2qJ2 8J6,20J8,40 J2J4J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='9), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) that χ0(1, 1, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,j≥0 q2i2+2j2−2ij (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q2−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � i,j≥0 q2i2+j2+j−2ij (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �� i≥0 qi2zi (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i � j≥0 qjz−j(−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j ∞ � k=−∞ z−kqk2 � = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−qz, −q/z, −q/z, −qz, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12) = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT � ∞ � n=0 qn (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ ziqi2 ∞ � j=−∞ zjqj2 � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 qn (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n � i+j=n qi2+j2 = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 qn2+n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ q2i2−2ni = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (S0(q) + S1(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13) Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 9 We have S0(q) = ∞ � n=0 q4n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2−4ni = ∞ � n=0 q2n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2(i−n)2 = ∞ � n=0 q2n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2 = J5 4J2,20J16,40 J3 2J2 8J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14) Similarly, S1(q) = ∞ � n=0 q4n2+6n+2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2−4ni−2i = ∞ � n=0 q2n2+4n+2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2(n−i)2+2(n−i) = ∞ � n=0 q2n2+4n+2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2+2i = 2q2J2 8J8,20J4,40 J2J4J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) that χ0(q2, q−2, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,j≥0 q2i2+2j2−2ij+2i−2j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q2−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � i,j≥0 q2i2+j2−j−2ij+2i (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �� i≥0 qi2+2izi (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i � j≥0 q−jz−j(−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j ∞ � k=−∞ z−kqk2 � = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−q3z, −1/(qz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞(−qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (1/(qz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16) = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT �(−q3z, −1/(qz), q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞(−qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (1/(qz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT � ∞ � n=0 q−nz−n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ qi2+2izi ∞ � j=−∞ qj2zj � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 q−n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n � i+j=n qi2+2i+j2 = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 qn2−n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ q2i2−2ni+2i = (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (S0(q) + S1(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='17) Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 10 ANTUN MILAS AND LIUQUAN WANG We have S0(q) = ∞ � n=0 q4n2−2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2−4ni+2i = ∞ � n=0 q2n2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2(i−n)2+2(i−n) = ∞ � n=0 q2n2 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2+2i = 2 J2 8J40 J2J8,40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='18) Similarly, S1(q) = ∞ � n=0 q4n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2−4in = ∞ � n=0 q2n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2(i−n)2 = ∞ � n=0 q2n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2 = J5 4J6,20J8,40 J3 2J2 8J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='19) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='19) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='17), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (4) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) that χ0(q−2, q2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,j≥0 q2i2+2j2−2ij−2i+2j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q1−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � i,j≥0 q2i2+j2−2ij−2i+2j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �� i≥0 qi2−2izi (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i � j≥0 q2jz−j(−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j ∞ � k=−∞ z−kqk2 � = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−z/q, −q3/z, −qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT �(−z/q, −q3/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞(−qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ CT � ∞ � n=0 q2nz−n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ ziqi2−2i ∞ � j=−∞ zjqj2 � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 q2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n � i+j=n qi2+j2−2i = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ ∞ � n=0 qn2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)n ∞ � i=−∞ q2i2−2ni−2i = (−q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (S0(q) + S1(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='20) Here S0(q) and S1(q) correspond to the sums with n being even and odd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We have S0(q) = ∞ � n=0 q4n2+4n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2−4in−2i = ∞ � n=0 q2n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2(n−i)2+2(n−i) MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 11 = ∞ � n=0 q2n2+2n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n ∞ � i=−∞ q2i2+2i = 2J2 8J2,20J16,40 J2J4J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='21) Similarly, S1(q) = ∞ � n=0 q4n2+8n+3 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2i2−4ni−4i = ∞ � n=0 q2n2+4n+3 (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2(n−i)2+4(n−i) = q ∞ � n=0 q2n2+4n (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)2n+1 ∞ � i=−∞ q2(i+1)2 = qJ5 4J8,20J4,40 J3 2J2 8J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='20), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (5) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) that χ0(q−2, q2, q−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,,j≥0 q2i2+2j2−2ij−2i+2j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ = (−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � i,j≥0 q2i2+j2+j−2ij−2i (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j = 2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �� i≥0 qi2−2izi (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i � j≥0 (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)jqjz−j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j ∞ � k=−∞ z−kqk2 � = 2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−q3/z, −z/q, −qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='23) Now if we replace z by q2z in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='23), which does not change the result by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6), and then compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16), after replacing q2 by q, we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (6) We have by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) that χ0(q−4, q4, q−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2) = � i,j≥0 q2i2+2j2−2ij−4i+4j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q−2j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ = (−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � i,j≥0 q2i2+j2−2ij−4i+3j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i(q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j = 2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �� i≥0 qi2−4izi (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)i � j≥0 q3jz−j(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j (q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)j ∞ � k=−∞ z−kqk2 � = 2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−q−3z, −q5/z, −qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q3/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)) = 2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−q−1z, −q3/z, −q3z, −1/(qz), q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � (replace z by q2z) = 2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(1 + q−1z)(1 + q−1z−1) (1 + qz)(1 + qz−1) (−qz, −q/z, −qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � 12 ANTUN MILAS AND LIUQUAN WANG = 2q−2(−q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞CT �(−qz, −q/z, −qz, −q/z, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ (q/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q2)∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='24) Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12) differ only by the factor 2q−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' After replacing q2 by q, this proves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3 In this section, we provide two different proofs for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the first proof, we treat the functions ˜Fi(τ) (1 ≤ i ≤ 6) together by viewing them as vector-valued modular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the second proof, we treat these identities one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Theta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In this subsection, we shall make some preparations for the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recall the full modular group SL(2, Z) = �� a b c d � : a, b, c, d ∈ Z, ad − bc = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This group is also conveniently denoted as Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It is generated by the matrices S = � 0 −1 1 0 � , T = � 1 1 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For any congruence subgroup G of SL(2, Z) and Dirichlet character χ, we use Mk(G, χ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Sk(G, χ)) to denote the space of modular forms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' cusp forms) on G with weight k and multiplier χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' When χ is trivial, we omit it and write the space as Mk(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Sk(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Besides Γ(1) itself, we will mainly work with the congruence subgroups Γ1(N) := �� a b c d � ∈ SL(2, Z), � a b c d � ≡ � 1 ∗ 0 1 � (mod N) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1) Γ0(N) := �� a b c d � ∈ SL(2, Z), � a b c d � ≡ � ∗ ∗ 0 ∗ � (mod N) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) It is well-known that the Weber modular functions f(τ), f1(τ) and f2(τ) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='21) transform as a vector-valued modular form of rank three under Γ(1): f(−1/τ) = f(τ), f2(−1/τ) = 1 √ 2 f1(τ), f1(−1/τ) = √ 2f2(τ) f(τ + 1) = e−πi/24f1(τ), f1(τ + 1) = e−πi/24f(τ), f2(τ + 1) = eπi/12f2(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We also require the Dedekind η-functions η(τ) and its transformation properties: η(−1/τ) = √ −iτη(τ), η(τ + 1) = eπi/12η(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recall the series (∂Θ)j,k(τ) and (∂G)j,k(τ) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The follow- ing lemma summarizes some useful properties and especially some transformation formulas for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1) For any k > 0 and j we have (∂Θ)0,k(τ) = 0, (∂Θ)k,k(τ) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3) (∂Θ)j, k 2 (τ) = −(∂Θ)k−j, k 2 (τ), (∂G)j, k 2 (τ) = (∂G)k−j, k 2 (τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4) MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 13 (∂Θ)j,k(τ) = 1 2(∂Θ)2j,4k(τ) + 1 2(∂Θ)2j+4k,4k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5) (∂G)j,k(τ) = 1 2(∂Θ)2j,4k(τ) − 1 2(∂Θ)2j+4k,4k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) (2) For k ∈ N + 1 2 and j ∈ N we have (∂Θ)j,k(τ + 1) = eiπj2/2k(∂G)j,k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='7) (∂G)j,k(τ + 1) = eiπj2/2k(∂Θ)j,k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) (∂Θ)j,k(τ + 2) = eiπj2/k(∂Θ)j,k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='9) (∂G)j,k(τ + 2) = eiπj2/k(∂G)j,k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='10) For k ∈ N + 1 2 and j ∈ N + 1 2 we have (∂Θ)j,k(τ + 1) = eiπj2/2k(∂Θ)j,k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11) (∂G)j,k(τ + 1) = eiπj2/2k(∂G)j,k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12) (3) For k ∈ 1 2N and j ∈ N we have (∂Θ)j,k(−1/τ) = (−τ) � −iτ/2k 2k−1 � j′=1 eiπjj′/k(∂Θ)j′,k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13) (∂G)j,k(−1/τ) = (−τ) � −iτ/2k 2k � j′=1 eiπj(2j′−1)/k(∂Θ) 2j′−1 2 ,k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14) (4) For k ∈ 1 2N and j ∈ N + 1 2 we have (∂Θ)j,k(−1/τ) = (−τ) � −iτ/2k 2k−1 � j′=1 eiπjj′/k(∂G)j′,k(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15) (∂G)j,k(−1/τ) = (−τ) � −iτ/2k 2k � j′=1 eiπj(2j′−1)/k(∂G) 2j′−1 2 ,k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Replacing n by −n in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22), we deduce that (∂Θ)0,k(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Replacing n by −n − 1, we obtain (∂Θ)k,k(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the same way we can prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Splitting the sum according to the parity of n, we have (∂Θ)j,k(τ) = � n∈Z (4kn + j)q(4kn+j)2/4k + � n∈Z (4kn + 2k + j)q(4kn+2k+j)2/(4k) = 1 2 � n∈Z (8kn + 2j)q(8kn+2j)2/(16k) + 1 2 � n∈Z (8kn + 4k + 2j)q(8kn+4k+2j)2/(16k) = 1 2(∂Θ)2j,4k(τ) + 1 2(∂Θ)2j+4k,4k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Similarly, we can prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) and hence finish the proof of part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Part (2) follows from definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 14 ANTUN MILAS AND LIUQUAN WANG It remains to prove parts (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' First, the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13) follows from [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4)] and the fact (∂Θ)0,k(τ) = 0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It also appears as [1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Now assume that k ∈ 1 2N and j ∈ N + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Replacing τ by −1/τ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' we deduce that (∂Θ)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='k(−1/τ) = 1 2(−τ) � −iτ 8k 8k−1 � j′=1 eiπjj′/2k(∂Θ)j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) + 1 2(−τ) � −iτ 8k 8k−1 � j′=1 eiπ(2k+j)j′/2k(∂Θ)j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) = −1 4τ � −iτ 2k 8k−1 � j′=1 (1 + eiπj′)eiπjj′/2k(∂Θ)j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) = −1 2τ � −iτ 2k 4k−1 � j′=1 eiπjj′/k(∂Θ)2j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) = −1 2τ � −iτ 2k 2k−1 � j′=1 � eiπjj′/k(∂Θ)2j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) + eiπj(j′+2k)/k)(∂Θ)2j′+4k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) � = −1 2τ � −iτ 2k 2k−1 � j′=1 eiπjj′/k� (∂Θ)2j′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) − (∂Θ)2j′+4k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4k(τ) � = −τ � −iτ 2k 2k−1 � j′=1 eiπjj′/k(∂G)j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='k(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Here for the last third line we used the fact that (∂Θ)4k,4k(τ) = 0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For k ∈ 1 2N and j ∈ 1 2N, replacing τ by −1/τ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13), we deduce that (∂G)j,k(−1/τ) = 1 2(−τ) � −iτ 8k 8k−1 � j′=1 (eiπjj′/2k − eiπ(j+2k)j′/2k)(∂Θ)j′,4k(τ) = (−τ) � −iτ 8k 4k � j′=1 eiπj(2j′−1)/2k(∂Θ)2j′−1,4k(τ) = (−τ) � −iτ 8k 2k � j′=1 � eiπj(2j′−1)/2k(∂Θ)2j′−1,4k(τ) + eiπj(2j′+4k−1)/2k(∂Θ)2j′+4k−1,4k(τ) � = (−τ) � −iτ 8k 2k � j′=1 eiπj(2j′−1)/2k� (∂Θ)2j′−1,4k(τ) + e2iπj(∂Θ)2j′+4k−1,4k(τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Discussing according to j ∈ N or j ∈ N + 1 2 and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' □ Combining these formulas give MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 15 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then the following functions: f(τ)3 η(τ)3(∂Θ)i, 2k+1 2 (τ), 1 ≤ i ≤ k, f1(τ)3 η(τ)3 (∂G)i, 2k+1 2 (τ), 1 ≤ i ≤ k, f2(τ)3 η(τ)3 (∂Θ) 2i−1 2 , 2k+1 2 (τ), 1 ≤ i ≤ k, combine into a vector-valued modular function under Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' First notice that all components are modular functions on an appropriate congruence subgroup (see [5] for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Thus it suffices to prove that the span of the functions (which are linearly independent) closes under τ → τ + 1 and τ → − 1 τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For τ → τ + 1 this is clear and follows immediately from the formulas above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Under τ → −1/τ we see using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4) that the span of f3(τ) η3(τ)(∂Θ)i, 2k+1 2 (τ) transforms to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Similarly, formulas f2(−1/τ) = 1 √ 2f1(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='15) show that the span of f3 2(τ) η3(τ)(∂Θ) 2i−1 2 , 2k+1 2 (τ) transforms under τ → −1/τ to the span of f3 1(τ) η3(τ)(∂G)i, 2k+1 2 (τ) and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We proved the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' □ As a corollary we record Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1) The vector-valued function � f(τ)3 η(τ)3(∂Θ)1, 5 2(τ), f(τ)3 η(τ)3(∂Θ)2, 5 2(τ) � transforms as a vector-valued modular form (of weight zero) for Γθ = ⟨S, T 2⟩, the sugbroup of Γ(1) generated by S and T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2) The vector-valued function � f(τ)3 η(τ)3(∂Θ)1, 5 2(τ), f(τ)3 η(τ)3(∂Θ)2, 5 2(τ), f1(τ)3 η(τ)3 (∂G)1, 5 2(τ), f1(τ)3 η(τ)3 (∂G)2, 5 2(τ), f2(τ)3 η(τ)3 (∂Θ) 1 2, 5 2(τ), f2(τ)3 η(τ)3 (∂Θ) 3 2, 5 2(τ) � transforms as a vector-valued modular function for Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Part (2) of this corollary proves the second assertion in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Therefore, to finish the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3, it suffices to prove the six identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Below we present two different proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' First proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We first recall several known facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recall the theta functions θ2(τ) and θ3(τ) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The pair (W1(τ), W2(τ)) := �θ3(τ) η(τ) , θ2(τ) η(τ) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='17) 16 ANTUN MILAS AND LIUQUAN WANG defines a two-dimensional vector-valued modular form ρ1 of weight zero whose S- matrix is given by � W1(−1/τ) W2(−1/τ) � = 1 √ 2 � 1 1 1 −1 � � W1(τ) W2(τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' These two series are precisely the level one characters of standard A(1) 1 -modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let also chr,s 3,5(τ) := 1 η(τ) � n∈Z � q (30n+5r−3s)2 60 − q (30n+5r+3s)2 60 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This notation indicates that ch1,1 3,5(τ), ch1,2 3,5(τ), ch2,1 3,5(τ), ch2,2 2,5(τ) are precisely char- acters of four irreducible (3, 5) Virasoro minimal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We need the following known identities obtained using the quintuple product identity [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='17] that connect Rogers’ series (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='8) with these characters (see also [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 170]): Z1(τ) := q1/40 ∞ � n=0 qn2+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n = ch1,1 3,5(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='18) Z2(τ) := q31/40 ∞ � n=0 qn2+2n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n+1 = ch2,1 3,5(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='19) Z3(τ) := q9/40 ∞ � n=0 qn2+n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n+1 = ch2,2 3,5(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='20) Z4(τ) := q−1/40 ∞ � n=0 qn2 (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' q)2n = ch1,2 3,5(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='21) These expressions transforms under SL(2, Z), and they define a 4-dimensional vector-valued modular form ρ2 of weight zero with a well-known S-matrix: \uf8eb \uf8ec \uf8ec \uf8ed Z1(−1/τ) Z2(−1/τ) Z3(−1/τ) Z4(−1/τ) \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � 2 5 \uf8eb \uf8ec \uf8ec \uf8ed sin(2π/5) − sin(2π/5) − sin(π/5) sin(π/5) − sin(2π/5) − sin(2π/5) sin(π/5) sin(π/5) − sin(π/5) sin(π/5) − sin(2π/5) sin(2π/5) sin(π/5) sin(π/5) sin(2π/5) sin(2π/5) \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed Z1(τ) Z2(τ) Z3(τ) Z4(τ) \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' See for instance [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 172, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (15)] except for a typo in the (3, 3)-entry of this matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' One can easily write the diagonal T-matrices for ρ1 and ρ2 using the leading terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Observe that ρ1⊗ρ2, that is (WiZj)1≤i≤2,1≤j≤4, defines a 8-dimensional vector- valued modular form on Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' However, we also have a proper subspace in ρ1 ⊗ ρ2 that also closes under the full modular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let Wi and Zi be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We have � ˜F1(q), ˜F2(q), ˜F3(q), ˜F4(q), ˜F5(q), ˜F6(q) � = (f(τ)(W1Z4 + W2Z3), f2(τ)(W1Z1 + W2Z2), f2(τ)(W1Z3 + W2Z4), f(τ)(W1Z2 + W2Z1), f1(τ)(W1Z4 − W2Z3), f1(τ)(W2Z1 − W1Z2)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) and it defines a vector-valued modular form ˜ρ1 of weight zero on Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 17 In fact, the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) follows directly from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' One can also see this in a more straightforward way from the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For instance, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11) (after replacing q2 by q) we immediately conclude that ˜F1(q) = f(τ)(W1Z4 + W2Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='23) The second assertion of this proposition follows by direct computations using the S and T-matrices of W1, W2 and Zi (1 ≤ i ≤ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Denote by ˜ρ2 the vector-valued modular form of weight zero: (h1(q), h2(q), h3(q), h4(q), h5(q), h6(q)) := � f(τ)3 η(τ)3(∂Θ)1, 5 2(τ), 2f2(τ)3 η(τ)3 (∂Θ) 1 2, 5 2(τ), 2f2(τ)3 η(τ)3 (∂Θ) 3 2 , 5 2(τ), f(τ)3 η(τ)3(∂Θ)2, 5 2(τ), f1(τ)3 η(τ)3 (∂G)1, 5 2(τ), f1(τ)3 η(τ)3 (∂G)2, 5 2(τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='24) This is precisely the vector-valued modular form appearing in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3(2), except that we added the factor 2 for the second and third entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' So far we constructed two 6-dimensional vector valued modular forms: ˜ρ1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='22) and ˜ρ2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It is easy to see that (by direct computations) the S and T matrices of them agree with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We claim that ˜ρ1 = ˜ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' To prove the above claim, we switch to new bases of vector-valued modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recall the basis gi(q) from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Similarly we let ˜g1(q) := h1(q) + h5(q) = q−7/80(2 + 12q + 30q2 + · · · ) ∈ q−7/80C[[q]], ˜g2(q) := h1(q) − h5(q) = q33/80(6 + 18q + 54q2 + · · · ) ∈ q33/80C[[q]], ˜g3(q) := h4(q) + h6(q) = q17/80(4 + 6q + 30q2 + · · · ) ∈ q17/80C[[q]], ˜g4(q) := h4(q) − h6(q) = q57/80(6 + 16q + 42q2 + · · · ) ∈ q57/80C[[q]], ˜g5(q) := h2(q) = q1/40(1 + 6q + 15q2 + · · · ) ∈ q1/40C[[q]], ˜g6(q) := h3(q) = q9/40(3 + 11q + 30q2 + · · · ) ∈ q9/40C[[q]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then (gi(q) − ˜gi(q))6 i=1 also transforms as a vector-valued modular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Using their q-expansions it is easy to see (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Remark 3) that the Wronskian of gi(h)−˜gi(q), 1 ≤ i ≤ 6 has the order of vanishing that is strictly bigger than 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' But then, according to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3, the Wronskian is identically zero and thus (gi(q) − ˜gi(q)) are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The last sentence in Remark 1 implies that the linear dependence is equivalent to gj(q) = ˜gj(q) for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Thus we obtain a 5-dimensional vector-valued modular form (gi(q) − ˜gi(q))i̸=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Applying the same type of argument to it yields gk(q) = ˜gk(q) for k ̸= j, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Thus we conclude gi(q) = ˜gi(q) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' That clearly implies ˜Fi(q) = hi(q) and completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The claim ˜ρ1(τ) = ˜ρ2(τ) can also be proved by a Sturm-type criterion of vector-valued modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' From [12, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1] we know that it suffices to compare the first two terms in their q-expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Interestingly, in the above proof using Wronskian, we also only need to compare the first two terms of gi(q) and ˜gi(q), which guarantees that the Wronskian of gi(q) − ˜gi(q), 1 ≤ i ≤ 6 has the order of vanishing > 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 18 ANTUN MILAS AND LIUQUAN WANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Second proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We will need the order of a function f(τ) with respect to a congruence subgroup G at the cusp p ∈ Q ∪ {∞} and denote it as ord(f, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recall the theta function θ3(τ) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It is known that [16, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='41] θ3(τ) ∈ M 1 2(Γ0(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For any odd primitive Dirichlet character ψ with conductor N, it is known that [16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='44] θ(ψ, τ) := ∞ � n=1 ψ(n)nqn2 ∈ S 3 2(Γ0(4N2, ψχ−4), where χ−4 is the nontrivial Dirichlet character modulo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For a ∈ {1, 2, 3, 4}, we have � n∈Z n≡a (mod 5) nqn2 ∈ M 3 2(Γ1(100)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let ψk (k = 0, 1) be the primitive Dirichlet character with conductor 5 sat- isfying ψk(2) = (−1)ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then θ(ψ0, τ) = � n∈Z ψ0(n)nqn2 = � n∈Z (5n + 1)q(5n+1)2 − � n∈Z (5n + 4)q(5n+4)2 + i �� n∈Z (5n + 2)q(5n+2)2 − � n∈Z (5n + 3)q(5n+3)2 � = 2 � n∈Z (5n + 1)q(5n+1)2 + 2i � n∈Z (5n + 2)q(5n+2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='26) Here for the last equality we used the fact that for a ∈ {1, 2}, � n∈Z (5n + a)q(5n+a)2 = − � n∈Z (5n + 5 − a)q(5n+5−a)2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='27) which can be proved easily by changing n to −n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the same way, we have θ(ψ1, τ) = 2 � n∈Z (5n + 1)q(5n+1)2 − 2i � n∈Z (5n + 2)q(5n+2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='28) Since θ(ψk, τ) ∈ M 3 2(Γ1(100)) (k = 0, 1), from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='28) we deduce that � n∈Z (5n + 1)q(5n+1)2 = 1 4(θ(ψ0, τ) + θ(ψ1, τ)) ∈ M 3 2(Γ1(100)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='29) � n∈Z (5n + 2)q(5n+2)2 = 1 4i(θ(ψ0, τ) − θ(ψ1, τ)) ∈ M 3 2(Γ1(100)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='30) This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='27) proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' □ We also need the following identity J2 1 = J2J5 8 J2 4J2 16 − 2qJ2J2 16 J8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='31) MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 19 This appeared frequently in the literature and is a direct consequence of [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 40, Entry 25(v),(vi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Following the notion in [8], we define the generalized Dedekine eta function ηδ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='g(τ) := q 1 2 δP2(g/δ) � m≡±g (mod δ) (1 − qm), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='32) where P2(t) = {t}2 − {t} + 1 6 is the second periodic Bernoulli polynomial, {t} is the fractional part of t, g, δ, m ∈ Z+ and 0 < g < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Now we are ready to give our second proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' By the definition given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='19) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='20), we know that the identities for �F5(q) and �F6(q) follow from those of �F1(q) and �F4(q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Thus, it suffices to prove the first four identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14), the first four identities are equivalent to T1(q) := � n∈Z (5n + 1)q5n2+2n = J2 1J8 4J40 J5 2J2 8J8,40 + 2qJ2 1J4J2 8J6,20J8,40 J3 2J40 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33) T2(q) := � n∈Z (5n + 2)q5n2+4n = 2J2 1J4J2 8J2,20J16,40 J3 2J40 + qJ2 1J7 4J8,20J4,40 J5 2J2 8J40 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='34) T3(q) := � n∈Z (10n + 1)q5n2+n = J2J3 4J2,20J16,40 J2 8J40 + 2q2J3 2J2 8J8,20J4,40 J3 4J40 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='35) T4(q) := � n∈Z (10n + 3)q5n2+3n = 2J3 2J2 8J40 J2 4J8,40 + J2J3 4J6,20J8,40 J2 8J40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='36) We will prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33) first and then deduce the other identities from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' After replac- ing q by q5 and multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33) by qθ3(5τ), we see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33) is equivalent to θ3(5τ) � n∈Z (5n + 1)q(5n+1)2 = θ4 3(5τ) (f1(τ) + f2(τ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='37) Here f1(τ) := q J8 5J14 20J200 J20 10J2 40J40,200 = η8(5τ)η14(20τ) η20(10τ)η2(40τ)η200,40(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='38) f2(τ) := 2q6J8 5J7 20J2 40J30,100J40,200 J18 10J200 = η8(5τ)η7(20τ)η2(40τ)η(100τ)η100,30(τ)η200,40(τ) η18(10τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='39) With the help of Maple and the algorithm in [8], it is easy to check that both f1(τ) and f2(τ) are modular functions on Γ1(200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Their poles and corresponding orders of them are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' On the other hand, it is easy to see that θ4 3(5τ) ∈ M2(Γ0(20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Hence θ4 3(5τ) ∈ M2(Γ1(200)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We can evaluate the orders of zeros of θ4 3(5τ) at any cusp for Γ1(200) (using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3 and the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='12) in [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In Table 1 we only list the orders of zeros of θ4 3(5τ) at the poles of fi(τ) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It turns out that after multiplying 20 ANTUN MILAS AND LIUQUAN WANG cusp p ord(fi(τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' p) ord(θ4 3(5τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' p) 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 42,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 54,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 58,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 74,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 78,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 82,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 86,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 94,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 98 −16 20 1 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 1 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 3 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 3 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 7 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 7 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 7 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 9 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 9 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 23 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 23 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 29 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 29 90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 67 70 −80 100 1 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 9 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 11 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 19 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 21 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 29 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 31 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 39 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 41 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 49 50 −16 20 3 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 7 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 13 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 17 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 23 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 27 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 33 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 37 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 43 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 47 50 −14 20 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Orders of poles and zeros at cusps for Γ1(200) by θ4 3(5τ), all the poles of fi(τ) will be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Hence θ4 3(5τ)fi(τ) ∈ M2(Γ1(200)) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' So far we have proved that both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='37) belong to M2(Γ1(200)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' By Sturm’s criterion (see [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 185, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='14], for example), to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='37), it suffices to verify that both sides agree for the first 1 + 2 12 · 1 2[SL(2, Z) : Γ1(200)] = 2401 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We have checked this with Maple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='37) holds and we finish the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Next, we are going to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='34)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='36) based on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We aim to find a 2-dissection formula for T1(q): T1(q) = � n∈Z (5n + 1)q5n2+2n = L0(q2) + qL1(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='40) On the one hand, since 5n2 + 2n has the same parity with n, we have L0(q2) = � n even (5n + 1)q5n2+2n = � n∈Z (10n + 1)q20n2+4n = T3(q4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='41) qL1(q2) = � n odd (5n + 1)q5n2+2n = � n∈Z (5(−2n − 1) + 1)q5(−2n−1)2+2(−2n−1) = −2q3 � n∈Z (5n + 2)q20n2+16n = −2q3T2(q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='42) On the other hand, substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='31) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='33), we obtain � n∈Z (5n + 1)q5n2+2n = � J2J5 8 J2 4J2 16 − 2qJ2J2 16 J8 � � J8 4J40 J5 2J2 8J8,40 + 2qJ4J2 8J6,20J8,40 J3 2J40 � = J6 4J3 8J40 J4 2J2 16J8,40 − 4q2J4J8J2 16J6,20J2,40 J2 2J40 + 2q �J7 8J6,20J8,40 J2 2J4J2 16J40 − J8 4J2 16J40 J4 2J3 8J8,40 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='43) MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 21 Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='40)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='43), we deduce that L0(q) = T3(q2) = J6 2J3 4J20 J4 1J2 8J4,20 − 4qJ2J4J2 8J3,10J4,20 J2 1J20 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='44) L1(q) = −2qT2(q2) = J7 4J3,10J4,20 J2 1J2J2 8J20 − J8 2J2 8J20 J4 1J3 4J4,20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='45) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='44) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='45) and the method in [8], it is easy to verify that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='35) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It remains to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' For this we make a 2-dissection for T2(q): T2(q) = � n∈Z (5n + 2)q5n2+4n = H0(q2) + qH1(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='46) In the same way as we did for T1(q), we can prove that H0(q) = 2T1(q2) = 2J7 4J1,10J8,20 J2 1J2J2 8J20 − 2qJ7 2J2 8J4,10J2,20 J4 1J3 4J20 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='47) H1(q) = −T4(q2) = J5 2J3 4J4,10J2,20 J4 1J2 8J20 − 4J2J4J2 8J1,10J8,20 J2 1J20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='48) Now from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='48) and using the method in [8], it is easy to verify that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='36) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' General conjecture and concluding remarks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Non-modularity of some Nahm sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In [6, Section 5] in addition to χ0(1, 1, 1) and χ0(1, 1, q 1 2) two additional specializations are considered: χ0(q, 1, 1) and χ0(1, q, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Thus it seems natural to ask whether these two series are also modular after addition of a suitable multiplicative factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' We next show that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Recall that we denote by fA,B,C(q) the Nahm sum associated to T3 matrix with B = (B1, B2, B3) as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Using this notation, we can write fA,(1,0,0),C(q) = qCχ0(q, 1, 1) and fA,(0,1,1),C(q) = qCχ0(1, q, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Denote by Q1 = 1 2(3 − √ 5), Q2 = −2 + √ 5, Q3 = 1 4(3 − √ 5) the unique solution inside the interval (0, 1) of the TBA system [20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1]: 1 − Q1 = Q2 1Q−1 2 , 1 − Q2 = Q−1 1 Q2 2Q−1 3 , 1 − Q3 = Q3Q−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The Nahm sums fA,(1,0,0),C(q) and fA,(0,1,0),C(q) are not modular for any rational number C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' To prove this, it suffices to argue that the two Nahm sums do not have expected asymptotic expansion around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Letting q = e−ǫ, then according to [20, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1] we have the following asymptotic behavior (as ǫ → 0+): fA,B,C(e−ǫ)e− α ǫ ∼ βe−γǫ(1 + � p≥1 cpǫp) where α is a positive constant, γ = C + 1 24 �r i=1 1+Qi 1−Qi, cp are some hard-to-compute coefficients expressed using generalized 3-fold Gaussian integrals and β is a nonzero 22 ANTUN MILAS AND LIUQUAN WANG constant not needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' As a necessary condition for modularity we notice rela- tions [20, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1]: γp p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' − cp = 0, p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In our situation, the condition c1 − γ = 0 is equivalent to C = 9B1 2 4 √ 5 − 3B1 2 4 + 3B1B2 √ 5 − B1B2 − B1B3 2 √ 5 + B1B3 2 + 2B1 √ 5 − 9B1 10 + B2 2 √ 5 + B2B3 2 √ 5 + B2B3 2 + 7B2 4 √ 5 − 17B2 20 + B3 2 √ 5 + B3 2 4 − B3 2 √ 5 + B3 10 − 7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Plugging in B = (1, 0, 0) gives the value C = 1 80(−139 + 68 √ 5) which is irrational and similarly for B = (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Therefore the two Nahm sums in question cannot be modular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' □ Although χ0(q, 1, 1) and χ0(1, q, 1) cannot be made modular their sum satisfies χ0(q, 1, 1) + χ0(1, q, 1) = χ0(q−1, q, 1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1) which is modular after multiplying with q 17 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1) follows from a slightly more general statement: � i,j,k≥0 qi2+j2+k2/2−ij−jk+j−ixi 1xj 2xk 3 (q)i(q)j(q)k = � i,j,k≥0 qi2+j2+k2/2−ij−jkxi 1xj 2xk 3(x1qi + qj) (q)i(q)j(q)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) Comparing the coefficients of xi 1xj 2xk 3 of both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2), we see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2) is equivalent to qi2+j2+k2/2−ij−jk+j−i (q)i(q)j(q)k = qi2+j2+k2/2−ij−jk+j (q)i(q)j(q)k + q(i−1)2+j2+k2/2−(i−1)j−jk+(i−1) (q)i−1(q)j(q)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The last formula is trivial to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' General conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In this part we discuss the general conjecture on the modularity of the rank n Nahm sum χ0(1) associated to Tn as in the introduction, where for brevity we let 1 := (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' It is possible to formulate a slightly stronger conjecture result analogous to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3 but we omit discussing it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=',fℓ be any holomorphic functions in the upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Denote by D = � q d dq � = 1 2πi ∂ ∂τ Ramanujan’s derivative and by ∂k := D − k 12E2 where E2(q) = 1−24 � n≥1 nqn 1−qn is the second Eisenstein series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' This map is known to send the space of modular forms of weight k (on some congruence subgroup) into the space of modular forms of weight k+2 for the same congruence subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Denote by WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) the Wronskian determinant with respect to the D-derivation which is again a holomorphic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Additionally, denote by W∂k(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) the Wronskian with respect to the ∂k derivation, where the r-th derivative is defined as ∂r k := ∂k+2r−2 ◦ · · · ◦ ∂k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Suppose that each fi admits a q-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then WD also has a q-expansion so we can denote by � WD the Wronskian normalized such that the MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 23 leading coefficient in the q-expansion is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then we have a known result (see for instance [5,14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' ,fℓ be modular forms of weight k with respect to a congru- ence subgroup, then the Wronskian WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) = W∂k(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) is a modular form of weight ℓ(ℓ + k − 1) on the same congruence subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In [5] a more precise information about the automorphy factor and congruence subgroups for the Wronskian modular form WD is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Now we specialize Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2 to a situation where V = Span(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) define an ℓ-dimensional modular invariant space under Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (1) Let f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='.,fℓ be a basis of the modular invariant space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' If ord(WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ), i∞) = λ then WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) = η(τ)24λG(τ), where G(τ) is a nonzero holomorphic modular form of weight ℓ(ℓ−1)−12λ on Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In particular, if λ = ℓ(ℓ−1) 12 , then � WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) = η(τ)2ℓ(ℓ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2) If ord(WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ), i∞) > ℓ(ℓ − 1) 12 then the Wronskian is identically zero and fi are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Part (1) of this proposition can be found in [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' see also [14, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4] and [2, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Part (2) follows from part (1) and the fact that Mk(Γ(1)) = 0 for k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' See also [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' In the situation when fi(q) = qri(a(i) 0 + a(i) 1 q + · · ·), a(i) 0 ̸= 0 for every i and ri ̸= rj for i ̸= j, it is easy to see that the order of vanishing of WD(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', fℓ) is �ℓ i=1 ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Using this fact and Remark 1 we immediately see that the order of vanishing of WD( ˜F1, ˜F2, ˜F3, ˜F4, ˜F5, ˜F6) is 3 2, so in our case the Wronskian is not an η-power and instead we have � WD( ˜F1, ˜F2, ˜F3, ˜F4, ˜F5, ˜F6) = (5892480)−1·η(τ)36(70027513E4(τ)3−64135033E6(τ)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Now we are ready to state a general conjecture which is based on some numerical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Let χ0(1) be the Nahm sum associated to Tn, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Then we have (1) For n = 2k ≥ 2 even: qakχ0(1) = f(τ)2k � WD(R2k,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='., R2k,k) η(τ)k(2k−1) , 24 ANTUN MILAS AND LIUQUAN WANG where ak = −k(1+4k) 48(1+k) and R2k,i(τ) = � n∈Z (−1)nq(k+1)(n− (2i−1) 4(k+1) )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2) For n = 2k − 1 ≥ 3 odd: qakχ0(1) = f(τ)2k−1 � WD((∂Θ)1, 2k+1 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='., (∂Θ)k−1, 2k+1 2 ) η(τ)(k−1)(2k−1) , where ak = −1+6k−8k2 96k+48 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' More precisely, qakχ0(1) is a modular function which is a component of a 3k- dimensional vector valued modular function under Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Part (1) of the conjecture is known to hold for n = 2 [6] and we also verified n = 4 numerically for high powers of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Part (2) for n = 3 was proven in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' The second author was supported by the National Natural Science Foundation of China (12171375).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Adamovi´c and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Milas, The N = 1 triplet vertex operator superalgebras: twisted sec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' SIGMA Symmetry, Integrability and Geometry: Methods and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 2008 Dec 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='4:087.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Arike, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Kaneko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Nagatomo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Sakai, Affine vertex operator algebras and modular linear differential equations, Letters in Mathematical Physics 106, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 5 (2016): 693-718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Berndt, Number Theory in the Spirit of Ramanujan, the American Mathematical So- ciety, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Bringmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Calinescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Folsom, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Kimport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Graded dimensions of princi- pal subspaces and modular Andrews–Gordon-type series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Communications in Contemporary Mathematics 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 04 (2014): 1350050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Calinescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Milas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Penn, Vertex algebraic structure of principal subspaces of basic A(2) 2n -modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Journal of Pure and Applied Algebra, 220 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='1752-1784.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Cohen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Str¨omberg, Modular Forms: a Classical Approach, Graduate Studies in Mathematics 179, the American Mathematical Society, Providence, RI, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Garvan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Liang, Automatic proof of theta-function identities, arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='08051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Kawasetsu, The intermediate vertex subalgebras of the lattice vertex operator algebras, Letters in Mathematical Physics 104, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 2 (2014): 157-178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Lepowsky and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Wilson, The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 77 (1984), 199-290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Lepowsky and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Wilson, The structure of standard modules II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' the case A(1) 1 , principal gradation, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 79 (1985), 417–442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Magnusson and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Raum, On the Computation of General Vector-valued Modular Forms, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='06676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Mason, Vector-valued modular forms and linear differential operators, International Jour- nal of Number Theory 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 03 (2007): 377-390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Milas, On certain automorphic forms associated to rational vertex operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Moonshine - The First Quarter Century and Beyond: Proceedings of a Workshop, Edinburgh 2004, (2010): 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' MODULARITY OF NAHM SUMS FOR THE TADPOLE DIAGRAM 25 [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Milas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Penn, Lattice vertex algebras and combinatorial bases: general case and W-algebras, New York Journal of Mathematics 18 (2012): 621-650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q- series, CBMS Regional Conference Series in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 102, Published for the Confer- ence Board of the Mathematical Sciences, Washington, DC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' by the American Mathematical Society, Providence, RI, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Rogers, Second memoir on the expansion of certain infinite products, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 25 (1894), 318–343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Rogers, On two theorems of combinatory analysis and some allied identities, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 16 (1917), 315–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Shimura, On modular forms of half integral weight, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' (2) 97 (1973), 440–481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Vlasenko and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Zwegers, Nahm’s conjecture: asymptotic computations and counterex- amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Communications in Number Theory and Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='5(3):617-42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Wakimoto, Infinite-dimensional Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=', 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Wang, Identities on Zagier’s rank two examples for Nahm’s conjecture, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='10748v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Wang, Explict forms and proofs of Zagier’s rank three examples for Nahm’s problem, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='04375v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Weber, Lehrbuch der Algebra, Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='3, Elliptische Funktionen and Algebraische Zahlen, Braunschweig, 1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' [25] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Zagier, The dilogarithm function, in Frontiers in Number Theory, Physics and Geometry, II, Springer, 2007, 3–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content=' Department of Mathematics and Statistics, University at Albany (SUNY), Al- bany, NY 12222, United States Email address: amilas@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='albany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='edu School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, People’s Republic of China Email address: wanglq@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='mathlqwang@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE3T4oBgHgl3EQfdQoZ/content/2301.04532v1.pdf'} diff --git a/bdFPT4oBgHgl3EQfAzS0/content/tmp_files/2301.12983v1.pdf.txt b/bdFPT4oBgHgl3EQfAzS0/content/tmp_files/2301.12983v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c0229f38c3f7b090def777ecce92cc382bf3110 --- /dev/null +++ b/bdFPT4oBgHgl3EQfAzS0/content/tmp_files/2301.12983v1.pdf.txt @@ -0,0 +1,1260 @@ +arXiv:2301.12983v1 [math.DG] 30 Jan 2023 +Metric SYZ conjecture for certain toric Fano +hypersurfaces +Yang Li +January 31, 2023 +Abstract +We prove the metric version of the SYZ conjecture for a class of Calabi- +Yau hypersurfaces inside toric Fano manifolds, by solving a variational +problem whose minimizer may be interpreted as a global solution of the +real Monge-Amp`ere equation on certain polytopes. This does not rely on +discrete symmetry. +1 +Introduction +The metric aspect of the Strominger-Yau-Zaslow (SYZ) conjecture [28] asks for +the following: +Conjecture 1.1. (cf. [18][21][28]) Let Xt be a 1-parameter maximally degener- +ate family of polarized d-dimensional Calabi-Yau manifolds over the punctured +disc D∗ +t , then for any given 0 < δ ≪ 1, and all t small enough depending on +δ, there exists a special Lagrangian T d-fibration on an open subset of Xt for +0 < ∣t∣ ≪ 1, whose normalized Calabi-Yau measure is at least 1 − δ. +The case of Abelian varieties is classical, while the case of K3 surfaces is +known through the trick of hyperk¨ahler rotation [12][26]. The first nontrivial +example in all dimensions is the Fermat family of Calabi-Yau hypersurfaces [17] +Xs = {Z0 ... Zd+1 + e−s +d+1 +∑ +0 +Zd+2 +i += 0} ⊂ CPd+1, +s ≫ 1. +The proof of the SYZ conjecture in this case exploits the large discrete symmetry +group, to simplify some rather delicate combinatorial problems. +A turn of thinking came about in [18], where we reduced the SYZ conjecture +to a problem of algebraic geometry. In brief, there is a non-archimedean (NA) +analogue of the complex Monge-Amp`ere (MA) equation, defined in terms of +intersection theory, and the deep work of Boucksom et al. [4][3][2][5] proved the +existence and uniqueness of the solution. To deduce the SYZ conjecture, what +is left to prove is a ‘comparison property’ [18], which morally asserts the NA +MA solution is not too wild. +1 + +For this new strategy to be convincing, it is requisite to verify the compar- +ison property in examples. The classical case of Abelian varieties was checked +by Goto-Odaka [10]. The recent work of Pille-Schneider [27] and Hultgren et al. +[14] independently verified the comparison property for some generalisation of +the Fermat family, but both works still rely heavily on the large discrete sym- +metry group, and thus still only apply to certain hypersurfaces inside CPd+1. +A notable contribution in [14] is to introduce a global variational problem. +The existence and uniqueness of the minimizer is quite transparent, but the local +PDE nature of the minimizer is not manifest. It is essentially known in [14] that +if one can deduce a local real Monge-Amp`ere equation from the minimizer, then +it would naturally produce the unique solution of the NA MA solution, and the +‘comparison property’ would be a corollary. The present paper aims to make +progress on this strategy, by proving the expected properties of the minimizer +for more examples without explicit reliance on the discrete symmetry. +Let ∆ ⊂ MR be an integral reflexive Delzant polytope, and X∆ be the asso- +ciated smooth toric Fano manifold of dimension d+1, with an ample polarization +L → X∆, which needs not be the anticanonical polarization. The origin 0 ∈ ∆ +corresponds to a distinguished section Xcan ∈ H0(X∆,−KX∆), which defines +the toric boundary of X∆. Let F ∈ H0(X∆,−KX∆) be a generic section, and +by assumption the divisor {F = 0} ⊂ X∆ is smooth, and intersects all the toric +boundary strata transversely, and in particular does not pass through the finite +number of toric fixed points on X∆. We will consider the family of Calabi-Yau +hypersurfaces as t → 0: +Xt = {Xcan + tF = 0} ⊂ X∆ × C∗ +t . +(1) +Algebro-geometrically Xt degenerate to the toric boundary. +We will assume one extra property on the reflexive polytope ∆, whose dual +polytope is denoted as ∆∨: +⟨m,n⟩ ≠ 0, +∀m ∈ vertex(∆), +∀n ∈ vertex(∆∨). +(2) +The main outcome is +Theorem 1.2. The SYZ conjecture 1.1 holds for the family Xt as t → 0. +The main new content of this paper is contained in section 2, which es- +tablishes a structure theory for the minimizer of the variational problem, and +has the flavour of convex analysis, variational calculus, and real MA equation. +This part does not require prior knowledge of K¨ahler geometry, and may be +of independent interest to analysts. In section 3 we briefly sketch the complex +geometric aspect towards the application to the SYZ conjecture, relying on the +aforementioned results in [17][19][18][14]. +Remark 1.3. J. Hultgren informs the author that together with R. Andreas- +son they have independently found an equivalent condition in terms of optimal +transport theory, for the minimizer of the variational problem to admit the +expected PDE interpretation [1]. Their upcoming paper will also contain coun- +terexamples, demonstrating in particular that some nontrivial condition on the +reflexive polytope is necessary in this strategy. +2 + +Acknowledgement. The author is a current Clay Research Fellow based at +MIT. He would like to especially thank J. Hultgren for pointing out his up- +coming work, and for helpful comments on this manuscript. He also thanks S. +Boucksom, L. Pille-Schneider, and S. Sun for discussions in the past. +2 +The variational problem +2.1 +Motivation: A global real Monge-Amp`ere equation? +Let MR ≃ Zd+1 ⊗Z R and NR = M ∨ +R be a pair of dual vector spaces, and let +∆ ⊂ MR be an integral reflexive polytope, with dual polytope ∆∨ ⊂ NR, so that +∆∨ = {x ∈ NR∣⟨m,x⟩ ≤ 1,∀m ∈ vertex(∆)}, +∆ = {p ∈ MR∣⟨n,p⟩ ≤ 1,∀n ∈ vertex(∆∨)}, +both ∆,∆∨ contain the origin, and all the vertices are integral points. Given neg- +ative real numbers µ(n) for all n ∈ vertex(∆∨), and λ(m) for all m ∈ vertex(∆), +we can define the convex functions on NR and MR, +Lλ(x) = max +m ⟨x,m⟩ + λ(m), +Lµ(p) = max +n ⟨p,n⟩ + µ(n). +These in turn define the convex polytopes containing the origin in the interior, +∆∨ +λ = {Lλ(x) ≤ 0} ⊂ NR, +∆µ = {Lµ(p) ≤ 0} ⊂ MR. +For instance, the choice λ = −1 recovers ∆∨, and the choice µ = −1 recovers ∆. +The normal vectors to the top dimensional faces are labelled by the vertices +m,n, which are primitive integral vectors. This integral structure on the faces +induce a canonical Lebesgue measure dx and dp on ∂∆∨ +λ and ∂∆µ respectively, +which we normalize to probability measures +dL∨ = +1 +∫∂∆∨ +λ dxdx, +dL = +1 +∫∂∆µ dpdp. +(3) +We shall use the notation ∣E∣ to denote the measure of subsets E with respect +to dL or dL∨. +Our secret goal is to formulate and solve a version of the real MA equation +on ∂∆∨ +λ and ∂∆µ. A direct attempt faces the following difficulties: +• The polytopes do not have natural global affine structures. On the overlap +of the candidate charts, the transition functions will only be piecewise +affine. +• Consequently, the condition of being a convex function depends on local +charts. +• The definition of the real MA equation via local charts will not be com- +patible with piecewise affine transition functions. +An insight of Hultgren et al. [14], carried out in a very symmetric special +case, is to adopt instead a global variational formulation. The price is that the +PDE nature of the problem is not manifest. +3 + +2.2 +Legendre transform on polytopes +We define the Legendre transforms (referred to as ‘c-transforms’ in [14], +which comes from optimal transport nomenclature), +L∞(∂∆µ) → L∞(∂∆∨ +λ), +L∞(∂∆∨ +λ) → L∞(∂∆µ), +such that for φ ∈ L∞(∂∆∨ +λ) and ψ ∈ L∞(∂∆µ), +ψ∗(x) = sup +p∈∂∆µ +⟨x,p⟩ − ψ(p), +φ∗(p) = sup +x∈∂∆∨ +λ +⟨x,p⟩ − φ(x). +(4) +The theory works symmetrically for φ,ψ. +Some of the immediate formal properties are listed below (cf. [14, section +3]): +Lemma 2.1. (Equicontinuity) The Legendre transforms φ∗,ψ∗ have uniformly +bounded Lipschitz constant. +Lemma 2.2. (Involution) For arbitrary φ ∈ L∞(∂∆∨ +λ), the double Legendre +transform φ∗∗ ≤ φ. If φ = ψ∗ for some ψ, then φ∗∗ = φ. +Lemma 2.3. (Monotonicity) If φ1 ≥ φ2, then φ∗ +1 ≤ φ∗ +2. For any constant c ∈ R, +we have (φ + c)∗ = φ∗ − c. Morever ∥φ∗ +1 − φ∗ +2∥C0 ≤ ∥φ1 − φ2∥C0. +We then introduce a class of functions P ⊂ C0(∂∆∨ +λ) and P∨ ⊂ C0(∂∆µ), +defined as the images of L∞ under the Legendre transform. By the involution +property, the Legendre transform sets up a canonical isomorphism P ≃ P∨, +which is isometric with respect to the C0-norms by Lemma 2.3. The formula +φ(x) = max +p∈∂∆µ⟨x,p⟩ − φ∗(p), +φ∗(p) = max +x∈∂∆∨ +λ +⟨x,p⟩ − φ(x) +provides a canonical extension of functions in P,P∨ to convex functions on +the ambient Euclidean spaces NR and MR. +We can thus think of P,P∨ as +‘global convex functions’, which in particular have gradient contained in ∆µ,∆∨ +λ +respectively. +2.3 +Conjugate sets and gradients +We begin with a classical analogy. In the classical Legendre transform theory +for convex functions on Rn, for a given position variable x, the gradient ∇φ(x) +is recovered by the conjugate momentum p achieving φ(x) = ⟨x,p⟩ − φ∗(p). In +the global context of polytopes, finding a reasonable analogue of the conjugate +p to x is a rather delicate question, and there are at least two natural notions. +Definition 2.4. Given a function φ ∈ P, let x ∈ ∂∆∨ +λ, then the conjugate set is +¯∇φ(x) ∶= {p ∈ ∂∆µ∣φ∗(p) + φ(x) = ⟨x,p⟩}. +For E ⊂ ∂∆∨ +λ, the conjugate set is +¯∇φ(E) ∶= {p ∈ ∂∆µ∣φ∗(p) + φ(x) = ⟨x,p⟩, for some x ∈ E}. +We leave the reader to define the Legendre dual version ¯∇φ∗. +4 + +In the polytope setting, there is a more subtle new notion, which better +captures the classical intuition of gradient. +To motivate this, we first delve +a little into polyhedral geometry. The top dimensional faces ∆∨ +m of ∂∆∨ +λ are +labelled by the vertices m of ∆, and concretely +∆∨ +m = {⟨x,m⟩ = −λ(m)} ⊂ ∂∆∨ +λ. +Likewise the vertices n of ∆∨ label the faces ∆n ⊂ ∂∆µ. +Given φ ∈ P, we can restrict φ to a convex function on Int(∆∨ +m), which +is just a convex domain in some Euclidean hyperplane. The classical gradient +of this restricted function naturally lies in the quotient space MR/Rm, and a +moment of thought reveals the gradient is contained in the image of ∆µ under +the quotient map MR → MR/Rm. +Lemma 2.5. Each fibre under the quotient map +∆µ ⊂ MR → Image(∆µ) ⊂ MR/Rm +intersects each of the following two unions of faces H+ +m,H− +m ⊂ ∂∆µ uniquely at +one point: +H+ +m = ⋃{∆n ∶ vertex n ∈ ∂∆∨ with ⟨n,m⟩ = 1}, +H− +m = ⋃{∆n ∶ vertex n ∈ ∂∆∨ with ⟨n,m⟩ ≤ −1}, +In particular H+ +m,H− +m each projects homeomorphically onto Image(∆µ) ⊂ MR/Rm. +Remark 2.6. We leave the reader to write down the Legendre dual statement +involving the self explanatory notations H+ +n,H− +n. +Proof. For every point p ∈ ∆µ, the line p + Rm intersects the convex polytope +∆µ in a line segment, with two endpoints (unless we are in the degenerate +setting where the line segment becomes a point). Now the vertices n prescribe +the normal vectors to the faces. In order for the line segment to exit ∆µ in +the positive (resp. negative) m direction, we need ⟨n,m⟩ > 0 (resp. < 0). By +assumption, all n,m are integral vectors, so ⟨m,n⟩ is an integer. Furthermore +by assumption ⟨m,n⟩ ≤ 1 for any choice of vertices m,n. The result follows. +It is desirable that the gradient p at x is contained in H+ +m so that it has a +unique natural identification with a point in MR/Rm. +Definition 2.7. Let x ∈ ∂∆∨ +λ, which may lie on possibly several faces. Then +the gradient set of x is the intersection +∇φ(x) = ¯∇φ(x) ∩ ⋂ +m +{H+ +m ∶ x ∈ ∆∨ +m}. +We say x has anomalous conjugate points if ¯∇φ(x) ∖ ∇φ(x) is nonempty. Like- +wise we define ∇φ(E) for subsets E, and the Legendre dual versions etc. +Remark 2.8. If p ∈ ¯∇φ(x) is a conjugate point, then tautologically x ∈ ¯∇φ∗(p). +However p ∈ ∇φ(x) does not automatically imply x ∈ ∇φ∗(p); one sufficient +condition is that p lies on only one face. +5 + +While conjugate points exist quite obviously by the definition of the Leg- +endre transform, the existence of at least one gradient p for any given x requires +a little more thought: +Lemma 2.9. If x ∈ ∂∆∨ +λ has an anomalous conjugate point p′, then it also +admits a gradient p ∈ ∇φ(x) (so in particular gradients always exist). +Proof. Suppose x lies on the intersection of some faces ∆∨ +mi. +By the defi- +nition of conjugate point φ∗(p′) + φ(x) = ⟨x,p⟩. Consider the polyhedral set +(p′ + ∑i R≥0mi) ∩ ∆µ, which must have some extremal point p such that (p + +∑i R≥0mi) ∩ ∆µ consists of only one point p. Then for each mi, there must be +some ni with ⟨mi,ni⟩ > 0 such that p ∈ ∆ni, hence p ∈ ⋂H+ +mi. +We claim p is also a conjugate point of x. We write p = p′ + ∑ simi with +si ≥ 0. Now since φ∗ as a function on MR has gradient contained inside ∆∨ +λ, we +have +φ∗(p) − φ∗(p′) ≤ max +y∈∆∨ +λ +⟨p − p′,y⟩ ≤ ∑si max +y∈∆∨ +λ +⟨mi,y⟩ ≤ −∑siλ(mi). +Hence +φ∗(p)+φ(x) ≤ φ∗(p′)+φ(x)−∑siλ(mi) = ⟨x,p′⟩−∑siλ(mi) = ⟨x,p′+∑simi⟩ = ⟨x,p⟩. +This combined with the tautological inequality φ∗(p) + φ(x) ≥ ⟨x,p⟩ shows that +the equalities hold, so p is a conjugate point, hence a gradient. +Remark 2.10. The arguments above will not work if we attempt to define the +gradient to lie in H− +m instead of H+ +m. In fact, very often there is no conjugate +point in H− +m. +Corollary 2.11. Let p ∈ ¯∇φ(x), such that there is an anomalous conjugate +point p′ with p = p′ + ∑ simi and si ≥ 0 as above. +Then for each mi with +x ∈ ∆∨ +mi, either si = 0 or ¯∇φ∗(p) ⊂ ∆∨ +mi. +Proof. Since the equalities are achieved in the proof of Lemma 2.9, we have +φ∗(p) − φ∗(p′) = −∑siλ(mi). For any y ∈ ∂∆∨ +λ, we have +φ(y) ≥ ⟨y,p′⟩ − φ∗(p′) = ⟨y,p′⟩ − φ∗(p) − ∑siλ(mi) += ⟨y,p⟩ − φ∗(p) − ∑si(λ(mi) + ⟨y,mi⟩). +Now suppose y is any conjugate point of p, then φ(y) = ⟨y,p⟩ − φ∗(p), whence +∑si(λ(mi) + ⟨y,mi⟩) ≥ 0. +By construction λ(mi) + ⟨y,mi⟩ ≤ 0 for any y ∈ ∆∨ +λ. For any i, as long as si > 0, +then ⟨y,mi⟩ = −λ(mi), namely y ∈ ∆∨ +mi. +Remark 2.12. In the special case that x lies on the interior of one face, then +there is only one m = mi involved, with p = p′ + sm, and since p ≠ p′, we are +forced to have s > 0, and ¯∇φ∗(p) ⊂ ∆∨ +m. +Proposition 2.13. (Comparison of gradient notions) Let x ∈ Int(∆∨ +m), then +p ∈ H+ +m is a gradient in ∇φ(x), if and only if p ∈ MR/Rm under the identification +H+ +m ≃ Image(∆µ) ⊂ MR/Rm is a classical gradient of the convex function φ on +Int(∆∨ +m). +6 + +Proof. If p ∈ ∇φ(x), then restricted to Int(∆∨ +m) we have +φ(y) ≥ ⟨y,p⟩ − φ∗(p) = φ(x) + ⟨y − x,p⟩, +∀y ∈ Int(∆∨ +m), +which means p is a classical gradient. +For the converse, we suppose p ∈ MR/Rm is a classical gradient at x, which +is identified uniquely with p ∈ ∆n ⊂ H+ +m. Since ∆∨ +λ lies in the half space ⟨m,x⟩+ +λ(m) ≤ 0, and ⟨n,m⟩ = 1, we can write any y ∈ ∂∆∨ +λ as +y = y′ − sn, +s ≥ 0, +⟨m,y′⟩ + λ(m) = 0. +Recall φ has a canonical extension to a convex function on NR. Restricted to +the hyperplane ⟨m,⋅⟩ + λ(m) = 0, the classical gradient property implies +φ(y′) ≥ φ(x) + ⟨y′ − x,p⟩. +But since the convex function φ on NR has gradient contained in ∆µ, +φ(y′) ≤ φ(y) + max +∆µ ⟨y′ − y,⋅⟩ ≤ φ(y) − sµ(n). +Combining the above, +φ(y) ≥ φ(x) + ⟨y′ − x,p⟩ + sµ(n) = φ(x) + ⟨y′ − x − sn,p⟩ = φ(x) + ⟨y − x,p⟩. +Since y is arbitrary, we deduce φ(x) + φ∗(p) = ⟨x,p⟩, namely p ∈ ¯∇φ(x). Since x +lies on only one face, and p ∈ H+ +m, we conclude p ∈ ∇φ(x). +2.4 +Variational problem +We now set up a global functional on P ≃ P∨: +F(φ) = ∫∂∆∨ +λ +φdL∨ + ∫∂∆µ +φ∗dL. +(5) +This functional is manifestly symmetric in terms of the Legendre transform; this +‘mirror symmetry’ is fundamental to our approach. In this variational problem, +the existence of a minimizer is quite transparent, but the local PDE nature of +the minimizer is less obvious, since the definition of the Legendre transform is +not local, and the differentiability of the functional at the critical point is not a +priori clear. +Theorem 2.14. The functional admits a minimizer in P, henceforth denoted +as φ. +Proof. (cf. [14, Thm 5.2]) By Lemma 2.3 and the fact that dL,dL∨ are both +probability measures, for any c ∈ R we have F(φ + c) = F(φ). +This allows +us to impose without loss that max∂∆∨ +λ φ = 0. Then by the uniform Lipschitz +estimate in P, we can envoke Arzela-Ascoli to take a C0 limit of a minimizing +sequence of the functional. By Lemma 2.3, the Legendre transformed functions +also converge in C0, so the limit attains the minimum. +Theorem 2.15. The minimizer is unique up to an additive constant. +7 + +Proof. (compare [14, Thm 5.2]) Suppose ˜φ is another minimizer, and we consider +φ′ = 1 +2(φ+ ˜φ), which still lies in P by the convexity of the function class P. Now +for any p ∈ ∂∆µ, +φ′∗(p) = max⟨x,p⟩ − φ′(x) +≤1 +2 max(⟨x,p⟩ − φ(x)) + 1 +2 max(⟨x,p⟩ − ˜φ(x)) = 1 +2(φ∗(p) + ˜φ∗(p)), +whence F(φ′) ≤ 1 +2(F(φ)+F(˜φ)). This forces φ′ to be also a minimizer, and the +equality is achieved. If p ∈ ∇φ′(x), then the equality φ′∗(p) = 1 +2(φ∗(p) + ˜φ∗(p)) +implies +(φ(x) + φ∗(p) − ⟨x,p⟩) + (˜φ(x) + ˜φ∗(p) − ⟨x,p⟩) = 0, +which forces p ∈ ∇φ(x) and p ∈ ∇˜φ(x). +When we restrict to any open face Int(∆∨ +m) of ∂∆∨ +λ, the functions φ, ˜φ are +convex functions on convex domains, and ∇φ(x),∇˜φ(x) are identified with the +classical gradients of convex functions, which naturally lie inside the quotient +space MR/Rm (cf. section 2.3). Since ∇φ = ∇˜φ Lebesgue-a.e, the difference +φ − ˜φ is a constant on the face (cf. [14, Lem 5.3]). By the continuity of the +functions, and matching the functions on the intersection of different faces, we +see that the constant must be independent of the face. +2.5 +Variational inequality +We now derive a fundamental variational inequality for the minimizer. +Proposition 2.16. For any measurable E ⊂ ∂∆∨ +λ, we have ∣¯∇φ(E)∣ ≥ ∣E∣. +Completely analogously ∣¯∇φ∗(E)∣ ≥ ∣E∣ for any measurable subset E ⊂ ∂∆µ. +Proof. We consider the function φ − t1E for 0 < t ≪ 1, where 1E denotes the +characteristic function of E. By Lemma 2.3, +φ∗ ≤ (φ − t1E)∗ ≤ φ∗ + t. +Morever (φ − t1E)∗ > φ∗ at p ∈ ∂∆µ, only when +sup +x∈E +(⟨p,x⟩ − φ(x)) ≥ φ∗(p) − t. +Hence +t∣{p ∶ sup +x∈E +(⟨p,x⟩ − φ(x)) ≥ φ∗(p) − t}∣ ≥ ∫∂∆µ +((φ − t1E)∗ − φ∗)dL. +Since φ is a minimizer, plugging in (φ − t1E)∗∗ as a competitor (the caveat +being that (φ − t1E) may not lie in P), we get +∫∂∆µ +(φ − t1E)∗ − φ∗ ≥ −∫∂∆∨ +λ +(φ − t1E)∗∗ − φ ≥ −∫∂∆∨ +λ +(φ − t1E) − φ = t∣E∣. +Here the second inequality uses Lemma 2.2. Combining the above and cancelling +the t factor, +∣{p ∶ sup +x∈E +(⟨p,x⟩ − φ(x)) ≥ φ∗(p) − t}∣ ≥ ∣E∣, +∀0 < t ≪ 1. +8 + +Taking the t → 0 limit, and recalling φ∗(p) ≥ ⟨x,p⟩ − φ(x), we get +∣{p ∶ sup +x∈E +(⟨p,x⟩ − φ(x)) = φ∗(p)}∣ ≥ ∣E∣. +For closed subsets E, the sup can be replaced by max by the continuity of +φ, so the LHS subset is the conjugate set ¯∇φ(E). For general Borel subsets E, +we take a compact exhaustion of E. For any compact K ⊂ E, we observe +∣¯∇φ(E)∣ ≥ ∣¯∇φ(K)∣ ≥ ∣K∣. +Since the Lebesgue measure is inner regular, we obtain ∣¯∇φ(E)∣ ≥ ∣E∣ by taking +the limit K ↑ E. +Corollary 2.17. If no point in the measurable subset E ⊂ ∂∆∨ +λ has any anoma- +lous conjugate point, then ∣∇φ(E)∣ ≥ ∣E∣. +Proof. If there is no anomalous conjugate point, then ∇φ(E) = ¯∇φ(E). We +then apply the variational inequality. +Remark 2.18. We can now explain the heuristic why the variational problem +should be related to the real MA equation. In the ideal situation, there is no +anomalous conjugate point, and ∇φ,∇φ∗ are both bijective on points, and define +mutually inverse maps. Then +∣∇φ(E)∣ ≥ ∣E∣ = ∣∇φ∗(∇φ(E))∣ ≥ ∣∇φ(E)∣. +This forces equality everywhere. On each open face of ∂∆∨ +λ the gradient agrees +with the classical notion in convex function theory, and ∣∇φ(E)∣ = ∣E∣ is just +the weak formulation of the real MA equation. To make this argument work +more rigorously, one needs to control the anomalous conjugate points, and the +multivalued nature of gradient. +2.6 +Anomalous conjugate points +The remaining technical difficulty mainly comes from the possible existence of +anomalous conjugate points. +We introduce the subset S∨ ⊂ ∂∆∨ +λ as S∨ = S∨ +1 ∪ S∨ +2 , where +S∨ +1 = ⋃{lower dim faces of ∂∆∨ +λ}, +S∨ +2 = ⋃ +m +{x∣¯∇φ(x) ∩ H+ +m contains at least two points}. +In particular ∇φ(x) is single valued outside S∨. Similarly the reader may write +down the Legendre dual analogue S. We now recall a basic fact in classical +convex function theory. +Lemma 2.19. The subset S∨ ⊂ ∂∆∨ +λ has Lebesgue measure zero. A similar +statement holds for S ⊂ ∂∆µ. +9 + +Proof. (cf. [13, section 1.1.1]) We restrict φ to the the open face Int(∆∨ +m), to +obtain a classical convex function on a convex domain. As explained in section +2.3, the notion of gradient inside H+ +m is naturally identified with the classical +notion which takes value in MR/Rm. Since our convex functions are Lipschitz, +they are differentiable Lebesgue-a.e. The multi-valued gradient problem hap- +pens on the non-differentiability locus, which has measure zero. Now the global +statement follows by taking the union of all the faces. +The following lemmas say that the anomalous conjugate has controlled +effect on the gradient set. In this section, let E be a measurable subset of ∆∨ +m, +such that any x ∈ E admits some anomalous conjugate point in H− +m. Let F ′ = +¯∇φ(E)∩H− +m, and let F ⊂ H+ +m be the image of F ′ under the projection H− +m → H+ +m +(cf. section 2.3). By the proof of Lemma 2.9, we have F ⊂ ¯∇φ(E) ∩ H+ +m. +Lemma 2.20. ∣F∣ ≤ ∣E∣. +Proof. By construction every p ∈ F ∖ S is of the form p = p′ + sm with s > 0 +and p′ ∈ F ′. By Cor. 2.11 concerning the anomalous conjugate points, we must +have ¯∇φ∗(p) ⊂ ∆∨ +m. Since p ∈ H+ +m ∖ S, it lies on only one face ∆n ⊂ H+ +m with +⟨n,m⟩ = 1, whence ∆∨ +m ⊂ H+ +n. We then have ¯∇φ∗(p) ⊂ ∆∨ +m ⊂ H+ +n, whence +¯∇φ∗(p) = ¯∇φ∗(p) ∩ H+ +n = ∇φ∗(p). +Since p ∉ S, we know that ∇φ∗(p) consists of only one point, so in fact ∇φ∗(p) = +{x}. But p ∈ F ⊂ ¯∇φ(E), so x ∈ E and ¯∇φ∗(p) ⊂ E. +In summary we get the inclusion ¯∇φ∗(F ∖ S) ⊂ E. Now applying the vari- +ational inequality Prop. 2.16, +∣E∣ ≥ ∣¯∇φ∗(F ∖ S)∣ ≥ ∣F ∖ S∣ = ∣F∣. +The last equality uses that S has zero measure, by Lemma 2.19. +Lemma 2.21. ∣F ′∣ ≤ ∣F∣. +When the equality is achieved, then F ′ has zero +measure except perhaps on the faces ∆n with ⟨n,m⟩ = −1. +Proof. By Lemma 2.5, under the quotient map MR → MR/Rm, the sets H+ +m +and H− +m each project homeomorphically to the image of ∆µ. The projection +map is piecewise affine, and we focus on some face ∆n. With respect to the +natural Lebesgue measures dp induced by the integral structure, the quotient +map ∆n → MR/Rm has Jacobian factor ∣⟨n,m⟩∣. For ∆n ⊂ H+ +m this factor is +equal to one, and for ∆n ⊂ H− +m this factor is a positive integer (cf. Lemma +2.5). Thus under the projection H− +m → H+ +m, the Jacobian factor is ≥ 1. Since +the normalized measure dL is a constant multiple of dp, the Jacobian factor is +still ≥ 1 with respect to dL. This shows ∣F ′∣ ≤ ∣F∣. The equality case forces the +Jacobian factor to be equal to one, which means ∣⟨n,m⟩∣ = 1. +Proposition 2.22. ∣E∣ = ∣F∣ = ∣F ′∣. In particular the equality forces that F ′ +has zero measure except perhaps on the faces ∆n with ⟨n,m⟩ = −1. +Proof. By Lemma 2.20, 2.21, we already know ∣E∣ ≥ ∣F∣ ≥ ∣F ′∣, so it suffices to +prove ∣F ′∣ ≥ ∣E∣. We write H− +m = ⋃ ∆ni. For each i, we have the set inclusion +10 + +∆∨ +m ⊂ H− +ni, and since any point of E is conjugate to some point in H− +m∩ ¯∇φ(E) = +F ′, we conclude +E ⊂ ¯∇φ∗(F ′) ∩ ∆∨ +m ⊂ ⋃ +i +¯∇φ∗(F ′ ∩ ∆ni) ∩ H− +ni, +whence ∣E∣ ≤ ∑i ∣¯∇φ∗(F ′ ∩ ∆ni) ∩ H− +ni∣. +Recall that our setup has the Legendre duality symmetry. For any fixed i, +we now let F ′ ∩ ∆ni play the role of E in the Legendre dual version of Lemma +2.20, 2.21. The role of F ′ is then played by ¯∇φ∗(F ′∩∆ni)∩H− +ni, and we conclude +∣¯∇φ∗(F ′ ∩ ∆ni) ∩ H− +ni∣ ≤ ∣F ′ ∩ ∆ni∣. +Summing over i and combining the above, +∣E∣ ≤ ∑ +i +∣¯∇φ∗(F ′ ∩ ∆ni) ∩ H− +ni∣ ≤ ∑ +i +∣F ′ ∩ ∆ni∣ = ∣F ′∣. +This is the promised reverse inequality. +Remark 2.23. The equality forces a very strong rigidity, and it is an interesting +question whether it forces ∣E∣ = 0 in fact. +Corollary 2.24. ∣E∣ = ∣∇φ(E) ∩ F∣ ≤ ∣∇φ(E)∣. +Proof. Since ∣E∣ = ∣F∣ by Prop. 2.22, it suffices to prove F ∖ ∇φ(E) ∪ S has +measure zero. Now every point in p ∈ F ∖S has a unique gradient x = ∇φ∗(p) ∈ E, +and if x ∉ S∨ then p = ∇φ(x). This shows that +F ∖ ∇φ(E) ∪ S ⊂ Image of (¯∇φ(S∨ ∩ E) ∩ H− +m) under H− +m → H+ +m. +By applying Prop. 2.22 again to E ∩ S∨, we see +∣F ∖ ∇φ(E) ∪ S∨∣ ≤ ∣Image(¯∇φ(S∨ ∩ E) ∩ H− +m)∣ = ∣E ∩ S∨∣ = 0, +so the result follows. +2.7 +Anomalous conjugate points II +In this section we aim to prove +Proposition 2.25. For any E ⊂ ∂∆∨ +λ, we have ∣∇φ(E)∣ ≤ ∣E∣. +Lemma 2.26. Assume the subset E ⊂ ∆∨ +m has the property that any x ∈ E +admits some conjugate in a face ∆n with ⟨m,n⟩ = 0. Then ∣E ∩ ∇φ∗(∂∆µ)∣ = 0. +Proof. Consider a point x ∈ (E∩∇φ∗(∂∆µ))∖S∨. Since x ∈ ∇φ∗(∂∆µ), we write +x = ∇φ∗(p). Since x does not lie on S∨, it has a unique gradient, so p = ∇φ(x). +Now by assumption x has some anomalous conjugate p′ in a face ∆n with +⟨m,n⟩ = 0. By Lemma 2.14, there exists a gradient of the form p′ +sm for s > 0. +The uniqueness of the gradient then forces p = p′ + sm. Since p′ ∈ ∆n, we have +⟨p′,n⟩ + µ(n) = 0, hence +⟨p,n⟩ + µ(n) = ⟨p − p′,n⟩ = s⟨m,n⟩ = 0. +11 + +This means p ∈ ∆n. Since x lies on ∆∨ +m but not on S∨, it must be in the interior +of ∆∨ +m, so by the definition of gradient x = ∇φ∗(p) forces ⟨m,n⟩ = 1. +This +contradicts ⟨m,n⟩ = 0. This contradiction shows that x cannot exist, which +means (E ∩ ∇φ∗(∂∆∨ +µ)) ∖ S∨ is empty. Since S∨ has measure zero by Lemma +2.19, we see ∣E ∩ ∇φ∗(∂∆µ)∣ = 0. +We also record the Legendre dual statement. +Lemma 2.27. Suppose the subset G ⊂ ∆n has the property that any p ∈ G +admits a conjugate point in ∆∨ +m with ⟨m,n⟩ = 0. Then ∣G ∩ ∇φ(∂∆∨ +λ)∣ = 0. +We can now prove Prop. 2.25. +Proof. (Prop. 2.25) First, we notice it suffices to prove Prop. 2.25 only for +E ⊂ ∆∨ +m. This is because we can decompose E into the union of Em ⊂ ∆∨ +m for +all the possible m, and once we know ∣∇φ(Em)∣ ≤ ∣Em∣ for every m, it would +follow that +∣∇φ(E)∣ ≤ ∑∣∇φ(Em)∣ ≤ ∑∣Em∣ = ∣E∣. +The same argument shows more generally that if we can partition E into a union +of subsets, and the intersections have measure zero, then it suffices to prove the +statement for the subsets. +We will partition ∇φ(E) ∖ S into several subsets. +1. Let G1 be the union over n of all the p ∈ ∇φ(E) ∩ ∆n ∖ S, which admit +an anomalous conjugate point in some ∆∨ +m′ with ⟨m′,n⟩ = 0. By Lemma +2.27, its size ∣G1∣ = 0. +2. Let G2 consist of all the p ∈ ∇φ(E) ∖ S which admit an anomalous conju- +gate point, but not covered by the first case. We decompose G2 ⊂ ∇φ(E) ⊂ +H+ +m into the union of G2,n = G2 ∩ ∆n for all ∆n ⊂ H+ +m. By the Legendre +dual version of Cor. 2.24 applied to G2,n, we have ∣G2,n∣ ≤ ∣∇φ∗(G2,n)∣, +whence +∣G2∣ = ∑∣G2,n∣ ≤ ∑∣∇φ∗(G2,n)∣. +Next, for any p ∈ ∇φ(E) ∖ S, we can write p = ∇φ(x) for x ∈ E, and +since p is not in S, it has a unique gradient point which is x ∈ E. Thus +∇φ(∇φ∗(p)) contains p, and ∇φ∗(p) ⊂ E ⊂ ∆∨ +m. Since the subset of points +in ∆∨ +m with multiple gradients has zero measure (cf. Lemma 2.19), we +see that the intersections between the sets ∇φ∗(G2,n) have zero measure, +whence +∣G2∣ ≤ ∑∣∇φ∗(G2,n)∣ = ∣∇φ∗(G2)∣. +3. Let G3 consist of all the p ∈ ∇φ(E)∖S with no anomalous conjugate point. +Any such p lies outside of S, hence has a unique gradient, which must then +be its unique conjugate point. The set ∇φ∗(G3) ⊂ E is the union of these +conjugate points. +By the Legendre dual version of Cor. +2.17 we have +∣∇φ∗(G3)∣ ≥ ∣G3∣. +12 + +By construction ∇φ(E) ∖ S = G1 ∪ G2 ∪ G3, hence by the above discussions +∣∇φ(E) ∖ S∣ ≤ ∣G1∣ + ∣G2∣ + ∣G3∣ ≤ ∣∇φ∗(G2)∣ + ∣∇φ∗(G3)∣. +The same argument in item 2 above shows that ∣∇φ∗(G2) ∩ ∇φ∗(G3)∣ = 0, so +∣∇φ(E) ∖ S∣ ≤ ∣∇φ∗(G2)∣ + ∣∇φ∗(G3)∣ = ∣∇φ∗(G2 ∪ G3)∣ ≤ ∣E∣. +The last inequality here is because ∇φ∗(G2 ∪G3) ⊂ E. Now since S has measure +zero, we get +∣∇φ(E)∣ = ∣∇φ(E) ∖ S∣ ≤ ∣E∣, +as required. +2.8 +Structure theorems for the minimizer +In this section we will list certain structural properties of the minimizer φ, which +in the best situation lead to the real MA equation. +Proposition 2.28. The set function E ↦ ∣∇φ(E)∣ defines a Borel measure +on ∂∆∨ +λ, which is absolutely continuous with respect to the Lebesgue measure. +Likewise with the Legendre dual analogue. +Proof. Since ∣∇φ(E)∣ ≤ ∣E∣ by Prop. 2.25, we see that ∣E∣ = 0 implies ∣∇φ(E)∣ = 0. +Next we justify the countable additivity axiom. Let E = ⋃Ei be a disjoint +partition, then clearly ∣∇φ(E)∣ ≤ ∑ ∣∇φ(Ei)∣. To see that the equality holds, +without loss we may assume E is disjoint from the measure zero set S∨, so +that the gradient is a single valued map. Morever, the mutual intersections of +∇φ(Ei) is contained in S, which again has measure zero, so +∑∣∇φ(Ei)∣ = ∣⋃ +i +∇φ(E)∣ = ∣∇φ(E)∣ +as required. +We now introduce a good-bad decomposition on ∂∆∨ +λ. The good set G∨ ⊂ +∂∆∨ +λ consists of the following two types of points x ∈ ∂∆∨ +λ: +• Type I good point: x has no anomalous conjugate point, so that all con- +jugate points are gradients of x; +• Type II good point: x is contained in some face ∆∨ +m and x admits an +anomalous conjugate point p ∈ ¯∇φ(x) ∩ H− +m. +The bad set B∨ ⊂ ∆∨ +λ consists of all x contained in some face ∆∨ +m, and which +admits some anomalous conjugate point p ∈ ∆n with ⟨m,n⟩ = 0. Clearly ∂∆∨ +λ = +G∨ ∪ B∨. We leave the reader to write down the Legendre dual version ∂∆µ = +G ∪ B. +Theorem 2.29. (Property of good-bad decomposition) If E ⊂ G∨, then ∣∇φ(E)∣ = +∣E∣. If E ⊂ B∨, then ∣∇φ(E)∣ = 0. In particular ∣G∨ ∩ B∨∣ = 0. +13 + +Proof. Since the measure E ↦ ∣∇φ(E)∣ has finite additivity, it suffices to con- +sider E of single types. For type I good points, we use ∣∇φ(E)∣ ≤ ∣E∣ from Prop. +2.25 and ∣∇φ(E)∣ ≥ ∣E∣ from Cor. 2.17. For type II good points, we use Cor. +2.24, Prop. 2.25 and finite additivity. +Now we consider the bad points. By finite additivity, without loss E ⊂ ∆∨ +m. +Since ∣S∨∣ = ∣∇φ(S∨)∣ = 0, without loss we can suppose E is disjoint from S∨, so +every x ∈ E has a unique gradient p = ∇φ(x). By assumption E consists of bad +points, so there is an anomalous conjugate p′ ∈ ∆n with ⟨m,n⟩ = 0. By Lemma +2.9, the unique gradient is of the form p = p′ + sm for s > 0. Thus +⟨p,n⟩ = ⟨p′,n⟩ + s⟨n,m⟩ = −µ(n), +whence p lies on ∆n as well. Since p is a gradient of x, it must also lie on H+ +m, +so it falls into H+ +m ∩ ∆n, which has measure zero. +One lesson of Prop. 2.29 is that ∇φ only sees the good set, and an analogous +version holds for ∇φ∗. In fact modulo measure zero sets, these two maps are in +some sense inverse to each other, when we restrict to the good set. +Theorem 2.30. ( Legendre duality) The maps ∇φ and ∇φ∗ have the following +properties: +1. For any E ⊂ G∨, we have ∣∇φ(E) ∩ B∣ = 0, and ∣∇φ(E) ∩ G∣ = ∣E∣. +2. We consider the maps between subsets of G∨ and G defined by +⎧⎪⎪⎨⎪⎪⎩ +E ↦ ∇φ(E) ∩ G, +∀E ⊂ G∨, +F ↦ ∇φ∗(F) ∩ G∨, +∀F ⊂ G. +The mutual compositions agree with E,F up to a measure zero subset. +3. In particular ∣G∣ = ∣G∨∣ and ∣B∣ = ∣B∨∣. +Proof. The claim that ∣∇φ(E) ∩ B∣ = 0 follows from the Legendre dual version +of Lemma 2.26. By Prop. 2.28 and Thm. 2.29, we have ∣G ∩ B∣ = 0, and +∣∇φ(E) ∩ G∣ = ∣∇φ(E)∣ − ∣∇φ(E) ∩ B∣ = ∣∇φ(E)∣ = ∣E∣. +This proves item 1. +Consequently, the two maps in item 2 preserve the measures. To prove the +composition statement, without loss E is disjoint from the measure zero set S∨. +For any p ∈ ∇φ(E) ∩ G ∖ S, the ∇φ∗(p) uniquely recovers the point x ∈ E with +p = ∇φ(x). Thus the composition of ∇φ and ∇φ∗ recovers a full measure subset +of E. This proves item 2. +Now by item 1, 2 applied to G∨, we get ∣G∣ = ∣G∨∣. Since ∂∆∨ +λ = G∨ ∪ B∨ and +∣G∨ ∩ B∨∣ = 0, we have +∣B∣ = 1 − ∣G∣ = 1 − ∣G∨∣ = ∣B∨∣, +which proves item 3. +14 + +The most important special case for us is the following: +Theorem 2.31. The following conditions are equivalent: +1. The size of the bad set ∣B∣ = 0, or equivalently ∣B∨∣ = 0. (For instance, this +works if ⟨n,m⟩ = 0 never holds for any vertices of ∆,∆∨.) +2. The images of ∇φ and ∇φ∗ have full measure. +When this holds, then ∣∇φ(E)∣ = ∣E∣ for any E ⊂ ∂∆∨ +λ, and likewise with ∇φ∗. +The two maps ∇φ and ∇φ∗ compose to the identity modulo measure zero sets. +Proof. By Thm. 2.30, the vanishing of ∣B∣ would force ∣B∨∣ = 0. Thus the good +sets have full measure, and we know from Thm. 2.30 that ∇φ and ∇φ∗ on the +good sets are measure preserving, mutually inverse maps modulo zero measure. +In particular the images of ∇φ and ∇φ∗ have full measure. +Conversely, suppose the images of ∇φ and ∇φ∗ have full measure. By Thm. +2.29, +∣∇φ(∂∆∨ +λ)∣ = ∣∇φ(G∨)∣ + ∣∇φ(B∨)∣ = ∣G∨∣, +which then forces ∣G∨∣ = 1, namely ∣B∨∣ = 0. This shows that item 2 implies item +1, and the rest follow from Thm. 2.30. +When the equivalence conditions in Thm. 2.31 is satisfied, then the con- +clusions admit classical interpretations. When we restrict φ to an open face +Int(∆∨ +m) to obtain a convex function still denoted as φ, then ∇φ is identified +with the classical gradient (cf. +section 2.3). +Recall from (3) how the nor- +malized measure is related to the Lebesgue measure induced by the integral +structure. Morever, the projection map H+ +m → MR/Rm has Jacobian factor one +with respect to the integral Lebesgue measure (cf. the proof of Lemma 2.21). +In summary, the fact that ∣∇φ(E)∣ = ∣E∣ with respect to the normalized mea- +sures, translates into the weak Alexandrov formulation of the following real MA +equation over Int(∆∨ +m): +det(D2φ) = C0 = ∫∂∆µ dp +∫∂∆∨ +λ dx. +(6) +Completely analogously, we have the weak Alexandrov formulation of the fol- +lowing real MA equation over the open faces Int(∆n): +det(D2φ∗) = C−1 +0 . +(7) +This expresses the local PDE nature of the variational problem. +2.9 +Some examples +We first give some simple examples to which Theorem 2.31 applies. +15 + +Example 2.32. We revisit the standard simplex example [14, section 1.1][17]. +We take the standard basis e0,e1,... ed+1 inside Zd+2, and define +M = {p ∈ Zd+2∣∑pi = 0}, +N = M ∨ = Zd+2/Z(1,... 1), +and MR = M ⊗ R,NR = N ⊗ R. The polytope ∆ ⊂ MR is the convex hull of +mi = (d + 2)ei − +d+1 +∑ +0 +ej, +i = 0,... d + 1. +Its dual polytope ∆∨ is also a simplex, with vertices given by +n0 = (−1,0,... ,0),... nd+1 = (0,... ,0,−1). +Thus +⟨mi,nj⟩ = +⎧⎪⎪⎨⎪⎪⎩ +−(d + 1), +i = j, +1, +i ≠ j. +In particular ⟨n,m⟩ = 0 never occurs, and Thm 2.31 applies. +This example +corresponds to hypersurfaces inside CPd+1. +Example 2.33. Suppose we have k integral reflexive polytopes ∆i ⊂ MR,i, +with dual polytopes ∆∨ +i ⊂ NR,i, such that the pairing ⟨m,n⟩ ≠ 0 for all possible +vertices m,n of ∆i and ∆∨ +i . Then we can take +NR = ⊕iNR,i, +MR = ⊕iMR,i, +∆ ⊂ MR, +∆∨ ⊂ NR. +The vertices of ∆∨ are of the form (0,... ,ni,0,...) with ni ∈ vertex(∆∨ +i ), while +the vertices of ∆ are of the form (m1,... mk) for mi ∈ vertex(∆i). It is im- +mediate to check that the pairing ⟨m,n⟩ ≠ 0 still holds for ∆,∆∨. Algebro- +geometrically, this is just taking the product of several toric Fano varieties. +We now give a simple example which shows that ∣B∣ = 0 might fail if ⟨m,n⟩ = +0 is allowed. +Example 2.34. This example lives inside R2. Let ∆ be the convex full of +(1,0),(0,1),(−1,1),(−1,0),(0,−1),(1,−1). +Then ∆∨ is the convex hull of ±(1,0),±(1,1),±(0,1). Notice for instance that +m0 = (1,−1) is orthogonal to n0 = (1,1). We can take ∆∨ +λ = ∆∨, and let +∆µ = {(x,y) ∈ R2 ∶ ∣x∣ ≤ 1,∣y∣ ≤ 1,∣x + y∣ ≤ ǫ}, +for some given constant 0 < ǫ < 1. When ǫ ≪ 1, then ∆µ is concentrated along +a line segment. +We claim that ∣B∣ ≠ 0. Otherwise by Thm 2.31, we have ∣∇φ(E)∣ = ∣E∣ for +any E. We take E = ∆∨ +m, so that by the definition of gradient ∇φ(E) ⊂ H+ +m. +Thus +∣∆∨ +m∣ = ∣∇φ(∆∨ +m)∣ ≤ ∣H+ +m∣. +For the choice m = m0 = (1,−1), this leads to a contradition for small enough ǫ. +16 + +Algebro-geometrically, the corresponding Fano manifold X∆ is the blow +up of P1 × P1 at two toric fixed points, so there are two disjoint exceptional +divisors E1,E2. Adjusting ǫ amounts to changing the K¨ahler class on X∆ in +a 1-parameter family, with ǫ = 1 corresponding to c1(π∗ +1[OP1] + π∗ +2[OP1]), and +decreasing ǫ corresponding to subtracting multiplies of E1 + E2. The family +Xt of Calabi-Yau hypersurfaces (cf. +the introduction) is here just a pencil +of elliptic curves. The second cohomology of an elliptic curve has rank one, +so the restriction of all these K¨ahler classes to the elliptic curve will all be +proportional. +A plausible interpretation of the above example, is that once +an ample polarization L → X on the degenerating Calabi-Yau hypersurfaces is +fixed, there is still some flexibility to extend this polarization to L → X∆, and +for an inappropriate choice of L → X∆, the variational problem may not be +helpful for the SYZ conjecture. +2.10 +Open questions +Our work suggests a number of natural questions, which are relevant for further +applications to the SYZ conjecture, and which may be of some independent +interest to PDE theorists. +1. Thm. +2.31 makes it clear that the failure for the minimizer to satisfy +∣∇φ(E)∣ = ∣E∣ is measured by the size of the bad set ∣B∣. Is it possible to +compute or estimate ∣B∣ in general, when we allow ⟨m,n⟩ = 0? +2. Assume ∣B∣ = 0. Is it possible for the anomalous conjugate point to exist? +3. Assume ∣B∣ = 0. When can we prove that the gradient is unique everywhere +(instead of Lebesgue-a.e), namely ∇φ(x) consists of only one point for +every x ∈ ∂∆∨ +λ? +4. The Legendre dual version of the above question asks if ∇φ∗(p) consists +of only one point for every p ∈ ∂∆µ. +This question is closely related to the strict convexity of the restriction +φ to a convex function on the open faces. This has implication on the +regularity question of φ, since a strictly convex solution of the real MA +equation is known to be smooth. +If both ∇φ and ∇φ∗ are single valued, they would define mutually inverse +maps setting up a homeomorphism between ∂∆∨ +λ and ∂∆µ. This global +version of Legendre duality is desirable from the viewpoint of potential +applications to mirror symmetry. +5. The polytopes ∂∆∨ +λ and ∂∆µ have no natural global smooth structure; +the only a priori information is that the open faces Int(∆∨ +m) and Int(∆n) +have natural affine structures, and in particular smooth structures. +Now suppose the previous questions have positive answers, so that φ and +φ∗ are smooth on the respective open faces, and the gradient maps set +up a global homeomorphism. Then we can transfer Int(∆n) to the open +subset ∇φ∗(Int(∆n)) ⊂ ∂∆∨ +λ, and on the overlap with the open faces of +∂∆∨ +λ the smooth structures are compatible. This suggests that there is +17 + +a hidden smooth structure with singularity on ∂∆∨ +λ, where the singular +locus lies on the subset of the lower dimensional faces of ∂∆∨ +λ, which maps +under ∇φ to the lower dimensional faces of ∂∆µ. Morever, this smooth +structure with singularity is compatible with Legendre duality. +It would be desirable for mirror symmetry (cf. the Kontsevich-Soibelman +conjecture [15]) that the singular locus is of Hausdorff codimension two. +Remark 2.35. J. Hultgren informs the author that their upcoming work +[1] will contain more discussions on the affine structure with singularity. +3 +Application to the SYZ conjecture +For backgrounds on the non-archimedean (NA) geometry, and the previous lit- +erature on its relation to K¨ahler geometry, the reader may refer to [5] [27, +section 1][18]. A recent survey on the progress in the metric aspects of the SYZ +conjecture can be found in [21]. +We now sketch how the result of section 2 implies Theorem 1.2. +This +argument is essentially known, and consists of assembling ingredients from the +literature. +3.1 +Complex geometry setup +We work in the context of the introduction, so X∆ is a smooth toric Fano +manifold associated to the reflexive Delzant polytope ∆, and X = ∪Xt (cf. (1)) +is a union of Calabi-Yau hypersurfaces degenerating to the toric boundary. This +degeneration family admits a natural model +X = {Xcan + tF = 0} ⊂ X∆ × C. +Upon base change, X can be regarded as a family over SpecC((t)), and X +defines a model over SpecC[[t]], which is divisorial log terminal (dlt) since by +assumption the singularity structure on the central fibre is ´etale locally of the +form (cf. [14, Page 25]) +X ≃ V (x1 ... xk − ty) ⊂ Ak+2 +xi,t,y × Ad−k, +d = dimC Xt. +The central fibres is identified with the toric boundary of X∆, and consists +of many divisors labelled by the vertices n ∈ ∆∨, each with multiplicity one. +The dual complex ∆X of the dlt model ∆X has vertices corresponding to these +divisors, and the tropicalization map gives a natural identification +trop ∶ ∆X ≃ ∆∨ ⊂ NR. +The model X is minimal, meaning that the logarithmic relative canonical bundle +is trivial. As a caveat, our dlt model X is not Q-factorial. +Any ample line bundle L → X∆ corresponds to a moment polytope ∆µ +for some choice of µ(n) < 0 for n ∈ vertex(∆∨), such that the piecewise linear +function µ ∶ NR → R extending the values µ(n) is a ‘strictly concave’ function. +This induces a polarization line bundle L → X, which in particular prescribes +the K¨ahler class c1(L) on Xt. We denote (Ld) = ∫Xt c1(L)d. +18 + +Remark 3.1. The concavity condition on µ is related to ampleness, and if we +drop it, the line bundle L → X∆ will only be big, even though the induced +line bundle L → X might still be ample. On the other hand, in section 2 this +condition is not needed. Once we drop it, then for some n ∈ vertex(∆∨) the face +∆n may be empty, but this does not affect the arguments. +By the adjunction formula, X over some small punctured disc admits a +nowhere vanishing holomorphic volume form Ω, which induces holomorphic vol- +ume forms Ωt on the hypersurfaces Xt, hence normalized Calabi-Yau measures +on Xt +µt = +Ωt ∧ Ωt +∫Xt Ωt ∧ Ωt +. +(8) +The metric SYZ conjecture concerns the Calabi-Yau metrics (Xt,ωt) in the class +c1(L), satisfying the complex MA equation +ωd +t = (Ld)µt. +(9) +The SYZ conjecture predicts that given any small 0 < δ ≪ 1, for all t small +enough depending on δ, then (Xt,ωt) contains an open subset with µt-measure +at least 1 − δ, which admits a special Lagrangian torus fibration. +3.2 +Non-archimedean geometry +The work of Boucksom-Jonsson [4] established the following picture. The fam- +ily X viewed as a smooth projective variety over SpecC((t)) gives rise to the +Berkovich space Xan, which can be regarded as the limit of Xt as t → 0 un- +der the hybrid topology. The Berkovich space contains the essential skeleton +Sk(X), and under the hybrid topology convergence, the family of probability +measures µt converge to a Lebesgue type probability measure µ0 supported on +Sk(X) ⊂ Xan. This has a concrete description. Given a dlt model X, there is +a canonical embedding of its dual complex ∆X into Xan, and if X is minimal, +then +emb ∶ ∆X ≃ Sk(X) ⊂ Xan. +In our case of Calabi-Yau hypersurfaces, under the canonical isomorphisms +∆X +emb +��→ Sk(X) +trop +��→ ∂∆∨ ⊂ NR, +the probability measure µ0 is up to a global constant the Lebesgue measure dx +induced by the integral structure on the open faces of ∂∆∨. When the dust +settles, µ0 = dL∨ (cf. (3)), where we choose λ = −1 so that ∆∨ +λ = ∆∨. +Given the model X, every point on Xan has a centre cX (x) which is a +scheme theoretic point in X0. +The map cX ∶ Xan → X0 is anticontinuous, +which means the preimage of closed subsets of X0 are open subsets of Xan. In +particular, the preimage of the finitely many toric fixed points in X0 is an open +subset U ⊂ Xan, and the evaluation of the logarithmic coordinates provides a +natural retraction map from U to the open faces of ∆X , which is identified with +the tropicalization map U → ∂∆∨ ⊂ NR. +19 + +Boucksom-Favre-Jonsson [3][2][5] developed a NA pluripotential theory, +which concerns the analogue of semipositive metrics and Monge-Amp`ere mea- +sures on Xan. Its central result concerns the solution of the NA Calabi conjec- +ture. The special case relevant to us is the following: +Theorem 3.2. Let L → X be an ample line bundle, then there is a unique up +to constant semipositive metric ∥⋅∥CY on L → Xan, whose NA MA measure is +equal to the Lebesgue measure (Ld)µ0 supported on Sk(X) ⊂ Xan. +It is convenient to think about a semipositive metric ∥⋅∥ in terms of a +potential function ϕ. Concretely in our case, we can choose any local trivialising +section s of the line bundle L → X∆, and evaluate ϕ = −log ∥s∥. We say ∥⋅∥ +satisfies the ‘weak comparison property’, if under the retraction map from U +to the open faces of ∆X , the potential ϕ is constant on fibres. Notice that the +pair (X,X0) is a semistable SNC pair near the toric fixed points. Vilsmeier [29] +implies that +Theorem 3.3. [29][14, Lem. 6.1][27, Thm 3.17] Under the weak comparison +property, then the NA MA measure MANA on U ⊂ Xan is related to the real +MA measure MAR on the open faces of ∆X ≃ Sk(X) ≃ ∂∆∨, by the equation +1Sk(X)MANA(∥⋅∥) = d!MAR(ϕ). +A key link to the SYZ conjecture is provided by [18]. +Theorem 3.4. The weak comparison property for the metric ∥⋅∥CY implies the +SYZ conjecture 1.1 for the given family Xt. +Remark 3.5. The semistable SNC setting (instead of the dlt setting) was +assumed in [18], but the proof only uses the semistable SNC condition on those +divisor intersection strata contributing to Sk(X). If the reader prefers to work +with SNC models, one can apply the discussion to any SNC resolution of X +isomorphic to X on its smooth locus. The choice of the resolution does not +matter because the blow up along the singular locus of X has no effect on +U ⊂ Xan. +Our current strategy, which follows [14], is to produce ∥⋅∥CY via solving the +variational problem. We are given the polytopes ∆,∆∨, ∆∨ +λ = ∆∨ and ∆µ, and +we assume that the equivalent conditions in Thm. 2.31 applies (eg. when (2) +holds). Then section 2 produces the minimizer φ on ∂∆∨, and by Thm. 2.31, +this implies that φ restricted to the open faces Int(∆∨ +m) satisfies the real MA +equation in the weak formulation. +This minimizer φ extends canonically to a convex function on NR by +φ(x) = max +p∈∂∆µ⟨x,p⟩ − φ∗(p). +By [27, Thm. C], this convex function φ corresponds canonically to a semiposi- +tive toric metric on the Berkovich analytification of the toric Fano manifold X∆, +hence induces by restriction a semipositive metric Φ on L → Xan. This con- +cretely works as follows. The integral points m ∈ MR correspond to monomial +sections sm which induce logarithms log ∣sm∣ on Xan, and by linear interpolation +20 + +we can make sense of log ∣sp∣ = ∑ pi log ∣si∣, where si corresponds to a Z-basis of +MR. Then the semipositive metric is defined by +Φ = max +p∈∂∆µ log ∣sp∣ − φ∗(p). +(10) +Lemma 3.6. The semipositive metric Φ on L → Xan satisfies the weak com- +parison property. +Proof. Over the open face Int(∆∨ +m), upon choosing a local trivialising section as +the monomial s = sm, then log ∣sp∣ is identified with the local function on Xan +log ∣sp +s ∣ = ⟨x,p⟩ − ⟨x,m⟩ = ⟨x,p⟩ + λ(m) = ⟨x,p⟩ − 1. +Here the point x ∈ NR is the image under the tropicalization map U → ∂∆∨ ⊂ +NR. Thus the local potential of Φ over the open face factorizes through the +tropicalization map, and is identified with the function φ − 1. +Corollary 3.7. The NA MA measure of Φ is supported on Sk(X), and agrees +with the Lebesgue type measure (Ld)µ0. +Proof. Over any open face Int(∆∨ +m), the convex function φ satisfies the weak +formulation of the real MA equation (6), +det(D2φ) = C0 = ∫∂∆µ dp +∫∂∆∨ +λ dx. +Applying Theorem 3.3, the NA MA measure over these open faces is supported +inside Sk(X) ⊂ Xan, and can be identified as d! times the real MA measure. +We need to figure out the normalisation constants. Tracing through the +conventions (cf. (3)), the NA MA measure is equal to +d!MAR(φ) = d!C0∣dx∣ = (d!C0 ∫∂∆∨ +λ +dx)dL∨ = (d!∫∂∆µ +dp)dL∨ = (d!∫∂∆µ +dp)µ0. +(11) +Claim 3.8. The factor d!∫∂∆µ dp = (Ld). +To see the claim, we start with the asymptotic Riemann-Roch formula +h0(Xt,kL) ∼ (Ld) +d! kd, +k ≫ 1. +On the other hand, by the short exact sequence on the toric Fano manifold X∆ +0 → kL ⊗ O(−Xt) → kL → kL∣Xt → 0 +together with the Serre vanishing of h1(Xt,kL ⊗ O(−Xt)) for k ≫ 1 (since L is +ample), +h0(Xt,kL) = h0(X∆,kL)−h0(X∆,kL⊗O(−Xt)) = h0(X∆,kL)−h0(X∆,kL⊗O(−X0)). +21 + +Now the global sections of H0(X∆,kL) are spanned by the monomial sections, +which correspond to the integral points on k∆µ, while H0(X∆,kL⊗O(−X0)) are +spanned by those monomials which vanish on the toric boundary of X∆. Hence +their difference counts the integral points on ∂(k∆µ). Now these correspond to +the rational points in +1 +k ∂(k∆µ) ∩ 1 +k Zd+1 = ∂∆µ ∩ 1 +k Zd+1. +and as k → +∞, they equidistribute with respect to the Lebesgue measure on +∂∆µ. Hence +h0(Xt,kL) = #(∂∆µ ∩ 1 +k Zd+1) ∼ kd ∫∂∆µ +dp. +Comparing the asymptotes proves the claim. +Applying the claim to (11), we see the identification of the NA MA measure +with (Ld)µ0, over all the open faces of Sk(X) ≃ ∂∆∨. On the other hand, the +total measure of the semipositive metric Φ on L → Xan is (Ld). This forces the +open faces to contain the full measure. +Now by the uniqueness of the NA MA equation, the semipositive metric Φ +must agree with the Boucksom-Favre-Jonsson solution ∥⋅∥CY . Since Φ is known +to satisfy the weak comparison property, by applying Thm. 3.4 we deduce that +the SYZ conjecture holds for the family Xt, whence Theorem 1.2 is proved. +References +[1] Rolf Andreasson, Jakob Hultgren. in preparation. +[2] Boucksom, S´ebastien; Favre, Charles; Jonsson, Mattias. Singular semipos- +itive metrics in non-Archimedean geometry. J. Algebraic Geom. 25 (2016), +no. 1, 77–139. +[3] Boucksom, S´ebastien; Jonsson, Mattias. Tropical and non-Archimedean lim- +its of degenerating families of volume forms. J. ´Ec. polytech. Math. 4 (2017), +87–139. +[4] Boucksom, S´ebastien; Favre, Charles; Jonsson, Mattias. Solution to a non- +Archimedean Monge-Amp`ere equation. J. Amer. Math. Soc. 28 (2015), no. +3, 617–667. +[5] Boucksom, +S´ebastien; +Favre, +Charles; +Jonsson, +Mattias. +The +non- +Archimedean Monge-Amp`ere equation. Nonarchimedean and tropical geom- +etry, 31–49, Simons Symp., Springer, [Cham], 2016. +[6] Caffarelli, L. A. A localization property of viscosity solutions to the Monge- +Amp`ere equation and their strict convexity. Ann. of Math. (2) 131 (1990), +no. 1, 129–134. +[7] Caffarelli, Luis A. Interior W 2,p estimates for solutions of the Monge-Amp`ere +equation. Ann. of Math. (2) 131 (1990), no. 1, 135–150. +22 + +[8] Caffarelli, Luis A. A note on the degeneracy of convex solutions to Monge +Amp`ere equation. Comm. Partial Differential Equations 18 (1993), no. 7-8, +1213–1217. +[9] Chambert-Loir, Antoine. Heights and measures on analytic spaces. A survey +of recent results, and some remarks. Motivic integration and its interactions +with model theory and non-Archimedean geometry. Volume II, 1–50, London +Math. Soc. Lecture Note Ser., 384, Cambridge Univ. Press, Cambridge, 2011. +[10] Keita Goto, Yuji Odaka. Special Lagrangian fibrations, Berkovich retrac- +tion, and crystallographic groups. arXiv:2206.14474. +[11] Gross, Mark. Mirror symmetry and the Strominger-Yau-Zaslow conjecture. +Current developments in mathematics 2012, 133–191, Int. Press, Somerville, +MA, 2013. +[12] Gross, Mark; Wilson, P. M. H. Large complex structure limits of K3 sur- +faces. J. Differential Geom. 55 (2000), no. 3, 475–546. +[13] Guti´errez, Cristian E. The Monge-Amp`ere equation. Progress in Nonlin- +ear Differential Equations and their Applications, 89. Birkh¨auser/Springer, +[Cham], 2016. xiv+216 pp. +[14] Jakob Hultgren, Mattias Jonsson, Enrica Mazzon, Nicholas McCleerey. +Tropical and non-Archimedean Monge-Amp`ere equations for a class of +Calabi-Yau hypersurfaces. arXiv:2208.13697. +[15] Kontsevich, +Maxim; +Soibelman, +Yan. +Affine +structures +and +non- +Archimedean analytic spaces. The unity of mathematics, 321–385, Progr. +Math., 244, Birkh¨auser Boston, Boston, MA, 2006. +[16] Li, Y. SYZ geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa +type metrics. arXiv:1902.08770. accepted by AMS Memoir. +[17] Li, Yang. Strominger-Yau-Zaslow conjecture for Calabi-Yau hypersurfaces +in the Fermat family. Acta Math. 229 (2022), no. 1, 1–53. +[18] Li, +Y. +Metric +SYZ +conjecture +and +non-archimedean +geometry. +arXiv:2007.01384. accepted by Duke. +[19] Li, +Y. +Uniform +Skoda +integrability +and +Calabi-Yau +degeneration. +arXiv:2006.16961. +[20] Li, Y; Tosatti, Valentino. Diameter bounds for degenerating Calabi-Yau +metrics. arXiv:2006.13068. accepted by JDG. +[21] Li, Yang. Survey on the metric SYZ conjecture and non-Archimedean ge- +ometry. Internat. J. Modern Phys. A 37 (2022), no. 17, Paper No. 2230009, +44 pp. +[22] Mooney, Connor. Partial regularity for singular solutions to the Monge- +Amp`ere equation. Comm. Pure Appl. Math. 68 (2015), no. 6, 1066–1084. +[23] Mooney, Connor. Solutions to the Monge-Amp`ere equation with polyhedral +and Y-shaped singularities. J. Geom. Anal. 31 (2021), no. 10, 9509–9526. +23 + +[24] Nicaise, Johannes; Xu, Chenyang. The essential skeleton of a degeneration +of algebraic varieties. Amer. J. Math. 138 (2016), no. 6, 1645–1667. +[25] Nicaise, Johannes; Xu, Chenyang; Yu, Tony Yue. The non-archimedean +SYZ fibration. Compos. Math. 155 (2019), no. 5, 953–972. +[26] Odaka, Yuji; Oshima, Yoshiki. Collapsing K3 surfaces and Moduli com- +pactification. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 8, 81–86. +[27] L´eonard Pille-Schneider. Hybrid toric varieties and the non-archimedean +SYZ fibration on Calabi-Yau hypersurfaces. arXiv:2210.05578. +[28] Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric. Mirror symmetry is +T -duality. Nucl.Phys.B479:243-259,1996. +[29] Vilsmeier, Christian. A comparison of the real and non-archimedean +Monge–Amp`ere operator. Mathematische Zeitschrift (2020). +[30] Yau, Shing Tung. On the Ricci curvature of a compact K¨ahler manifold +and the complex Monge-Amp`ere equation. I. Comm. Pure Appl. Math. 31 +(1978), no. 3, 339–411. +24 + diff --git a/bdFPT4oBgHgl3EQfAzS0/content/tmp_files/load_file.txt b/bdFPT4oBgHgl3EQfAzS0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a568bc2f8845ea3a3aa54b67e099b375bb38b2bd --- /dev/null +++ b/bdFPT4oBgHgl3EQfAzS0/content/tmp_files/load_file.txt @@ -0,0 +1,923 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf,len=922 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='12983v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='DG] 30 Jan 2023 Metric SYZ conjecture for certain toric Fano hypersurfaces Yang Li January 31, 2023 Abstract We prove the metric version of the SYZ conjecture for a class of Calabi- Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Amp`ere equation on certain polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This does not rely on discrete symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1 Introduction The metric aspect of the Strominger-Yau-Zaslow (SYZ) conjecture [28] asks for the following: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [18][21][28]) Let Xt be a 1-parameter maximally degener- ate family of polarized d-dimensional Calabi-Yau manifolds over the punctured disc D∗ t , then for any given 0 < δ ≪ 1, and all t small enough depending on δ, there exists a special Lagrangian T d-fibration on an open subset of Xt for 0 < ∣t∣ ≪ 1, whose normalized Calabi-Yau measure is at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The case of Abelian varieties is classical, while the case of K3 surfaces is known through the trick of hyperk¨ahler rotation [12][26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The first nontrivial example in all dimensions is the Fermat family of Calabi-Yau hypersurfaces [17] Xs = {Z0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Zd+1 + e−s d+1 ∑ 0 Zd+2 i = 0} ⊂ CPd+1, s ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The proof of the SYZ conjecture in this case exploits the large discrete symmetry group, to simplify some rather delicate combinatorial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A turn of thinking came about in [18], where we reduced the SYZ conjecture to a problem of algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In brief, there is a non-archimedean (NA) analogue of the complex Monge-Amp`ere (MA) equation, defined in terms of intersection theory, and the deep work of Boucksom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [4][3][2][5] proved the existence and uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' To deduce the SYZ conjecture, what is left to prove is a ‘comparison property’ [18], which morally asserts the NA MA solution is not too wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1 For this new strategy to be convincing, it is requisite to verify the compar- ison property in examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The classical case of Abelian varieties was checked by Goto-Odaka [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The recent work of Pille-Schneider [27] and Hultgren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14] independently verified the comparison property for some generalisation of the Fermat family, but both works still rely heavily on the large discrete sym- metry group, and thus still only apply to certain hypersurfaces inside CPd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A notable contribution in [14] is to introduce a global variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The existence and uniqueness of the minimizer is quite transparent, but the local PDE nature of the minimizer is not manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' It is essentially known in [14] that if one can deduce a local real Monge-Amp`ere equation from the minimizer, then it would naturally produce the unique solution of the NA MA solution, and the ‘comparison property’ would be a corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The present paper aims to make progress on this strategy, by proving the expected properties of the minimizer for more examples without explicit reliance on the discrete symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let ∆ ⊂ MR be an integral reflexive Delzant polytope, and X∆ be the asso- ciated smooth toric Fano manifold of dimension d+1, with an ample polarization L → X∆, which needs not be the anticanonical polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The origin 0 ∈ ∆ corresponds to a distinguished section Xcan ∈ H0(X∆,−KX∆), which defines the toric boundary of X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let F ∈ H0(X∆,−KX∆) be a generic section, and by assumption the divisor {F = 0} ⊂ X∆ is smooth, and intersects all the toric boundary strata transversely, and in particular does not pass through the finite number of toric fixed points on X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We will consider the family of Calabi-Yau hypersurfaces as t → 0: Xt = {Xcan + tF = 0} ⊂ X∆ × C∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (1) Algebro-geometrically Xt degenerate to the toric boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We will assume one extra property on the reflexive polytope ∆, whose dual polytope is denoted as ∆∨: ⟨m,n⟩ ≠ 0, ∀m ∈ vertex(∆), ∀n ∈ vertex(∆∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (2) The main outcome is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The SYZ conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1 holds for the family Xt as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The main new content of this paper is contained in section 2, which es- tablishes a structure theory for the minimizer of the variational problem, and has the flavour of convex analysis, variational calculus, and real MA equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This part does not require prior knowledge of K¨ahler geometry, and may be of independent interest to analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In section 3 we briefly sketch the complex geometric aspect towards the application to the SYZ conjecture, relying on the aforementioned results in [17][19][18][14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hultgren informs the author that together with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Andreas- son they have independently found an equivalent condition in terms of optimal transport theory, for the minimizer of the variational problem to admit the expected PDE interpretation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Their upcoming paper will also contain coun- terexamples, demonstrating in particular that some nontrivial condition on the reflexive polytope is necessary in this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2 Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The author is a current Clay Research Fellow based at MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' He would like to especially thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hultgren for pointing out his up- coming work, and for helpful comments on this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' He also thanks S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Boucksom, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Pille-Schneider, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Sun for discussions in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2 The variational problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1 Motivation: A global real Monge-Amp`ere equation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let MR ≃ Zd+1 ⊗Z R and NR = M ∨ R be a pair of dual vector spaces, and let ∆ ⊂ MR be an integral reflexive polytope, with dual polytope ∆∨ ⊂ NR, so that ∆∨ = {x ∈ NR∣⟨m,x⟩ ≤ 1,∀m ∈ vertex(∆)}, ∆ = {p ∈ MR∣⟨n,p⟩ ≤ 1,∀n ∈ vertex(∆∨)}, both ∆,∆∨ contain the origin, and all the vertices are integral points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Given neg- ative real numbers µ(n) for all n ∈ vertex(∆∨), and λ(m) for all m ∈ vertex(∆), we can define the convex functions on NR and MR, Lλ(x) = max m ⟨x,m⟩ + λ(m), Lµ(p) = max n ⟨p,n⟩ + µ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' These in turn define the convex polytopes containing the origin in the interior, ∆∨ λ = {Lλ(x) ≤ 0} ⊂ NR, ∆µ = {Lµ(p) ≤ 0} ⊂ MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For instance, the choice λ = −1 recovers ∆∨, and the choice µ = −1 recovers ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The normal vectors to the top dimensional faces are labelled by the vertices m,n, which are primitive integral vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This integral structure on the faces induce a canonical Lebesgue measure dx and dp on ∂∆∨ λ and ∂∆µ respectively, which we normalize to probability measures dL∨ = 1 ∫∂∆∨ λ dxdx, dL = 1 ∫∂∆µ dpdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (3) We shall use the notation ∣E∣ to denote the measure of subsets E with respect to dL or dL∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Our secret goal is to formulate and solve a version of the real MA equation on ∂∆∨ λ and ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A direct attempt faces the following difficulties: The polytopes do not have natural global affine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' On the overlap of the candidate charts, the transition functions will only be piecewise affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Consequently, the condition of being a convex function depends on local charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The definition of the real MA equation via local charts will not be com- patible with piecewise affine transition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' An insight of Hultgren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14], carried out in a very symmetric special case, is to adopt instead a global variational formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The price is that the PDE nature of the problem is not manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2 Legendre transform on polytopes We define the Legendre transforms (referred to as ‘c-transforms’ in [14], which comes from optimal transport nomenclature), L∞(∂∆µ) → L∞(∂∆∨ λ), L∞(∂∆∨ λ) → L∞(∂∆µ), such that for φ ∈ L∞(∂∆∨ λ) and ψ ∈ L∞(∂∆µ), ψ∗(x) = sup p∈∂∆µ ⟨x,p⟩ − ψ(p), φ∗(p) = sup x∈∂∆∨ λ ⟨x,p⟩ − φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (4) The theory works symmetrically for φ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Some of the immediate formal properties are listed below (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14, section 3]): Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (Equicontinuity) The Legendre transforms φ∗,ψ∗ have uniformly bounded Lipschitz constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (Involution) For arbitrary φ ∈ L∞(∂∆∨ λ), the double Legendre transform φ∗∗ ≤ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If φ = ψ∗ for some ψ, then φ∗∗ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (Monotonicity) If φ1 ≥ φ2, then φ∗ 1 ≤ φ∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any constant c ∈ R, we have (φ + c)∗ = φ∗ − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Morever ∥φ∗ 1 − φ∗ 2∥C0 ≤ ∥φ1 − φ2∥C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We then introduce a class of functions P ⊂ C0(∂∆∨ λ) and P∨ ⊂ C0(∂∆µ), defined as the images of L∞ under the Legendre transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the involution property, the Legendre transform sets up a canonical isomorphism P ≃ P∨, which is isometric with respect to the C0-norms by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The formula φ(x) = max p∈∂∆µ⟨x,p⟩ − φ∗(p), φ∗(p) = max x∈∂∆∨ λ ⟨x,p⟩ − φ(x) provides a canonical extension of functions in P,P∨ to convex functions on the ambient Euclidean spaces NR and MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We can thus think of P,P∨ as ‘global convex functions’, which in particular have gradient contained in ∆µ,∆∨ λ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3 Conjugate sets and gradients We begin with a classical analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In the classical Legendre transform theory for convex functions on Rn, for a given position variable x, the gradient ∇φ(x) is recovered by the conjugate momentum p achieving φ(x) = ⟨x,p⟩ − φ∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In the global context of polytopes, finding a reasonable analogue of the conjugate p to x is a rather delicate question, and there are at least two natural notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Given a function φ ∈ P, let x ∈ ∂∆∨ λ, then the conjugate set is ¯∇φ(x) ∶= {p ∈ ∂∆µ∣φ∗(p) + φ(x) = ⟨x,p⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For E ⊂ ∂∆∨ λ, the conjugate set is ¯∇φ(E) ∶= {p ∈ ∂∆µ∣φ∗(p) + φ(x) = ⟨x,p⟩, for some x ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We leave the reader to define the Legendre dual version ¯∇φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 4 In the polytope setting, there is a more subtle new notion, which better captures the classical intuition of gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' To motivate this, we first delve a little into polyhedral geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The top dimensional faces ∆∨ m of ∂∆∨ λ are labelled by the vertices m of ∆, and concretely ∆∨ m = {⟨x,m⟩ = −λ(m)} ⊂ ∂∆∨ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Likewise the vertices n of ∆∨ label the faces ∆n ⊂ ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Given φ ∈ P, we can restrict φ to a convex function on Int(∆∨ m), which is just a convex domain in some Euclidean hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The classical gradient of this restricted function naturally lies in the quotient space MR/Rm, and a moment of thought reveals the gradient is contained in the image of ∆µ under the quotient map MR → MR/Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Each fibre under the quotient map ∆µ ⊂ MR → Image(∆µ) ⊂ MR/Rm intersects each of the following two unions of faces H+ m,H− m ⊂ ∂∆µ uniquely at one point: H+ m = ⋃{∆n ∶ vertex n ∈ ∂∆∨ with ⟨n,m⟩ = 1}, H− m = ⋃{∆n ∶ vertex n ∈ ∂∆∨ with ⟨n,m⟩ ≤ −1}, In particular H+ m,H− m each projects homeomorphically onto Image(∆µ) ⊂ MR/Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We leave the reader to write down the Legendre dual statement involving the self explanatory notations H+ n,H− n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For every point p ∈ ∆µ, the line p + Rm intersects the convex polytope ∆µ in a line segment, with two endpoints (unless we are in the degenerate setting where the line segment becomes a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now the vertices n prescribe the normal vectors to the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In order for the line segment to exit ∆µ in the positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' negative) m direction, we need ⟨n,m⟩ > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By assumption, all n,m are integral vectors, so ⟨m,n⟩ is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Furthermore by assumption ⟨m,n⟩ ≤ 1 for any choice of vertices m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' It is desirable that the gradient p at x is contained in H+ m so that it has a unique natural identification with a point in MR/Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let x ∈ ∂∆∨ λ, which may lie on possibly several faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then the gradient set of x is the intersection ∇φ(x) = ¯∇φ(x) ∩ ⋂ m {H+ m ∶ x ∈ ∆∨ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We say x has anomalous conjugate points if ¯∇φ(x) ∖ ∇φ(x) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Like- wise we define ∇φ(E) for subsets E, and the Legendre dual versions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If p ∈ ¯∇φ(x) is a conjugate point, then tautologically x ∈ ¯∇φ∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' However p ∈ ∇φ(x) does not automatically imply x ∈ ∇φ∗(p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' one sufficient condition is that p lies on only one face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 5 While conjugate points exist quite obviously by the definition of the Leg- endre transform, the existence of at least one gradient p for any given x requires a little more thought: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If x ∈ ∂∆∨ λ has an anomalous conjugate point p′, then it also admits a gradient p ∈ ∇φ(x) (so in particular gradients always exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Suppose x lies on the intersection of some faces ∆∨ mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the defi- nition of conjugate point φ∗(p′) + φ(x) = ⟨x,p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Consider the polyhedral set (p′ + ∑i R≥0mi) ∩ ∆µ, which must have some extremal point p such that (p + ∑i R≥0mi) ∩ ∆µ consists of only one point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then for each mi, there must be some ni with ⟨mi,ni⟩ > 0 such that p ∈ ∆ni, hence p ∈ ⋂H+ mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We claim p is also a conjugate point of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We write p = p′ + ∑ simi with si ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now since φ∗ as a function on MR has gradient contained inside ∆∨ λ, we have φ∗(p) − φ∗(p′) ≤ max y∈∆∨ λ ⟨p − p′,y⟩ ≤ ∑si max y∈∆∨ λ ⟨mi,y⟩ ≤ −∑siλ(mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hence φ∗(p)+φ(x) ≤ φ∗(p′)+φ(x)−∑siλ(mi) = ⟨x,p′⟩−∑siλ(mi) = ⟨x,p′+∑simi⟩ = ⟨x,p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This combined with the tautological inequality φ∗(p) + φ(x) ≥ ⟨x,p⟩ shows that the equalities hold, so p is a conjugate point, hence a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The arguments above will not work if we attempt to define the gradient to lie in H− m instead of H+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In fact, very often there is no conjugate point in H− m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let p ∈ ¯∇φ(x), such that there is an anomalous conjugate point p′ with p = p′ + ∑ simi and si ≥ 0 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then for each mi with x ∈ ∆∨ mi, either si = 0 or ¯∇φ∗(p) ⊂ ∆∨ mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since the equalities are achieved in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='9, we have φ∗(p) − φ∗(p′) = −∑siλ(mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any y ∈ ∂∆∨ λ, we have φ(y) ≥ ⟨y,p′⟩ − φ∗(p′) = ⟨y,p′⟩ − φ∗(p) − ∑siλ(mi) = ⟨y,p⟩ − φ∗(p) − ∑si(λ(mi) + ⟨y,mi⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now suppose y is any conjugate point of p, then φ(y) = ⟨y,p⟩ − φ∗(p), whence ∑si(λ(mi) + ⟨y,mi⟩) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By construction λ(mi) + ⟨y,mi⟩ ≤ 0 for any y ∈ ∆∨ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any i, as long as si > 0, then ⟨y,mi⟩ = −λ(mi), namely y ∈ ∆∨ mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In the special case that x lies on the interior of one face, then there is only one m = mi involved, with p = p′ + sm, and since p ≠ p′, we are forced to have s > 0, and ¯∇φ∗(p) ⊂ ∆∨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (Comparison of gradient notions) Let x ∈ Int(∆∨ m), then p ∈ H+ m is a gradient in ∇φ(x), if and only if p ∈ MR/Rm under the identification H+ m ≃ Image(∆µ) ⊂ MR/Rm is a classical gradient of the convex function φ on Int(∆∨ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If p ∈ ∇φ(x), then restricted to Int(∆∨ m) we have φ(y) ≥ ⟨y,p⟩ − φ∗(p) = φ(x) + ⟨y − x,p⟩, ∀y ∈ Int(∆∨ m), which means p is a classical gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For the converse, we suppose p ∈ MR/Rm is a classical gradient at x, which is identified uniquely with p ∈ ∆n ⊂ H+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since ∆∨ λ lies in the half space ⟨m,x⟩+ λ(m) ≤ 0, and ⟨n,m⟩ = 1, we can write any y ∈ ∂∆∨ λ as y = y′ − sn, s ≥ 0, ⟨m,y′⟩ + λ(m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Recall φ has a canonical extension to a convex function on NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Restricted to the hyperplane ⟨m,⋅⟩ + λ(m) = 0, the classical gradient property implies φ(y′) ≥ φ(x) + ⟨y′ − x,p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' But since the convex function φ on NR has gradient contained in ∆µ, φ(y′) ≤ φ(y) + max ∆µ ⟨y′ − y,⋅⟩ ≤ φ(y) − sµ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Combining the above, φ(y) ≥ φ(x) + ⟨y′ − x,p⟩ + sµ(n) = φ(x) + ⟨y′ − x − sn,p⟩ = φ(x) + ⟨y − x,p⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since y is arbitrary, we deduce φ(x) + φ∗(p) = ⟨x,p⟩, namely p ∈ ¯∇φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since x lies on only one face, and p ∈ H+ m, we conclude p ∈ ∇φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='4 Variational problem We now set up a global functional on P ≃ P∨: F(φ) = ∫∂∆∨ λ φdL∨ + ∫∂∆µ φ∗dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (5) This functional is manifestly symmetric in terms of the Legendre transform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' this ‘mirror symmetry’ is fundamental to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In this variational problem, the existence of a minimizer is quite transparent, but the local PDE nature of the minimizer is less obvious, since the definition of the Legendre transform is not local, and the differentiability of the functional at the critical point is not a priori clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The functional admits a minimizer in P, henceforth denoted as φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2]) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3 and the fact that dL,dL∨ are both probability measures, for any c ∈ R we have F(φ + c) = F(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This allows us to impose without loss that max∂∆∨ λ φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then by the uniform Lipschitz estimate in P, we can envoke Arzela-Ascoli to take a C0 limit of a minimizing sequence of the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3, the Legendre transformed functions also converge in C0, so the limit attains the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The minimizer is unique up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (compare [14, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2]) Suppose ˜φ is another minimizer, and we consider φ′ = 1 2(φ+ ˜φ), which still lies in P by the convexity of the function class P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now for any p ∈ ∂∆µ, φ′∗(p) = max⟨x,p⟩ − φ′(x) ≤1 2 max(⟨x,p⟩ − φ(x)) + 1 2 max(⟨x,p⟩ − ˜φ(x)) = 1 2(φ∗(p) + ˜φ∗(p)), whence F(φ′) ≤ 1 2(F(φ)+F(˜φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This forces φ′ to be also a minimizer, and the equality is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If p ∈ ∇φ′(x), then the equality φ′∗(p) = 1 2(φ∗(p) + ˜φ∗(p)) implies (φ(x) + φ∗(p) − ⟨x,p⟩) + (˜φ(x) + ˜φ∗(p) − ⟨x,p⟩) = 0, which forces p ∈ ∇φ(x) and p ∈ ∇˜φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When we restrict to any open face Int(∆∨ m) of ∂∆∨ λ, the functions φ, ˜φ are convex functions on convex domains, and ∇φ(x),∇˜φ(x) are identified with the classical gradients of convex functions, which naturally lie inside the quotient space MR/Rm (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since ∇φ = ∇˜φ Lebesgue-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='e, the difference φ − ˜φ is a constant on the face (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14, Lem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the continuity of the functions, and matching the functions on the intersection of different faces, we see that the constant must be independent of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='5 Variational inequality We now derive a fundamental variational inequality for the minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any measurable E ⊂ ∂∆∨ λ, we have ∣¯∇φ(E)∣ ≥ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Completely analogously ∣¯∇φ∗(E)∣ ≥ ∣E∣ for any measurable subset E ⊂ ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We consider the function φ − t1E for 0 < t ≪ 1, where 1E denotes the characteristic function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3, φ∗ ≤ (φ − t1E)∗ ≤ φ∗ + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Morever (φ − t1E)∗ > φ∗ at p ∈ ∂∆µ, only when sup x∈E (⟨p,x⟩ − φ(x)) ≥ φ∗(p) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hence t∣{p ∶ sup x∈E (⟨p,x⟩ − φ(x)) ≥ φ∗(p) − t}∣ ≥ ∫∂∆µ ((φ − t1E)∗ − φ∗)dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since φ is a minimizer, plugging in (φ − t1E)∗∗ as a competitor (the caveat being that (φ − t1E) may not lie in P), we get ∫∂∆µ (φ − t1E)∗ − φ∗ ≥ −∫∂∆∨ λ (φ − t1E)∗∗ − φ ≥ −∫∂∆∨ λ (φ − t1E) − φ = t∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Here the second inequality uses Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Combining the above and cancelling the t factor, ∣{p ∶ sup x∈E (⟨p,x⟩ − φ(x)) ≥ φ∗(p) − t}∣ ≥ ∣E∣, ∀0 < t ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 8 Taking the t → 0 limit, and recalling φ∗(p) ≥ ⟨x,p⟩ − φ(x), we get ∣{p ∶ sup x∈E (⟨p,x⟩ − φ(x)) = φ∗(p)}∣ ≥ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For closed subsets E, the sup can be replaced by max by the continuity of φ, so the LHS subset is the conjugate set ¯∇φ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For general Borel subsets E, we take a compact exhaustion of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any compact K ⊂ E, we observe ∣¯∇φ(E)∣ ≥ ∣¯∇φ(K)∣ ≥ ∣K∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since the Lebesgue measure is inner regular, we obtain ∣¯∇φ(E)∣ ≥ ∣E∣ by taking the limit K ↑ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If no point in the measurable subset E ⊂ ∂∆∨ λ has any anoma- lous conjugate point, then ∣∇φ(E)∣ ≥ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If there is no anomalous conjugate point, then ∇φ(E) = ¯∇φ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We then apply the variational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We can now explain the heuristic why the variational problem should be related to the real MA equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In the ideal situation, there is no anomalous conjugate point, and ∇φ,∇φ∗ are both bijective on points, and define mutually inverse maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then ∣∇φ(E)∣ ≥ ∣E∣ = ∣∇φ∗(∇φ(E))∣ ≥ ∣∇φ(E)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This forces equality everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' On each open face of ∂∆∨ λ the gradient agrees with the classical notion in convex function theory, and ∣∇φ(E)∣ = ∣E∣ is just the weak formulation of the real MA equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' To make this argument work more rigorously, one needs to control the anomalous conjugate points, and the multivalued nature of gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='6 Anomalous conjugate points The remaining technical difficulty mainly comes from the possible existence of anomalous conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We introduce the subset S∨ ⊂ ∂∆∨ λ as S∨ = S∨ 1 ∪ S∨ 2 , where S∨ 1 = ⋃{lower dim faces of ∂∆∨ λ}, S∨ 2 = ⋃ m {x∣¯∇φ(x) ∩ H+ m contains at least two points}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular ∇φ(x) is single valued outside S∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Similarly the reader may write down the Legendre dual analogue S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We now recall a basic fact in classical convex function theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The subset S∨ ⊂ ∂∆∨ λ has Lebesgue measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A similar statement holds for S ⊂ ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [13, section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1]) We restrict φ to the the open face Int(∆∨ m), to obtain a classical convex function on a convex domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' As explained in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3, the notion of gradient inside H+ m is naturally identified with the classical notion which takes value in MR/Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since our convex functions are Lipschitz, they are differentiable Lebesgue-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The multi-valued gradient problem hap- pens on the non-differentiability locus, which has measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now the global statement follows by taking the union of all the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The following lemmas say that the anomalous conjugate has controlled effect on the gradient set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In this section, let E be a measurable subset of ∆∨ m, such that any x ∈ E admits some anomalous conjugate point in H− m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let F ′ = ¯∇φ(E)∩H− m, and let F ⊂ H+ m be the image of F ′ under the projection H− m → H+ m (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='9, we have F ⊂ ¯∇φ(E) ∩ H+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ∣F∣ ≤ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By construction every p ∈ F ∖ S is of the form p = p′ + sm with s > 0 and p′ ∈ F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='11 concerning the anomalous conjugate points, we must have ¯∇φ∗(p) ⊂ ∆∨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since p ∈ H+ m ∖ S, it lies on only one face ∆n ⊂ H+ m with ⟨n,m⟩ = 1, whence ∆∨ m ⊂ H+ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We then have ¯∇φ∗(p) ⊂ ∆∨ m ⊂ H+ n, whence ¯∇φ∗(p) = ¯∇φ∗(p) ∩ H+ n = ∇φ∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since p ∉ S, we know that ∇φ∗(p) consists of only one point, so in fact ∇φ∗(p) = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' But p ∈ F ⊂ ¯∇φ(E), so x ∈ E and ¯∇φ∗(p) ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In summary we get the inclusion ¯∇φ∗(F ∖ S) ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now applying the vari- ational inequality Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='16, ∣E∣ ≥ ∣¯∇φ∗(F ∖ S)∣ ≥ ∣F ∖ S∣ = ∣F∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The last equality uses that S has zero measure, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ∣F ′∣ ≤ ∣F∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When the equality is achieved, then F ′ has zero measure except perhaps on the faces ∆n with ⟨n,m⟩ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='5, under the quotient map MR → MR/Rm, the sets H+ m and H− m each project homeomorphically to the image of ∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The projection map is piecewise affine, and we focus on some face ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' With respect to the natural Lebesgue measures dp induced by the integral structure, the quotient map ∆n → MR/Rm has Jacobian factor ∣⟨n,m⟩∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For ∆n ⊂ H+ m this factor is equal to one, and for ∆n ⊂ H− m this factor is a positive integer (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus under the projection H− m → H+ m, the Jacobian factor is ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since the normalized measure dL is a constant multiple of dp, the Jacobian factor is still ≥ 1 with respect to dL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This shows ∣F ′∣ ≤ ∣F∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The equality case forces the Jacobian factor to be equal to one, which means ∣⟨n,m⟩∣ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ∣E∣ = ∣F∣ = ∣F ′∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular the equality forces that F ′ has zero measure except perhaps on the faces ∆n with ⟨n,m⟩ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='20, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='21, we already know ∣E∣ ≥ ∣F∣ ≥ ∣F ′∣, so it suffices to prove ∣F ′∣ ≥ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We write H− m = ⋃ ∆ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For each i, we have the set inclusion 10 ∆∨ m ⊂ H− ni, and since any point of E is conjugate to some point in H− m∩ ¯∇φ(E) = F ′, we conclude E ⊂ ¯∇φ∗(F ′) ∩ ∆∨ m ⊂ ⋃ i ¯∇φ∗(F ′ ∩ ∆ni) ∩ H− ni, whence ∣E∣ ≤ ∑i ∣¯∇φ∗(F ′ ∩ ∆ni) ∩ H− ni∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Recall that our setup has the Legendre duality symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any fixed i, we now let F ′ ∩ ∆ni play the role of E in the Legendre dual version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='20, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The role of F ′ is then played by ¯∇φ∗(F ′∩∆ni)∩H− ni, and we conclude ∣¯∇φ∗(F ′ ∩ ∆ni) ∩ H− ni∣ ≤ ∣F ′ ∩ ∆ni∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Summing over i and combining the above, ∣E∣ ≤ ∑ i ∣¯∇φ∗(F ′ ∩ ∆ni) ∩ H− ni∣ ≤ ∑ i ∣F ′ ∩ ∆ni∣ = ∣F ′∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This is the promised reverse inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The equality forces a very strong rigidity, and it is an interesting question whether it forces ∣E∣ = 0 in fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ∣E∣ = ∣∇φ(E) ∩ F∣ ≤ ∣∇φ(E)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since ∣E∣ = ∣F∣ by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='22, it suffices to prove F ∖ ∇φ(E) ∪ S has measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now every point in p ∈ F ∖S has a unique gradient x = ∇φ∗(p) ∈ E, and if x ∉ S∨ then p = ∇φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This shows that F ∖ ∇φ(E) ∪ S ⊂ Image of (¯∇φ(S∨ ∩ E) ∩ H− m) under H− m → H+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By applying Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='22 again to E ∩ S∨, we see ∣F ∖ ∇φ(E) ∪ S∨∣ ≤ ∣Image(¯∇φ(S∨ ∩ E) ∩ H− m)∣ = ∣E ∩ S∨∣ = 0, so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='7 Anomalous conjugate points II In this section we aim to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any E ⊂ ∂∆∨ λ, we have ∣∇φ(E)∣ ≤ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Assume the subset E ⊂ ∆∨ m has the property that any x ∈ E admits some conjugate in a face ∆n with ⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then ∣E ∩ ∇φ∗(∂∆µ)∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Consider a point x ∈ (E∩∇φ∗(∂∆µ))∖S∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since x ∈ ∇φ∗(∂∆µ), we write x = ∇φ∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since x does not lie on S∨, it has a unique gradient, so p = ∇φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now by assumption x has some anomalous conjugate p′ in a face ∆n with ⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='14, there exists a gradient of the form p′ +sm for s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The uniqueness of the gradient then forces p = p′ + sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since p′ ∈ ∆n, we have ⟨p′,n⟩ + µ(n) = 0, hence ⟨p,n⟩ + µ(n) = ⟨p − p′,n⟩ = s⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 11 This means p ∈ ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since x lies on ∆∨ m but not on S∨, it must be in the interior of ∆∨ m, so by the definition of gradient x = ∇φ∗(p) forces ⟨m,n⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This contradicts ⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This contradiction shows that x cannot exist, which means (E ∩ ∇φ∗(∂∆∨ µ)) ∖ S∨ is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since S∨ has measure zero by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='19, we see ∣E ∩ ∇φ∗(∂∆µ)∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We also record the Legendre dual statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Suppose the subset G ⊂ ∆n has the property that any p ∈ G admits a conjugate point in ∆∨ m with ⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then ∣G ∩ ∇φ(∂∆∨ λ)∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We can now prove Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25) First, we notice it suffices to prove Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25 only for E ⊂ ∆∨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This is because we can decompose E into the union of Em ⊂ ∆∨ m for all the possible m, and once we know ∣∇φ(Em)∣ ≤ ∣Em∣ for every m, it would follow that ∣∇φ(E)∣ ≤ ∑∣∇φ(Em)∣ ≤ ∑∣Em∣ = ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The same argument shows more generally that if we can partition E into a union of subsets, and the intersections have measure zero, then it suffices to prove the statement for the subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We will partition ∇φ(E) ∖ S into several subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let G1 be the union over n of all the p ∈ ∇φ(E) ∩ ∆n ∖ S, which admit an anomalous conjugate point in some ∆∨ m′ with ⟨m′,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='27, its size ∣G1∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let G2 consist of all the p ∈ ∇φ(E) ∖ S which admit an anomalous conju- gate point, but not covered by the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We decompose G2 ⊂ ∇φ(E) ⊂ H+ m into the union of G2,n = G2 ∩ ∆n for all ∆n ⊂ H+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the Legendre dual version of Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='24 applied to G2,n, we have ∣G2,n∣ ≤ ∣∇φ∗(G2,n)∣, whence ∣G2∣ = ∑∣G2,n∣ ≤ ∑∣∇φ∗(G2,n)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Next, for any p ∈ ∇φ(E) ∖ S, we can write p = ∇φ(x) for x ∈ E, and since p is not in S, it has a unique gradient point which is x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus ∇φ(∇φ∗(p)) contains p, and ∇φ∗(p) ⊂ E ⊂ ∆∨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since the subset of points in ∆∨ m with multiple gradients has zero measure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='19), we see that the intersections between the sets ∇φ∗(G2,n) have zero measure, whence ∣G2∣ ≤ ∑∣∇φ∗(G2,n)∣ = ∣∇φ∗(G2)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let G3 consist of all the p ∈ ∇φ(E)∖S with no anomalous conjugate point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Any such p lies outside of S, hence has a unique gradient, which must then be its unique conjugate point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The set ∇φ∗(G3) ⊂ E is the union of these conjugate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the Legendre dual version of Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='17 we have ∣∇φ∗(G3)∣ ≥ ∣G3∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 12 By construction ∇φ(E) ∖ S = G1 ∪ G2 ∪ G3, hence by the above discussions ∣∇φ(E) ∖ S∣ ≤ ∣G1∣ + ∣G2∣ + ∣G3∣ ≤ ∣∇φ∗(G2)∣ + ∣∇φ∗(G3)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The same argument in item 2 above shows that ∣∇φ∗(G2) ∩ ∇φ∗(G3)∣ = 0, so ∣∇φ(E) ∖ S∣ ≤ ∣∇φ∗(G2)∣ + ∣∇φ∗(G3)∣ = ∣∇φ∗(G2 ∪ G3)∣ ≤ ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The last inequality here is because ∇φ∗(G2 ∪G3) ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now since S has measure zero, we get ∣∇φ(E)∣ = ∣∇φ(E) ∖ S∣ ≤ ∣E∣, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='8 Structure theorems for the minimizer In this section we will list certain structural properties of the minimizer φ, which in the best situation lead to the real MA equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The set function E ↦ ∣∇φ(E)∣ defines a Borel measure on ∂∆∨ λ, which is absolutely continuous with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Likewise with the Legendre dual analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since ∣∇φ(E)∣ ≤ ∣E∣ by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25, we see that ∣E∣ = 0 implies ∣∇φ(E)∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Next we justify the countable additivity axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let E = ⋃Ei be a disjoint partition, then clearly ∣∇φ(E)∣ ≤ ∑ ∣∇φ(Ei)∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' To see that the equality holds, without loss we may assume E is disjoint from the measure zero set S∨, so that the gradient is a single valued map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Morever, the mutual intersections of ∇φ(Ei) is contained in S, which again has measure zero, so ∑∣∇φ(Ei)∣ = ∣⋃ i ∇φ(E)∣ = ∣∇φ(E)∣ as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We now introduce a good-bad decomposition on ∂∆∨ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The good set G∨ ⊂ ∂∆∨ λ consists of the following two types of points x ∈ ∂∆∨ λ: Type I good point: x has no anomalous conjugate point, so that all con- jugate points are gradients of x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Type II good point: x is contained in some face ∆∨ m and x admits an anomalous conjugate point p ∈ ¯∇φ(x) ∩ H− m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The bad set B∨ ⊂ ∆∨ λ consists of all x contained in some face ∆∨ m, and which admits some anomalous conjugate point p ∈ ∆n with ⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Clearly ∂∆∨ λ = G∨ ∪ B∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We leave the reader to write down the Legendre dual version ∂∆µ = G ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (Property of good-bad decomposition) If E ⊂ G∨, then ∣∇φ(E)∣ = ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If E ⊂ B∨, then ∣∇φ(E)∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular ∣G∨ ∩ B∨∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since the measure E ↦ ∣∇φ(E)∣ has finite additivity, it suffices to con- sider E of single types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For type I good points, we use ∣∇φ(E)∣ ≤ ∣E∣ from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25 and ∣∇φ(E)∣ ≥ ∣E∣ from Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For type II good points, we use Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='24, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='25 and finite additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now we consider the bad points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By finite additivity, without loss E ⊂ ∆∨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since ∣S∨∣ = ∣∇φ(S∨)∣ = 0, without loss we can suppose E is disjoint from S∨, so every x ∈ E has a unique gradient p = ∇φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By assumption E consists of bad points, so there is an anomalous conjugate p′ ∈ ∆n with ⟨m,n⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='9, the unique gradient is of the form p = p′ + sm for s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus ⟨p,n⟩ = ⟨p′,n⟩ + s⟨n,m⟩ = −µ(n), whence p lies on ∆n as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since p is a gradient of x, it must also lie on H+ m, so it falls into H+ m ∩ ∆n, which has measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' One lesson of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='29 is that ∇φ only sees the good set, and an analogous version holds for ∇φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In fact modulo measure zero sets, these two maps are in some sense inverse to each other, when we restrict to the good set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ( Legendre duality) The maps ∇φ and ∇φ∗ have the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any E ⊂ G∨, we have ∣∇φ(E) ∩ B∣ = 0, and ∣∇φ(E) ∩ G∣ = ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We consider the maps between subsets of G∨ and G defined by ⎧⎪⎪⎨⎪⎪⎩ E ↦ ∇φ(E) ∩ G, ∀E ⊂ G∨, F ↦ ∇φ∗(F) ∩ G∨, ∀F ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The mutual compositions agree with E,F up to a measure zero subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular ∣G∣ = ∣G∨∣ and ∣B∣ = ∣B∨∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The claim that ∣∇φ(E) ∩ B∣ = 0 follows from the Legendre dual version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='28 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='29, we have ∣G ∩ B∣ = 0, and ∣∇φ(E) ∩ G∣ = ∣∇φ(E)∣ − ∣∇φ(E) ∩ B∣ = ∣∇φ(E)∣ = ∣E∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This proves item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Consequently, the two maps in item 2 preserve the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' To prove the composition statement, without loss E is disjoint from the measure zero set S∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For any p ∈ ∇φ(E) ∩ G ∖ S, the ∇φ∗(p) uniquely recovers the point x ∈ E with p = ∇φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus the composition of ∇φ and ∇φ∗ recovers a full measure subset of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This proves item 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now by item 1, 2 applied to G∨, we get ∣G∣ = ∣G∨∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since ∂∆∨ λ = G∨ ∪ B∨ and ∣G∨ ∩ B∨∣ = 0, we have ∣B∣ = 1 − ∣G∣ = 1 − ∣G∨∣ = ∣B∨∣, which proves item 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 14 The most important special case for us is the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The size of the bad set ∣B∣ = 0, or equivalently ∣B∨∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (For instance, this works if ⟨n,m⟩ = 0 never holds for any vertices of ∆,∆∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The images of ∇φ and ∇φ∗ have full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When this holds, then ∣∇φ(E)∣ = ∣E∣ for any E ⊂ ∂∆∨ λ, and likewise with ∇φ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The two maps ∇φ and ∇φ∗ compose to the identity modulo measure zero sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='30, the vanishing of ∣B∣ would force ∣B∨∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus the good sets have full measure, and we know from Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='30 that ∇φ and ∇φ∗ on the good sets are measure preserving, mutually inverse maps modulo zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular the images of ∇φ and ∇φ∗ have full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Conversely, suppose the images of ∇φ and ∇φ∗ have full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='29, ∣∇φ(∂∆∨ λ)∣ = ∣∇φ(G∨)∣ + ∣∇φ(B∨)∣ = ∣G∨∣, which then forces ∣G∨∣ = 1, namely ∣B∨∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This shows that item 2 implies item 1, and the rest follow from Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When the equivalence conditions in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31 is satisfied, then the con- clusions admit classical interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When we restrict φ to an open face Int(∆∨ m) to obtain a convex function still denoted as φ, then ∇φ is identified with the classical gradient (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Recall from (3) how the nor- malized measure is related to the Lebesgue measure induced by the integral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Morever, the projection map H+ m → MR/Rm has Jacobian factor one with respect to the integral Lebesgue measure (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In summary, the fact that ∣∇φ(E)∣ = ∣E∣ with respect to the normalized mea- sures, translates into the weak Alexandrov formulation of the following real MA equation over Int(∆∨ m): det(D2φ) = C0 = ∫∂∆µ dp ∫∂∆∨ λ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (6) Completely analogously, we have the weak Alexandrov formulation of the fol- lowing real MA equation over the open faces Int(∆n): det(D2φ∗) = C−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (7) This expresses the local PDE nature of the variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='9 Some examples We first give some simple examples to which Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 15 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We revisit the standard simplex example [14, section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1][17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We take the standard basis e0,e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ed+1 inside Zd+2, and define M = {p ∈ Zd+2∣∑pi = 0}, N = M ∨ = Zd+2/Z(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1), and MR = M ⊗ R,NR = N ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The polytope ∆ ⊂ MR is the convex hull of mi = (d + 2)ei − d+1 ∑ 0 ej, i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Its dual polytope ∆∨ is also a simplex, with vertices given by n0 = (−1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ,0),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' nd+1 = (0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ,0,−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus ⟨mi,nj⟩ = ⎧⎪⎪⎨⎪⎪⎩ −(d + 1), i = j, 1, i ≠ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular ⟨n,m⟩ = 0 never occurs, and Thm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This example corresponds to hypersurfaces inside CPd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Suppose we have k integral reflexive polytopes ∆i ⊂ MR,i, with dual polytopes ∆∨ i ⊂ NR,i, such that the pairing ⟨m,n⟩ ≠ 0 for all possible vertices m,n of ∆i and ∆∨ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then we can take NR = ⊕iNR,i, MR = ⊕iMR,i, ∆ ⊂ MR, ∆∨ ⊂ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The vertices of ∆∨ are of the form (0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ,ni,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=') with ni ∈ vertex(∆∨ i ), while the vertices of ∆ are of the form (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' mk) for mi ∈ vertex(∆i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' It is im- mediate to check that the pairing ⟨m,n⟩ ≠ 0 still holds for ∆,∆∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Algebro- geometrically, this is just taking the product of several toric Fano varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We now give a simple example which shows that ∣B∣ = 0 might fail if ⟨m,n⟩ = 0 is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This example lives inside R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let ∆ be the convex full of (1,0),(0,1),(−1,1),(−1,0),(0,−1),(1,−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then ∆∨ is the convex hull of ±(1,0),±(1,1),±(0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Notice for instance that m0 = (1,−1) is orthogonal to n0 = (1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We can take ∆∨ λ = ∆∨, and let ∆µ = {(x,y) ∈ R2 ∶ ∣x∣ ≤ 1,∣y∣ ≤ 1,∣x + y∣ ≤ ǫ}, for some given constant 0 < ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When ǫ ≪ 1, then ∆µ is concentrated along a line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We claim that ∣B∣ ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Otherwise by Thm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31, we have ∣∇φ(E)∣ = ∣E∣ for any E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We take E = ∆∨ m, so that by the definition of gradient ∇φ(E) ⊂ H+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus ∣∆∨ m∣ = ∣∇φ(∆∨ m)∣ ≤ ∣H+ m∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' For the choice m = m0 = (1,−1), this leads to a contradition for small enough ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 16 Algebro-geometrically, the corresponding Fano manifold X∆ is the blow up of P1 × P1 at two toric fixed points, so there are two disjoint exceptional divisors E1,E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Adjusting ǫ amounts to changing the K¨ahler class on X∆ in a 1-parameter family, with ǫ = 1 corresponding to c1(π∗ 1[OP1] + π∗ 2[OP1]), and decreasing ǫ corresponding to subtracting multiplies of E1 + E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The family Xt of Calabi-Yau hypersurfaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' the introduction) is here just a pencil of elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The second cohomology of an elliptic curve has rank one, so the restriction of all these K¨ahler classes to the elliptic curve will all be proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A plausible interpretation of the above example, is that once an ample polarization L → X on the degenerating Calabi-Yau hypersurfaces is fixed, there is still some flexibility to extend this polarization to L → X∆, and for an inappropriate choice of L → X∆, the variational problem may not be helpful for the SYZ conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='10 Open questions Our work suggests a number of natural questions, which are relevant for further applications to the SYZ conjecture, and which may be of some independent interest to PDE theorists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31 makes it clear that the failure for the minimizer to satisfy ∣∇φ(E)∣ = ∣E∣ is measured by the size of the bad set ∣B∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Is it possible to compute or estimate ∣B∣ in general, when we allow ⟨m,n⟩ = 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Assume ∣B∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Is it possible for the anomalous conjugate point to exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Assume ∣B∣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When can we prove that the gradient is unique everywhere (instead of Lebesgue-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='e), namely ∇φ(x) consists of only one point for every x ∈ ∂∆∨ λ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The Legendre dual version of the above question asks if ∇φ∗(p) consists of only one point for every p ∈ ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This question is closely related to the strict convexity of the restriction φ to a convex function on the open faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This has implication on the regularity question of φ, since a strictly convex solution of the real MA equation is known to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If both ∇φ and ∇φ∗ are single valued, they would define mutually inverse maps setting up a homeomorphism between ∂∆∨ λ and ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This global version of Legendre duality is desirable from the viewpoint of potential applications to mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The polytopes ∂∆∨ λ and ∂∆µ have no natural global smooth structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' the only a priori information is that the open faces Int(∆∨ m) and Int(∆n) have natural affine structures, and in particular smooth structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now suppose the previous questions have positive answers, so that φ and φ∗ are smooth on the respective open faces, and the gradient maps set up a global homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then we can transfer Int(∆n) to the open subset ∇φ∗(Int(∆n)) ⊂ ∂∆∨ λ, and on the overlap with the open faces of ∂∆∨ λ the smooth structures are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This suggests that there is 17 a hidden smooth structure with singularity on ∂∆∨ λ, where the singular locus lies on the subset of the lower dimensional faces of ∂∆∨ λ, which maps under ∇φ to the lower dimensional faces of ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Morever, this smooth structure with singularity is compatible with Legendre duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' It would be desirable for mirror symmetry (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' the Kontsevich-Soibelman conjecture [15]) that the singular locus is of Hausdorff codimension two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hultgren informs the author that their upcoming work [1] will contain more discussions on the affine structure with singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3 Application to the SYZ conjecture For backgrounds on the non-archimedean (NA) geometry, and the previous lit- erature on its relation to K¨ahler geometry, the reader may refer to [5] [27, section 1][18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A recent survey on the progress in the metric aspects of the SYZ conjecture can be found in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We now sketch how the result of section 2 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This argument is essentially known, and consists of assembling ingredients from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1 Complex geometry setup We work in the context of the introduction, so X∆ is a smooth toric Fano manifold associated to the reflexive Delzant polytope ∆, and X = ∪Xt (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (1)) is a union of Calabi-Yau hypersurfaces degenerating to the toric boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This degeneration family admits a natural model X = {Xcan + tF = 0} ⊂ X∆ × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Upon base change, X can be regarded as a family over SpecC((t)), and X defines a model over SpecC[[t]], which is divisorial log terminal (dlt) since by assumption the singularity structure on the central fibre is ´etale locally of the form (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14, Page 25]) X ≃ V (x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' xk − ty) ⊂ Ak+2 xi,t,y × Ad−k, d = dimC Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The central fibres is identified with the toric boundary of X∆, and consists of many divisors labelled by the vertices n ∈ ∆∨, each with multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The dual complex ∆X of the dlt model ∆X has vertices corresponding to these divisors, and the tropicalization map gives a natural identification trop ∶ ∆X ≃ ∆∨ ⊂ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The model X is minimal, meaning that the logarithmic relative canonical bundle is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' As a caveat, our dlt model X is not Q-factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Any ample line bundle L → X∆ corresponds to a moment polytope ∆µ for some choice of µ(n) < 0 for n ∈ vertex(∆∨), such that the piecewise linear function µ ∶ NR → R extending the values µ(n) is a ‘strictly concave’ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This induces a polarization line bundle L → X, which in particular prescribes the K¨ahler class c1(L) on Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We denote (Ld) = ∫Xt c1(L)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 18 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The concavity condition on µ is related to ampleness, and if we drop it, the line bundle L → X∆ will only be big, even though the induced line bundle L → X might still be ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' On the other hand, in section 2 this condition is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Once we drop it, then for some n ∈ vertex(∆∨) the face ∆n may be empty, but this does not affect the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By the adjunction formula, X over some small punctured disc admits a nowhere vanishing holomorphic volume form Ω, which induces holomorphic vol- ume forms Ωt on the hypersurfaces Xt, hence normalized Calabi-Yau measures on Xt µt = Ωt ∧ Ωt ∫Xt Ωt ∧ Ωt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (8) The metric SYZ conjecture concerns the Calabi-Yau metrics (Xt,ωt) in the class c1(L), satisfying the complex MA equation ωd t = (Ld)µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (9) The SYZ conjecture predicts that given any small 0 < δ ≪ 1, for all t small enough depending on δ, then (Xt,ωt) contains an open subset with µt-measure at least 1 − δ, which admits a special Lagrangian torus fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2 Non-archimedean geometry The work of Boucksom-Jonsson [4] established the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The fam- ily X viewed as a smooth projective variety over SpecC((t)) gives rise to the Berkovich space Xan, which can be regarded as the limit of Xt as t → 0 un- der the hybrid topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The Berkovich space contains the essential skeleton Sk(X), and under the hybrid topology convergence, the family of probability measures µt converge to a Lebesgue type probability measure µ0 supported on Sk(X) ⊂ Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This has a concrete description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Given a dlt model X, there is a canonical embedding of its dual complex ∆X into Xan, and if X is minimal, then emb ∶ ∆X ≃ Sk(X) ⊂ Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In our case of Calabi-Yau hypersurfaces, under the canonical isomorphisms ∆X emb ��→ Sk(X) trop ��→ ∂∆∨ ⊂ NR, the probability measure µ0 is up to a global constant the Lebesgue measure dx induced by the integral structure on the open faces of ∂∆∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' When the dust settles, µ0 = dL∨ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (3)), where we choose λ = −1 so that ∆∨ λ = ∆∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Given the model X, every point on Xan has a centre cX (x) which is a scheme theoretic point in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The map cX ∶ Xan → X0 is anticontinuous, which means the preimage of closed subsets of X0 are open subsets of Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' In particular, the preimage of the finitely many toric fixed points in X0 is an open subset U ⊂ Xan, and the evaluation of the logarithmic coordinates provides a natural retraction map from U to the open faces of ∆X , which is identified with the tropicalization map U → ∂∆∨ ⊂ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 19 Boucksom-Favre-Jonsson [3][2][5] developed a NA pluripotential theory, which concerns the analogue of semipositive metrics and Monge-Amp`ere mea- sures on Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Its central result concerns the solution of the NA Calabi conjec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The special case relevant to us is the following: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Let L → X be an ample line bundle, then there is a unique up to constant semipositive metric ∥⋅∥CY on L → Xan, whose NA MA measure is equal to the Lebesgue measure (Ld)µ0 supported on Sk(X) ⊂ Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' It is convenient to think about a semipositive metric ∥⋅∥ in terms of a potential function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Concretely in our case, we can choose any local trivialising section s of the line bundle L → X∆, and evaluate ϕ = −log ∥s∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We say ∥⋅∥ satisfies the ‘weak comparison property’, if under the retraction map from U to the open faces of ∆X , the potential ϕ is constant on fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Notice that the pair (X,X0) is a semistable SNC pair near the toric fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Vilsmeier [29] implies that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [29][14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1][27, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='17] Under the weak comparison property, then the NA MA measure MANA on U ⊂ Xan is related to the real MA measure MAR on the open faces of ∆X ≃ Sk(X) ≃ ∂∆∨, by the equation 1Sk(X)MANA(∥⋅∥) = d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='MAR(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A key link to the SYZ conjecture is provided by [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The weak comparison property for the metric ∥⋅∥CY implies the SYZ conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='1 for the given family Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The semistable SNC setting (instead of the dlt setting) was assumed in [18], but the proof only uses the semistable SNC condition on those divisor intersection strata contributing to Sk(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' If the reader prefers to work with SNC models, one can apply the discussion to any SNC resolution of X isomorphic to X on its smooth locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The choice of the resolution does not matter because the blow up along the singular locus of X has no effect on U ⊂ Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Our current strategy, which follows [14], is to produce ∥⋅∥CY via solving the variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We are given the polytopes ∆,∆∨, ∆∨ λ = ∆∨ and ∆µ, and we assume that the equivalent conditions in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31 applies (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' when (2) holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then section 2 produces the minimizer φ on ∂∆∨, and by Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='31, this implies that φ restricted to the open faces Int(∆∨ m) satisfies the real MA equation in the weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This minimizer φ extends canonically to a convex function on NR by φ(x) = max p∈∂∆µ⟨x,p⟩ − φ∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' By [27, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' C], this convex function φ corresponds canonically to a semiposi- tive toric metric on the Berkovich analytification of the toric Fano manifold X∆, hence induces by restriction a semipositive metric Φ on L → Xan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This con- cretely works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The integral points m ∈ MR correspond to monomial sections sm which induce logarithms log ∣sm∣ on Xan, and by linear interpolation 20 we can make sense of log ∣sp∣ = ∑ pi log ∣si∣, where si corresponds to a Z-basis of MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Then the semipositive metric is defined by Φ = max p∈∂∆µ log ∣sp∣ − φ∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (10) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The semipositive metric Φ on L → Xan satisfies the weak com- parison property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Over the open face Int(∆∨ m), upon choosing a local trivialising section as the monomial s = sm, then log ∣sp∣ is identified with the local function on Xan log ∣sp s ∣ = ⟨x,p⟩ − ⟨x,m⟩ = ⟨x,p⟩ + λ(m) = ⟨x,p⟩ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Here the point x ∈ NR is the image under the tropicalization map U → ∂∆∨ ⊂ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Thus the local potential of Φ over the open face factorizes through the tropicalization map, and is identified with the function φ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The NA MA measure of Φ is supported on Sk(X), and agrees with the Lebesgue type measure (Ld)µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Over any open face Int(∆∨ m), the convex function φ satisfies the weak formulation of the real MA equation (6), det(D2φ) = C0 = ∫∂∆µ dp ∫∂∆∨ λ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='3, the NA MA measure over these open faces is supported inside Sk(X) ⊂ Xan, and can be identified as d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' times the real MA measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' We need to figure out the normalisation constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Tracing through the conventions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (3)), the NA MA measure is equal to d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='MAR(φ) = d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='C0∣dx∣ = (d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='C0 ∫∂∆∨ λ dx)dL∨ = (d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='∫∂∆µ dp)dL∨ = (d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='∫∂∆µ dp)µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (11) Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The factor d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='∫∂∆µ dp = (Ld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' To see the claim, we start with the asymptotic Riemann-Roch formula h0(Xt,kL) ∼ (Ld) d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' kd, k ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' On the other hand, by the short exact sequence on the toric Fano manifold X∆ 0 → kL ⊗ O(−Xt) → kL → kL∣Xt → 0 together with the Serre vanishing of h1(Xt,kL ⊗ O(−Xt)) for k ≫ 1 (since L is ample), h0(Xt,kL) = h0(X∆,kL)−h0(X∆,kL⊗O(−Xt)) = h0(X∆,kL)−h0(X∆,kL⊗O(−X0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 21 Now the global sections of H0(X∆,kL) are spanned by the monomial sections, which correspond to the integral points on k∆µ, while H0(X∆,kL⊗O(−X0)) are spanned by those monomials which vanish on the toric boundary of X∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hence their difference counts the integral points on ∂(k∆µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now these correspond to the rational points in 1 k ∂(k∆µ) ∩ 1 k Zd+1 = ∂∆µ ∩ 1 k Zd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' and as k → +∞, they equidistribute with respect to the Lebesgue measure on ∂∆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hence h0(Xt,kL) = #(∂∆µ ∩ 1 k Zd+1) ∼ kd ∫∂∆µ dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Comparing the asymptotes proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Applying the claim to (11), we see the identification of the NA MA measure with (Ld)µ0, over all the open faces of Sk(X) ≃ ∂∆∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' On the other hand, the total measure of the semipositive metric Φ on L → Xan is (Ld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' This forces the open faces to contain the full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Now by the uniqueness of the NA MA equation, the semipositive metric Φ must agree with the Boucksom-Favre-Jonsson solution ∥⋅∥CY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Since Φ is known to satisfy the weak comparison property, by applying Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='4 we deduce that the SYZ conjecture holds for the family Xt, whence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' References [1] Rolf Andreasson, Jakob Hultgren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [2] Boucksom, S´ebastien;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Favre, Charles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Jonsson, Mattias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Singular semipos- itive metrics in non-Archimedean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 25 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1, 77–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [3] Boucksom, S´ebastien;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Jonsson, Mattias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Tropical and non-Archimedean lim- its of degenerating families of volume forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' polytech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 4 (2017), 87–139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [4] Boucksom, S´ebastien;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Favre, Charles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Jonsson, Mattias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Solution to a non- Archimedean Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 28 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3, 617–667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [5] Boucksom, S´ebastien;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Favre, Charles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Jonsson, Mattias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The non- Archimedean Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Nonarchimedean and tropical geom- etry, 31–49, Simons Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=', Springer, [Cham], 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [6] Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A localization property of viscosity solutions to the Monge- Amp`ere equation and their strict convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (2) 131 (1990), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1, 129–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [7] Caffarelli, Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Interior W 2,p estimates for solutions of the Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' (2) 131 (1990), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1, 135–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 22 [8] Caffarelli, Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A note on the degeneracy of convex solutions to Monge Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Partial Differential Equations 18 (1993), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 7-8, 1213–1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [9] Chambert-Loir, Antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Heights and measures on analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A survey of recent results, and some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Motivic integration and its interactions with model theory and non-Archimedean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Volume II, 1–50, London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=', 384, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Press, Cambridge, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [10] Keita Goto, Yuji Odaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Special Lagrangian fibrations, Berkovich retrac- tion, and crystallographic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='14474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [11] Gross, Mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Mirror symmetry and the Strominger-Yau-Zaslow conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Current developments in mathematics 2012, 133–191, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Press, Somerville, MA, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [12] Gross, Mark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Wilson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Large complex structure limits of K3 sur- faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 55 (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3, 475–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [13] Guti´errez, Cristian E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Progress in Nonlin- ear Differential Equations and their Applications, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Birkh¨auser/Springer, [Cham], 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' xiv+216 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [14] Jakob Hultgren, Mattias Jonsson, Enrica Mazzon, Nicholas McCleerey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Tropical and non-Archimedean Monge-Amp`ere equations for a class of Calabi-Yau hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='13697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [15] Kontsevich, Maxim;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Soibelman, Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Affine structures and non- Archimedean analytic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The unity of mathematics, 321–385, Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=', 244, Birkh¨auser Boston, Boston, MA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [16] Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' SYZ geometry for Calabi-Yau 3-folds: Taub-NUT and Ooguri-Vafa type metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='08770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' accepted by AMS Memoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [17] Li, Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Strominger-Yau-Zaslow conjecture for Calabi-Yau hypersurfaces in the Fermat family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 229 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 1, 1–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [18] Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Metric SYZ conjecture and non-archimedean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='01384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' accepted by Duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [19] Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Uniform Skoda integrability and Calabi-Yau degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='16961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [20] Li, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Tosatti, Valentino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Diameter bounds for degenerating Calabi-Yau metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='13068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' accepted by JDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [21] Li, Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Survey on the metric SYZ conjecture and non-Archimedean ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Modern Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A 37 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 17, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 2230009, 44 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [22] Mooney, Connor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Partial regularity for singular solutions to the Monge- Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 68 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 6, 1066–1084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [23] Mooney, Connor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Solutions to the Monge-Amp`ere equation with polyhedral and Y-shaped singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 31 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 10, 9509–9526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 23 [24] Nicaise, Johannes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Xu, Chenyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The essential skeleton of a degeneration of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 138 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 6, 1645–1667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [25] Nicaise, Johannes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Xu, Chenyang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Yu, Tony Yue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' The non-archimedean SYZ fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 155 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 5, 953–972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [26] Odaka, Yuji;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Oshima, Yoshiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Collapsing K3 surfaces and Moduli com- pactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Japan Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 94 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 8, 81–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [27] L´eonard Pille-Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Hybrid toric varieties and the non-archimedean SYZ fibration on Calabi-Yau hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='05578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [28] Strominger, Andrew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Yau, Shing-Tung;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Zaslow, Eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Mirror symmetry is T -duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content='B479:243-259,1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [29] Vilsmeier, Christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' A comparison of the real and non-archimedean Monge–Amp`ere operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Mathematische Zeitschrift (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' [30] Yau, Shing Tung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' On the Ricci curvature of a compact K¨ahler manifold and the complex Monge-Amp`ere equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 31 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 3, 339–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFPT4oBgHgl3EQfAzS0/content/2301.12983v1.pdf'} diff --git a/dtAyT4oBgHgl3EQfjPjp/content/tmp_files/2301.00413v1.pdf.txt b/dtAyT4oBgHgl3EQfjPjp/content/tmp_files/2301.00413v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0d8fbee78d31de960c1cd66a32b8397831da797 --- /dev/null +++ b/dtAyT4oBgHgl3EQfjPjp/content/tmp_files/2301.00413v1.pdf.txt @@ -0,0 +1,1229 @@ +Sudden death of entanglement with Hamiltonian ensemble assisted by auxiliary qubits +Congwei Lu,1, ∗ Wanting He,1, ∗ Jun Wang,1, ∗ Haibo Wang,1 and Qing Ai1, † +1Department of Physics, Applied Optics Beijing Area Major Laboratory, +Beijing Normal University, Beijing 100875, China +(Dated: January 3, 2023) +In this paper, we theoretically propose a method to simulate the longitudinal relaxation of a single +qubit by coupling it to an auxiliary qubit. In order to mimic the finite-temperature relaxation, we +utilize the Hamiltonian-ensemble approach [Kropf, Gneiting, and Buchleitner, Phys. Rev. X 6, +031023 (2016)] and in each realization the auxiliary qubit possesses a random level spacing. The +longitudinal relaxation arises as a consequence of the ensemble average and the interaction between +the working qubit and the auxiliary qubit. Furthermore, we apply this approach to investigate the +influence of the longitudinal relaxation and the transverse relaxation on the entanglement dynamics +of two qubits. It is discovered that the sudden death of the entanglement will occur as long as the +longitudinal relaxation is present. The transverse relaxation assists the longitudinal relaxation and +thus accelerates the finite-time disentanglement. +I. +INTRODUCTION +In an open quantum system, the interaction between +the system and the environment will lead to the exchange +of information and energy between them. As a result, +the dynamic behavior of the system is quite different +from that of an isolated system [1–6]. Generally, there +are two types of relaxations, i.e., transverse relaxation +and longitudinal relaxation. +The latter will result in +population transfer and decay of the off-diagonal terms +of the density matrix, while the former will only decrease +the coherence. These two behaviors play a crucial role in +quantum information processing. +Recently, it was proposed that the quantum dynamics +of an open quantum system can be simulated by +the ensemble-averaged state of many random isolated +systems [7–13]. +The process of ensemble averaging +over each random realization will result in averaging +all random phases, thus inducing the loss of phase +information, i.e., dephasing. +However, due to the +classical property of noise, most of the previous quantum +simulation approaches can only simulate the longitudinal +relaxation at the high-temperature limit [12–16]. +The +quantum simulation of the longitudinal relaxation at +finite temperature is rarely studied. Since the dissipative +environment can cause finite-time disentanglement [17– +19], enhanced relaxation at avoided level-crossing [20–22], +and optical non-reciprocity by detailed balance [23], it +may be interesting to in depth understand the influence +of transverse relaxation and longitudinal relaxation on +the sudden death of entanglement and its dynamic +simulation. +In this paper, in order to simulate the longitudinal +relaxation +at +finite +temperature, +we +introduce +an +auxiliary qubit which interacts with the working qubit. +In order to mimic the longitudinal relaxation, +we +∗ These authors contributed equally to this work +† E-mail: aiqing@bnu.edu.cn +effectively prepare a large number of systems including +the auxiliary qubit and the working qubit. They evolve +from the same initial state but the auxiliary qubit +possesses a random level spacing. +By averaging over +the different realizations, we can effectively simulate +the longitudinal relaxation. +Our analytical results +demonstrate +that +this +approach +can +well +simulate +the finite-temperature longitudinal relaxation with the +dissipation rate linearly dependent on time. +Here, +the dissipation rate scales linearly with the variance +of the level spacing of the auxiliary qubit and the +interaction strength between the auxiliary qubit and +the working qubit. +The initial state of the auxiliary +qubit and the interaction strength jointly determine the +distribution of the steady state. We further investigate +the effects of longitudinal and transverse relaxation on +the entanglement of two-qubit. We let the first working +qubit interact with the auxiliary qubit to mimic the +longitudinal relaxation, and apply a random field on +the second working qubit to simulate the transverse +relaxation. These two working qubits are initialized in +the maximum-entangled state and the concurrence is +utilized to characterize the dynamics of entanglement. +Our simulations show that due to the longitudinal +relaxation, the sudden death of entanglement happens +at a finite time. In this case, the transverse relaxation +of the second working qubit will accelerate the finite- +time disentanglement of the two qubits. However, if the +longitudinal relaxation is absent, the entanglement will +go to zero when the time approaches infinity. +The rest of the paper is structured as follows. +In +Sec. II, we introduce the quantum-simulation approach +by Hamiltonian ensemble. +In Sec.III, we simulate the +longitudinal relaxation of a single qubit at a finite +temperature. +The effects of the noise fluctuation and +interaction strength on entanglement sudden death are +investigated in Sec. IV. Finally, we conclude our main +discoveries in Sec. V. +arXiv:2301.00413v1 [quant-ph] 1 Jan 2023 + +2 +… +… +A +B +A +B +A +B +FIG. 1. Schematic diagram for simulating the longitudinal +relaxation of a single qubit by the Hamiltonian-ensemble +approach. Each realization is composed of a working qubit A +and an auxiliary qubit B. They evolve independently from the +same initial state ρ(0). The ensemble-averaged state TrB ρε(t) +is then obtained by averaging over all reduced density matrix +TrB ρεi(t) (i = 1, 2, · · · , N) of the working qubit. +II. +HAMILTONIAN-ENSEMBLE APPROACH +First of all, we shall give a brief introduction to +the quantum-simulation approach by an ensemble of +Hamiltonians [9, 11–13], as schematically illustrated +in Fig. 1. +A general open quantum system can be +characterized by a total Hamiltonian +ˆHT += +ˆHS + +ˆHE + ˆHI [2], where +ˆHS is the system Hamiltonian, +ˆHE is the environment Hamiltonian, and ˆHI represents +their interaction. +The time evolution of the open +system can be described as ρT (t) = ˆUρT (0) ˆU †, with +ˆU += exp +� +−i ˆHT t/ℏ +� +. +Thus, the density matrix of +the system can be obtained by partially tracing over +the environmental degrees of freedom, i.e., ρS(t) = +TrE[ρT (t)]. +To simulate the open quantum dynamics, +we utilize the Hamiltonian ensemble +{( ˆHε, pε)}, +(1) +where the subscript ε denotes each realization in the +ensemble. +The single realization Hamiltonian +ˆHε +occurring with probability pε reads +ˆHε = ˆHS + ˆHε +E + ˆVε. +(2) +ˆHε +E and ˆVε are utilized to simulate the environment +and its interaction with system. We suppose that each +realization begins from the same initial state ρε(0) = +ρ(0). The corresponding evolution at time t is given by +ρε(t) = ˆUερ(0) ˆU † +ε, with ˆUε = exp +� +−i ˆHεt/ℏ +� +. Finally, we +trace over the environmental degree of freedom in each +realization and then average over all realizations, i.e., +⟨ρ(t)⟩ = TrE ρε(t) = +� +dεpε TrE ρε(t). +(3) +Hereafter, +all +ensemble-averaged +quantities +will +be +marked with a bar. In the next section, as an example, +we utilize the ensemble-averaged quantum dynamics of +the state ⟨ρ(t)⟩ to simulate the longitudinal relaxation +behavior of a single qubit. +III. +LONGITUDINAL RELAXATION OF A +SINGLE QUBIT +In the previous investigations, due to the classical +property of the noise, the quantum simulation approach +can only simulate the longitudinal relaxation at the +high-temperature limit [6, 12, 13]. +In this paper, we +introduce an auxiliary qubit, which interacts with the +working qubit as schematically illustrated in Fig. 1, +in order to simulate the longitudinal relaxation at +finite temperatures. In the Hamiltonian ensemble, the +Hamiltonian of a realization reads +ˆHε = ω0 +2 (ωAσA +z + εσB +z ) + f(ω0ε)σA ++σB +− + h.c., +(4) +where we set ℏ = 1, σz is the Pauli matrix, σ± are the +raising and lowering operators and ω0 is the unit for +frequency. +Hereafter, we set ω0 = 1 in the following +simulations. +f(ε) = f ∗(ε) is the coupling strength +between the working qubit and the auxiliary qubit. For +simplicity, the two-qubit product states are relabeled as +|1⟩AB = | + −⟩AB, |2⟩AB = | − +⟩AB, +|3⟩AB = | + +⟩AB, |4⟩AB = | − −⟩AB, +(5) +where | ± ±⟩AB ≡ |±⟩A ⊗ |±⟩B denote the eigenstates of +Pauli operator σA +z ⊗ σB +z . Since σA +z ⊗ σB +z is the conserved +quantity, i.e., [σA +z ⊗ σB +z , ˆHε] = 0, we can rewrite ˆHε as a +block-diagonal matrix in the basis listed in Eq. (5), +ˆHε = +� ˆH− +ε +0 +0 +ˆH+ +ε +� +, +(6) +where +ˆH− +ε = 1 +2(ωA − ε)σz + f(ε)σx, +ˆH+ +ε = 1 +2(ωA + ε)σz. +(7) +The +total +evolution +operator +exp +� +−i ˆHεt +� +can +be +represented as +ˆUε = +� ˆU − +ε +0 +0 +ˆU + +ε +� +, +(8) +where +ˆU − +ε = cos(Et) − isin(Et) +E +�ωA − ε +2 +σz + f(ε)σx +� +, +ˆU + +ε = cos (ωA + ε)t +2 +− i sin (ωA + ε)t +2 +σz, +(9) +with +E = +� +1 +4(ωA − ε)2 + f(ε)2. +(10) + +3 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.07 +0.14 +0.21 +0.28 +0.35 +0 +1 +2 +3 +4 +0 +0.07 +0.14 +0.21 +0.28 +0.35 +FIG. 2. The longitudinal relaxation against xB, ε2 and α for ωA = 0. The population of the subsystem A at |+⟩ under ensemble +average (a) for xB = 1, 0.8, 0.6, when ε2 = 1 and α = 5, (b) for ε2 = 10, 1, 0.5 when xB = 0.8 and α = 5, (c) for α = 3, 1, 0.6, +when xB = 0.8 and ε2 = 1. The analytic results are shown as solid lines, while the dotted lines are generated by N = 8000 +random samples with ε = 0. +We assume that the initial state is a product state +|ψ(0)⟩ = |φ(0)⟩A ⊗ |ϕ(0)⟩B, where the working qubit A +is in the state |φ(0)⟩A = |−⟩A, and the auxiliary qubit B +is in a superposition state |ϕ(0)⟩B = xB |+⟩B + yB |−⟩B. +The reduced density matrix of the working qubit A can +be obtained by partially tracing over B, +ρA(t) = TrB( ˆUε |ψ(0)⟩ ⟨ψ(0)| ˆU † +ε). +(11) +For simplicity, we assume that the coupling strength is +proportional to the detuning between the two-qubit, that +is, +f(ε) = +� +α2 − 1 +4(ε − ωA), +(12) +where α ≥ 1/2. Specifically, the matrix elements of ρA +read +ρ++ +A (t) ≡ ⟨+| ρA(t) |+⟩ += c2 +2 |xB|2 {1 − cos[2α(ε − ωA)t]} , +ρ+− +A (t) ≡ ⟨+| ρA(t) |−⟩ += −icxBy∗ +Be− i +2 (ε+ωA)t sin [α(ε − ωA)t] , +(13) +where c = +√ +4α2 − 1/(2α). Here, we assume that ε is +subject to a Gaussian distribution and the ensemble- +averaged state ⟨ρ(t)⟩ defined in Eq. (3) can be written +as +⟨ρ(t)⟩++ ≡ρ++ +A (t) +=1 +2c2|xB|2 � +1 − cos(2αωAt)e−2α2ε2t2� +, +⟨ρ(t)⟩+− ≡ρ+− +A (t) +(14) +=1 +2cxBy∗ +B +� +e− 1 +2 (α+ 1 +2 )2ε2t2ei(α− 1 +2 )ωAt +−e− 1 +2 (α− 1 +2 )2ε2t2e−i(α+ 1 +2 )ωAt� +. +Here, we have utilized the 2n-order moment identity +of a Gaussian distribution with mean zero, i.e., ε2n = +ε2(2n)!/(2nn!) [24]. +This result can be considered as the thermalization +of the working qubit A in a thermal bath, which is in a +thermal equilibrium at temperature T. When the system +A reaches the thermal equilibrium, its probability at +the excited state is P+ = e−β∆(1 + e−β∆)−1 [2], with +β = 1/kBT, and kB being the Boltzmann constant, +where ∆ is the energy-level difference between the two +levels. +The dissipation rate Γ(t) = 2α2ε2t is linear +with respect to time t, which can be utilized to improve +the quantum metrology and thus achieve Zeno limit +[25–27]. +To simulate this steady-state distribution at +arbitrary temperature T, we can effectively tune the +coupling strength, i.e., α, and the initial state of the +auxiliary qubit B, i.e., xB and yB, to fulfill that P+ = +ρ++ +A (t → ∞). Notice that for any temperature T, this +formula can always be satisfied because 0 ≤ c < 1 +and 0 ≤ xB ≤ 1, and thus 0 ≤ ρ++ +A (t → ∞) < 1/2. +Here, as a demonstration, we simulate a process of a +single qubit, initialized in the ground state, relaxation to +the equilibrium state at a finite temperature T through +interaction with the heat reservoir. In this simulation, +this process can be controlled by the initial state of the +auxiliary qubit, the properties of the noise characterized +by ε2 and the coupling strength characterized by α. +The behaviors of the longitudinal relaxation against the +parameters, i.e., xB, ε2 and α, are plotted in Fig. 2. +In Fig. 2(a), we leave ε2 and α unchanged and only +vary xB. We find that xB does not change the relaxation +time but the steady-state population. In contrast, we can +observe in Fig. 2(b), the relaxation time will decrease +with the increase of the noise variance ε2, which does +not change the steady-state population. +Interestingly, +the coupling strength between the subsystem A and +the auxiliary qubit B determines both the steady-state +population and the relaxation time. +As shown in +Fig. 2(c), the greater the coupling strength is, the shorter +the relaxation time becomes while the larger the steady- +state population reaches. +It is worth noting that in a +real Markovian open quantum system, the steady-state + +4 +0 +1 +2 +3 +4 +0 +0.2 +0.4 +0.6 +(a) +0 +2 +4 +6 +8 +10 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +(b) +FIG. 3. +Comparison of analytical and numerical results for +the longitudinal relaxation of a single qubit. In the numerical +calculation, we select N = 5000 random samples {ε} of +Gaussian distribution with variance 0.6 and expectation 0. +The quantum dynamics of (a) the excited-state population +⟨ρ(t)⟩++, (b) the real part of the coherence Re[⟨ρ(t)⟩+−], +when ωA = 4, xB = 0.9, ε2 = 0.6, and α = 1. +population is only subject to the temperature of the +environment and the energy-level difference of the system +[2], and thus is independent of the coupling strength. +However, in our model, the steady-state population of the +subsystem A is also determined by the coupling strength. +Similar discovery has also been observed in Ref. [28]. +In order to verify our numerical simulation, +the +analytical and numerical results of full elements of the +density matrix are compared in Fig. 3. We find that when +ωA ̸= 0, both the longitudinal and transverse relaxation +behaviors demonstrate an oscillatory decay, but the +decay of the transverse relaxation is slower. As shown in +Fig. 3(b), the interaction between the subsystem A and +the auxiliary qubit B will induce the coherence between +the ground state and the excited state, since the auxiliary +qubit is initially in a superposition. However, when the +steady state is reached, the coherence of the subsystem +A disappears and thus becomes a mixed state. Moreover, +since the numerical results agree with the analytical +results, our numerical simulations are reliable. +To +summarize, in this section, we utilize an auxiliary qubit +to effectively simulate the longitudinal relaxation process +of a single qubit at arbitrary temperature. It is found +that the initial state xB, frequency variance ε2 of the +auxiliary qubit, and the interaction strength between the +working qubit and the auxiliary qubit together determine +the relaxation time and steady-state population of the +longitudinal relaxation. +FIG. 4. +Schematic illustration of simulating sudden death +of entanglement in a two-qubit system. +Qubit A1 and B1 +are initialized in the maximum-entangled state and have +no interaction with each other. +The random energy level- +spacing characterized by εB is used to simulate the transverse +relaxation of B1. We simulate the longitudinal noise of A1 +through the interaction between A1 and the auxiliary qubit +A2. +IV. +FINITE-TIME DISENTANGLEMENT +In this section, we simulate the quantum dynamics +of two-qubit disentanglement and investigate the effects +of longitudinal and transverse relaxation on the disen- +tanglement behavior. The system includes two working +qubits, i.e., qubit A1 and B1, where the former interacts +with an auxiliary qubit, i.e., qubit A2, as schematically +demonstrated in Fig. 4. Thus, the total Hamiltonian can +be written in two parts as ˆHε = ˆHA +ε + ˆHB +ε , where +ˆHA +ε =ω0 +2 (ωAσA1 +z ++ εAσA2 +z ) + f(ω0εA)σA1 ++ σA2 +− + h.c., +ˆHB +ε =ω0 +2 (ωBσB1 +z ++ εBσB1 +z ). +(15) +The +composite +system +composed +of +A1 +and +B1 +is initialized in the maximum-entangled state, +i.e., +|ψ(0)⟩A1B1 += (|++⟩A1B1 + |−−⟩A1B1)/ +√ +2. +We let +A1 interact with an auxiliary qubit A2 to mimic the +longitudinal relaxation, as depicted in Sec. III, where +f(εA) is the coupling strength between A1 and A2. +For qubit B1, we apply a random energy-level spacing +described by εB to simulate the transverse relaxation +[9, 12, 13]. +Before the ensemble average, we first of all solve the +quantum dynamics of each realization. The initial state +of the three qubits reads +ρ(0) = x |ψ1(0)⟩ ⟨ψ1(0)| + y |ψ2(0)⟩ ⟨ψ2(0)| , +(16) +where +|ψ1(0)⟩ = +1 +√ +2 |+⟩A2 ⊗ (|++⟩A1B1 + |−−⟩A1B1), +|ψ2(0)⟩ = +1 +√ +2 |−⟩A2 ⊗ (|++⟩A1B1 + |−−⟩A1B1). +(17) +In other words, the two working qubits A1 and B1 are +in a maximum-entangled state, while the auxiliary qubit +A2 is in a mixed state. At time t, |ψ1(0)⟩ evolves into + +5 +FIG. 5. The critical disentanglement time tc against α and +ε2 +A for x = 0.2, ε2 +B = 0 and ωA = 0. +the state +|ψ1(t)⟩ = η1 |+ + +⟩A2A1B1 + η2 |− + −⟩A2A1B1 ++η3 |+ − −⟩A2A1B1 , +(18) +where +η1 += +exp [−i(ωA + εA + ωB + εB)t/2] / +√ +2, +because +|+ + +⟩A2A1B1 +is +the +eigenstate +of +ˆHε. +In +the +invariant +subspace +spanned +by +the +basis +{|− + −⟩A2A1B1 , |+ − −⟩A2A1B1}, the effective Hamilto- +nian can be simplified as +ˆHε = −1 +2(ωB + εB)I + 1 +2(ωA − εA)σz + f(εA)σx. (19) +And thus the evolution operator ˆUε = exp +� +−i ˆHεt +� +reads +ˆUε = +� +cos(EAt) + i sin(EAt) +EA +�εA − ωA +2 +σz − f(εA)σx +�� +× ei(εB+ωB)t/2, +(20) +where +EA = +� +1 +4(ωA − εA)2 + f(εA)2. +(21) +Since [η2, η3]T = ˆUε +� +0, 1/ +√ +2 +�T , the three coefficients of +|ψ1(t)⟩ are explicitly given as +η1= 1 +√ +2e− i +2 (ωA+εA+ωB+εB)t, +η2= −i +√ +2e +i +2 (εB+ωB)tf(εA)sin EAt +EA +, +η3=e +i +2 (εB+ωB)t +√ +2 +� +cos EAt + i sin EAt +2EA +(ωA − εA) +� +. +(22) +Suppose |ψ2(t)⟩ can be expanded as +|ψ2(t)⟩ = ξ1 |+ + +⟩A2A1B1 + ξ2 |− + −⟩A2A1B1 ++ξ3 |+ − −⟩A2A1B1 . +(23) +Following the above steps, we can obtain +ξ1= 1 +√ +2e +i +2 (ωA+εA+ωB+εB)t, +ξ2= −i +√ +2e− i +2 (εB+ωB)tf(εA)sin EAt +EA +, +ξ3=e− i +2 (εB+ωB)t +√ +2 +� +cos EAt − i sin EAt +2EA +(ωA − εA) +� +. +(24) +Because we have solved the quantum dynamics of +the three qubits, by tracing over the auxiliary qubit +A2, we can obtain the reduced density matrix of the +two working qubits ρA1B1(t) = TrA2 ρ(t) in the basis +{|+−⟩A1B1 , |++⟩A1B1 , |−−⟩A1B1 , |−+⟩A1B1} as +ρA1B1(t) = +� +� +� +a(t) +0 +0 +0 +0 +b(t) +z(t) +0 +0 +z∗(t) c(t) +0 +0 +0 +0 +d(t) +� +� +� . +(25) +As in the Sec. III, we define the coupling strength as +f(εA) = (εA − ωA) +� +α2 − 1/4, where α ≥ 1/2. The non- +vanishing matrix elements are given by +a(t) = xc2 +2 γ(t)2, +b(t) = x +2 + y +2 +� +1 − c2γ(t)2� +, +c(t) = y +2 + x +2 +� +1 − c2γ(t)2� +, +(26) +z(t) = x + y +2 +ζ(t) +� +1 − 2γ(t)2 − iγ(t) +2α +� +, +d(t) = yc2 +2 γ(t)2, +where ζ(t) = exp +� −i +2 (ωA + εA + 2ωB + 2εB)t +� +, γ(t) = +sin(α(ωA − εA)t) and c += +√ +4α2 − 1/(2α). +In our +numerical simulation, we assume that εA and εB are +subject to independent Gaussian distributions. After the +ensemble average, the non-vanishing matrix elements of +Eq. (25) are explicitly given as +a(t)=xc2 +4 +� +1 − cos(2αωAt)e−2α2ε2 +At2� +, +b(t)=x +2 + y +2 +� +1 − c2 +2 (1 − cos(2αωAt)e−2α2ε2 +At2) +� +, +c(t)=y +2 + x +2 +� +1 − c2 +2 (1 − cos(2αωAt)e−2α2ε2 +At2) +� +, +z(t)=1 +4e− 1 +2 ε2 +Bt2e− i +2 (ωA+2ωB)t[eiαωAte− 1 +2 (α+ 1 +2 )2ε2 +At2 +×(1 − 1 +2α) + e−iαωAte− 1 +2 (α− 1 +2 )2ε2 +At2(1 + 1 +2α)], +d(t)=yc2 +4 +� +1 − cos(2αωAt)e−2α2ε2 +At2� +. +(27) +To investigate the disentanglement behavior of the +composite +system +composed +of +A1 +and +B1 +under + +0.45 +7 +0.35 +6 +log(tewo) +5 +I 0.25 +4 +0.15 +3 +0.05 +2 +0.55 +0.65 +0.75 +0.85 +0.95 +a6 +0 +2 +4 +6 +0 +0.2 +0.4 +0.6 +0.8 +1 +FIG. 6. +The concurrence C(t) as a function of time t in +the presence of both longitudinal and transverse relaxation +for ε2 +B = 0, 0.5, 2 when x = 0.2, ε2 +A = 0.5, α = 1, and +ωA = 0. Notice that the transverse relaxation is turned off +when ε2 +B = 0. The analytic results are shown as solid lines. +The dotted lines are generated by N = 300 random samples +with εA = εB = 0. +both longitudinal and transverse relaxation, we utilize +the concurrence [29] to characterize the entanglement +property C(ρA1B1) = max(0, √κ1 − √κ2 − √κ3 − √κ4), +where κi’s are the eigenvalues of the matrix G in +decreasing order +G ≡ ρA1B1 +� +σA +y ⊗ σB +y +� +ρA1B1 +∗ � +σA +y ⊗ σB +y +� +, +(28) +where σα +y (α = A, B) are the Pauli operators. +When +C = 1, the two working qubits are in the maximum- +entangled state, while C = 0, they are disentangled with +each other. Here, we can simplify the concurrence as +C(ρA1B1) = 2max +� +0, |z| − +� +ad +� +. +(29) +At the beginning, A1 and B1 are initialized in the +maximum-entangled state with |z| = 1/2 and +√ +ad = +0. +In the following, +we will show two categories +of disentanglement in our simulation. +In the first +category, the entanglement tends to vanish only when +the time approaches infinite. +In the second category, +the entanglement decays exactly to zero at a critical +disentanglement time tc and remains zero thereafter. +We first investigate the influence of the longitudinal +relaxation on the entanglement properties when there is +no transverse relaxation, i.e., ε2 +B = 0, in the system. In +this case, the disentanglement behavior of the system +is dominated by the coupling strength α and the noise +fluctuation ε2 +A. +Obviously, when α = 1/2, i.e., the +subsystem A1 does not interact with the auxiliary qubit +A2, the system will always be in the maximum-entangled +state. When α > 1/2, the non-vanishing noise fluctuation +ε2 +A will determine whether the entanglement of the +system can disappear at a finite time. When α > 1/2 +0 +1 +2 +3 +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +FIG. 7. The concurrence C(t) as a function of time t in the +presence of both longitudinal and transverse relaxation for +ωA = 0, 3, 6, when x = 0.2, ε2 +A = 0.5, ε2 +B = 0.5, and α = 1. +The analytic results are shown as solid lines. The dotted lines +are generated by N = 300 random samples with εA = εB = 0. +and ε2 +A = 0, i.e., the longitudinal relaxation is turned off, +|z| and +√ +ad can be written as +|z| = +√ +2 +4 +� +1 + +1 +4α2 + +� +1 − +1 +4α2 +� +cos(2αωAt), +� +ad = √xy 4α2 − 1 +16α2 +[1 − cos(2αωAt)] . +(30) +There always exist t = nπ/(αωA) with n ∈ Z, such that +|z| = 1/2 and +√ +ad = 0, and thus the entanglement of +the system will not disappear persistently. However, if +we turn on the longitudinal relaxation, i.e., α > 1/2 and +ε2 +A > 0, |z| tends to zero and +√ +ad = √xy(4α2−1)/(16α2) +when the time approaches infinity. +Thus, as long as +xy > 0, there exists a finite tc, making the entanglement +disappear after time tc. +The critical disentanglement +time tc against ε2 +A and α without transverse relaxation is +shown in Fig. 5. We find that when α > 1/2 and ε2 +A > 0, +the entanglement will disappear at a finite time and tc +decays monotonically and rapidly as α and ε2 +A increase. +Now we consider the case when there is only transverse +relaxation, i.e., ε2 +B > 0 and ε2 +A = 0. The evolution of |z| +and +√ +ad can be written as +|z| = +√ +2 +4 e− 1 +2 ε2 +Bt2 +� +1 + +1 +4α2 + +� +1 − +1 +4α2 +� +cos(2αωAt), +� +ad = √xy 4α2 − 1 +16α2 +[1 − cos(2αωAt)] . +(31) +Obviously, |z| tends to zero when the time approaches +infinite. +No matter how long the time passes, there +always exist 2αωAt = 2kπ with k ∈ Z so that +√ +ad +vanishes. +Thus, the concurrence will not constantly + +7 +remain zero after a finite time but will go to zero at +infinite time. +We can conclude that when the longitudinal relaxation +exists, i.e., α > 1/2 and ε2 +A > 0, entanglement will +disappear at a finite time. +The larger the transverse +noise fluctuates, the faster the entanglement disappears. +And the oscillation frequency of A1 hardly affects the +time of disentanglement. The dynamics of concurrence +against ε2 +B and ωA in this case are shown in Figs. 6 +and 7, respectively. +From Fig. 6, we find that even +if ε2 +B += +0, the concurrence will remain zero after +the critical disentanglement time and decay faster with +the increase of ε2 +B. +From Fig. 7, we find that when +the noise fluctuation and interaction strength are kept +constant, the higher the frequency of subsystem A1 is, +the faster the entanglement C oscillates, but the critical +disentanglement time is almost the same. +When the +system is immune to the longitudinal relaxation, i.e., +ε2 +A = 0 or α = 1/2, the entanglement will not die at +a finite time but will disappear at infinite time due to +the transverse relaxation. +V. +CONCLUSION +We have utilized the Hamiltonian-ensemble approach +assisted by an auxiliary qubit to simulate the longitudinal +relaxation of a single qubit in open quantum system. +Concretely, many sets of auxiliary qubits with random +level spacings are used to simulate the environmental +noise. They are prepared in the same initial state and +interact with the working qubit. The theoretical results +show that the simulated dynamics of the working qubit +can be described by a real thermalization process. We +can simulate the equilibrium-state distribution at any +temperature, which is determined by the initial state +of the auxiliary qubit, noise fluctuation and interaction +strength. +Furthermore, we simulate the dynamics of +two-qubit entanglement initialized in the maximum- +entangled state. +We let the first qubit interact with +the auxiliary qubit and relax longitudinally, and let +the second qubit relax transversely. +We find that if +there is not longitudinal relaxation on the first qubit +but transverse relaxation on the second qubit, the +entanglement of the system will disappear when the +time approaches infinity. +However, if the longitudinal +relaxation exists, no matter whether the transverse +relaxation is present or not, the entanglement of the two- +qubit will disappear after a finite time. And the larger +the noise fluctuation is, the faster the entanglement will +decay. +ACKNOWLEDGMENTS +Q.A. thanks the financial support from Beijing Natural +Science Foundation under Grant No. 1202017 and the +National Natural Science Foundation of China under +Grant Nos. 11674033, 11505007, and Beijing Normal +University under Grant No. 2022129. +H.B.W. thanks +the financial support from National Natural Science +Foundation of China under Grant No. 61675028 and +National Natural Science Foundation of China under +Grant No. 12274037. +[1] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. +Fisher, A. Garg, +and W. Zwerger, “Dynamics of the +dissipative two-state system,” Rev. Mod. Phys. 59, 1 +(1987). +[2] H. P. Breuer and F. Petruccione, The Theory of Open +Quantum Systems (Oxford University Presss, 2002). +[3] W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W. Y. Tu, +and F. Nori, “General non-Markovian dynamics of open +quantum systems,” Phys. Rev. Lett. 109, 170402 (2012). +[4] H. P. Breuer, E. M. Laine, J. Piilo, +and B. Vacchini, +“Colloquium: Non-Markovian dynamics in open quan- +tum systems,” Rev. Mod. Phys. 88, 021002 (2016). +[5] I. de Vega and D. Alonso, “Dynamics of non-Markovian +open quantum systems,” Rev. Mod. Phys. 89, 015001 +(2017). +[6] X. Y. Chen, N. N. Zhang, W. T. He, X. Y. Kong, F. G. +Deng, Q. Ai, +and G. L. Long, “Global correlation and +local information flows in controllable non-Markovian +open quantum dynamics,” npj Quantum Inf. 8, 22 +(2022). +[7] I. Buluta and F. Nori, “Quantum simulators,” Science +326, 108 (2009). +[8] I. M. Georgescu, S. Ashhab, +and F. Nori, “Quantum +simulation,” Rev. Mod. Phys. 86, 153 (2014). +[9] C. M. Kropf, C. Gneiting, and A. Buchleitner, “Effective +dynamics of disordered quantum systems,” Phys. Rev. X +6, 031023 (2016). +[10] C. Gneiting and F. Nori, “Disorder-induced dephasing +in backscattering-free quantum transport,” Phys. Rev. +Lett. 119, 176802 (2017). +[11] H. B. Chen, C. Gneiting, P. Y. Lo, Y. N. Chen, +and F. Nori, “Simulating open quantum systems with +Hamiltonian ensembles and the nonclassicality of the +dynamics,” Phys. Rev. Lett. 120, 030403 (2018). +[12] B. X. Wang, M. J. Tao, Q. Ai, T. Xin, N. Lambert, +D. Ruan, Y. C. Cheng, F. Nori, F. G. Deng, and G. L. +Long, “Efficient quantum simulation of photosynthetic +light harvesting,” npj Quantum Inf. 4, 52 (2018). +[13] N. N. Zhang, M. J. Tao, W. T. He, X. Y. Chen, X. Y. +Kong, F. G. Deng, N. Lambert, +and Q. Ai, “Efficient +quantum simulation of open quantum dynamics at +various Hamiltonians and spectral densities,” Front. Phys +16, 51502 (2021). +[14] A. Soare, H. Ball, D. Hayes, J. Sastrawan, M. C. +Jarratt, J. J. McLoughlin, X. Zhen, T. J. Green, +and + +8 +M. J. Biercuk, “Experimental noise filtering by quantum +control,” Nat. Phys. 10, 825 (2014). +[15] A. Soare, H. Ball, D. Hayes, X. Zhen, M. C. Jarratt, +J. Sastrawan, H. Uys, and M. J. Biercuk, “Experimental +bath engineering for quantitative studies of quantum +control,” Phys. Rev. A 89, 042329 (2014). +[16] X. L. Zhen, F. H. Zhang, G. R. Feng, H. Li, and G. L. +Long, “Optimal experimental dynamical decoupling of +both longitudinal and transverse relaxations,” Phys. Rev. +A 93, 022304 (2016). +[17] T. Yu and J. H. Eberly, “Finite-time disentanglement +via spontaneous emission,” Phys. Rev. Lett. 93, 140404 +(2004). +[18] M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, +S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, +“Environment-induced sudden death of entanglement,” +Science 316, 579 (2007). +[19] T. Yu and J. H. Eberly, “Sudden death of entanglement,” +Science 323, 598 (2009). +[20] Z.-S. Yang, Y.-X. Wang, M.-J. Tao, W. Yang, M. Zhang, +Q. Ai, +and F.-G. Deng, “Longitudinal relaxation +of +a +nitrogen-vacancy +center +in +a +spin +bath +by +generalized cluster-correlation expansion method,” Ann. +Phys. (N.Y.) 413, 168063 (2020). +[21] M. Onizhuk, K. C. Miao, J. P. Blanton, H. Ma, C. P. +Anderson, A. Bourassa, D. D. Awschalom, and G. Galli, +“Probing the coherence of solid-state qubits at avoided +crossings,” PRX Quantum 2, 010311 (2021). +[22] K. Head-Marsden, +J. Flick, +C. J. Ciccarino, +and +P. Narang, “Quantum information and algorithms for +correlated quantum matter,” Chem. Rev. 121, 3061 +(2021). +[23] Y.-X. Yao and Q. Ai, “Optical non-reciprocity in coupled +resonators inspired by photosynthetic energy transfer,” +arXiv:2208.05841 (2022). +[24] J. W. Goodman, Statistical Optics (Wiley, Hoboken, NJ, +2015). +[25] A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum +metrology in non-Markovian environments,” Phys. Rev. +Lett. 109, 233601 (2012). +[26] Y. Matsuzaki, S. C. Benjamin, +and J. Fitzsimons, +“Magnetic field sensing beyond the standard quantum +limit under the effect of decoherence,” Phys. Rev. A 84, +012103 (2011). +[27] X. Y. Long, W. T. He, N. N. Zhang, K. Tang, Z. D Lin, +H. F. Liu, X. F. Nie, G. R. Feng, J. Li, T. Xin, Q. Ai, and +D. W. Lu, “Entanglement-enhanced quantum metrology +in colored noise by quantum Zeno effect,” Phys. Rev. +Lett. 129, 070502 (2022). +[28] H. Dong, S. Yang, X. F. Liu, and C. P. Sun, “Quantum +thermalization with couplings,” Phys. Rev. A 76, 044104 +(2007). +[29] W. K. Wootters, “Entanglement of formation of an +arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 +(1998). + diff --git a/dtAyT4oBgHgl3EQfjPjp/content/tmp_files/load_file.txt b/dtAyT4oBgHgl3EQfjPjp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fce5790be0d3c8b510556146ad6ff460a2d80a38 --- /dev/null +++ b/dtAyT4oBgHgl3EQfjPjp/content/tmp_files/load_file.txt @@ -0,0 +1,628 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf,len=627 +page_content='Sudden death of entanglement with Hamiltonian ensemble assisted by auxiliary qubits Congwei Lu,1, ∗ Wanting He,1, ∗ Jun Wang,1, ∗ Haibo Wang,1 and Qing Ai1, † 1Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China (Dated: January 3, 2023) In this paper, we theoretically propose a method to simulate the longitudinal relaxation of a single qubit by coupling it to an auxiliary qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In order to mimic the finite-temperature relaxation, we utilize the Hamiltonian-ensemble approach [Kropf, Gneiting, and Buchleitner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' X 6, 031023 (2016)] and in each realization the auxiliary qubit possesses a random level spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The longitudinal relaxation arises as a consequence of the ensemble average and the interaction between the working qubit and the auxiliary qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Furthermore, we apply this approach to investigate the influence of the longitudinal relaxation and the transverse relaxation on the entanglement dynamics of two qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' It is discovered that the sudden death of the entanglement will occur as long as the longitudinal relaxation is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The transverse relaxation assists the longitudinal relaxation and thus accelerates the finite-time disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' INTRODUCTION In an open quantum system, the interaction between the system and the environment will lead to the exchange of information and energy between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' As a result, the dynamic behavior of the system is quite different from that of an isolated system [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Generally, there are two types of relaxations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', transverse relaxation and longitudinal relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The latter will result in population transfer and decay of the off-diagonal terms of the density matrix, while the former will only decrease the coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' These two behaviors play a crucial role in quantum information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Recently, it was proposed that the quantum dynamics of an open quantum system can be simulated by the ensemble-averaged state of many random isolated systems [7–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The process of ensemble averaging over each random realization will result in averaging all random phases, thus inducing the loss of phase information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' However, due to the classical property of noise, most of the previous quantum simulation approaches can only simulate the longitudinal relaxation at the high-temperature limit [12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The quantum simulation of the longitudinal relaxation at finite temperature is rarely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Since the dissipative environment can cause finite-time disentanglement [17– 19], enhanced relaxation at avoided level-crossing [20–22], and optical non-reciprocity by detailed balance [23], it may be interesting to in depth understand the influence of transverse relaxation and longitudinal relaxation on the sudden death of entanglement and its dynamic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In this paper, in order to simulate the longitudinal relaxation at finite temperature, we introduce an auxiliary qubit which interacts with the working qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In order to mimic the longitudinal relaxation, we ∗ These authors contributed equally to this work † E-mail: aiqing@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='cn effectively prepare a large number of systems including the auxiliary qubit and the working qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' They evolve from the same initial state but the auxiliary qubit possesses a random level spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' By averaging over the different realizations, we can effectively simulate the longitudinal relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Our analytical results demonstrate that this approach can well simulate the finite-temperature longitudinal relaxation with the dissipation rate linearly dependent on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Here, the dissipation rate scales linearly with the variance of the level spacing of the auxiliary qubit and the interaction strength between the auxiliary qubit and the working qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The initial state of the auxiliary qubit and the interaction strength jointly determine the distribution of the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We further investigate the effects of longitudinal and transverse relaxation on the entanglement of two-qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We let the first working qubit interact with the auxiliary qubit to mimic the longitudinal relaxation, and apply a random field on the second working qubit to simulate the transverse relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' These two working qubits are initialized in the maximum-entangled state and the concurrence is utilized to characterize the dynamics of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Our simulations show that due to the longitudinal relaxation, the sudden death of entanglement happens at a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In this case, the transverse relaxation of the second working qubit will accelerate the finite- time disentanglement of the two qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' However, if the longitudinal relaxation is absent, the entanglement will go to zero when the time approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' II, we introduce the quantum-simulation approach by Hamiltonian ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='III, we simulate the longitudinal relaxation of a single qubit at a finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The effects of the noise fluctuation and interaction strength on entanglement sudden death are investigated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Finally, we conclude our main discoveries in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='00413v1 [quant-ph] 1 Jan 2023 2 … … A B A B A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Schematic diagram for simulating the longitudinal relaxation of a single qubit by the Hamiltonian-ensemble approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Each realization is composed of a working qubit A and an auxiliary qubit B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' They evolve independently from the same initial state ρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The ensemble-averaged state TrB ρε(t) is then obtained by averaging over all reduced density matrix TrB ρεi(t) (i = 1, 2, · · · , N) of the working qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' HAMILTONIAN-ENSEMBLE APPROACH First of all, we shall give a brief introduction to the quantum-simulation approach by an ensemble of Hamiltonians [9, 11–13], as schematically illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' A general open quantum system can be characterized by a total Hamiltonian ˆHT = ˆHS + ˆHE + ˆHI [2], where ˆHS is the system Hamiltonian, ˆHE is the environment Hamiltonian, and ˆHI represents their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The time evolution of the open system can be described as ρT (t) = ˆUρT (0) ˆU †, with ˆU = exp � −i ˆHT t/ℏ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Thus, the density matrix of the system can be obtained by partially tracing over the environmental degrees of freedom, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ρS(t) = TrE[ρT (t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' To simulate the open quantum dynamics, we utilize the Hamiltonian ensemble {( ˆHε, pε)}, (1) where the subscript ε denotes each realization in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The single realization Hamiltonian ˆHε occurring with probability pε reads ˆHε = ˆHS + ˆHε E + ˆVε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (2) ˆHε E and ˆVε are utilized to simulate the environment and its interaction with system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We suppose that each realization begins from the same initial state ρε(0) = ρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The corresponding evolution at time t is given by ρε(t) = ˆUερ(0) ˆU † ε, with ˆUε = exp � −i ˆHεt/ℏ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Finally, we trace over the environmental degree of freedom in each realization and then average over all realizations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ⟨ρ(t)⟩ = TrE ρε(t) = � dεpε TrE ρε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (3) Hereafter, all ensemble-averaged quantities will be marked with a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the next section, as an example, we utilize the ensemble-averaged quantum dynamics of the state ⟨ρ(t)⟩ to simulate the longitudinal relaxation behavior of a single qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' LONGITUDINAL RELAXATION OF A SINGLE QUBIT In the previous investigations, due to the classical property of the noise, the quantum simulation approach can only simulate the longitudinal relaxation at the high-temperature limit [6, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In this paper, we introduce an auxiliary qubit, which interacts with the working qubit as schematically illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 1, in order to simulate the longitudinal relaxation at finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the Hamiltonian ensemble, the Hamiltonian of a realization reads ˆHε = ω0 2 (ωAσA z + εσB z ) + f(ω0ε)σA +σB − + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', (4) where we set ℏ = 1, σz is the Pauli matrix, σ± are the raising and lowering operators and ω0 is the unit for frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Hereafter, we set ω0 = 1 in the following simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' f(ε) = f ∗(ε) is the coupling strength between the working qubit and the auxiliary qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' For simplicity, the two-qubit product states are relabeled as |1⟩AB = | + −⟩AB, |2⟩AB = | − +⟩AB, |3⟩AB = | + +⟩AB, |4⟩AB = | − −⟩AB, (5) where | ± ±⟩AB ≡ |±⟩A ⊗ |±⟩B denote the eigenstates of Pauli operator σA z ⊗ σB z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Since σA z ⊗ σB z is the conserved quantity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', [σA z ⊗ σB z , ˆHε] = 0, we can rewrite ˆHε as a block-diagonal matrix in the basis listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (5), ˆHε = � ˆH− ε 0 0 ˆH+ ε � , (6) where ˆH− ε = 1 2(ωA − ε)σz + f(ε)σx, ˆH+ ε = 1 2(ωA + ε)σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (7) The total evolution operator exp � −i ˆHεt � can be represented as ˆUε = � ˆU − ε 0 0 ˆU + ε � , (8) where ˆU − ε = cos(Et) − isin(Et) E �ωA − ε 2 σz + f(ε)σx � , ˆU + ε = cos (ωA + ε)t 2 − i sin (ωA + ε)t 2 σz, (9) with E = � 1 4(ωA − ε)2 + f(ε)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (10) 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='35 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The longitudinal relaxation against xB, ε2 and α for ωA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The population of the subsystem A at |+⟩ under ensemble average (a) for xB = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6, when ε2 = 1 and α = 5, (b) for ε2 = 10, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='5 when xB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8 and α = 5, (c) for α = 3, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6, when xB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8 and ε2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The analytic results are shown as solid lines, while the dotted lines are generated by N = 8000 random samples with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We assume that the initial state is a product state |ψ(0)⟩ = |φ(0)⟩A ⊗ |ϕ(0)⟩B, where the working qubit A is in the state |φ(0)⟩A = |−⟩A, and the auxiliary qubit B is in a superposition state |ϕ(0)⟩B = xB |+⟩B + yB |−⟩B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The reduced density matrix of the working qubit A can be obtained by partially tracing over B, ρA(t) = TrB( ˆUε |ψ(0)⟩ ⟨ψ(0)| ˆU † ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (11) For simplicity, we assume that the coupling strength is proportional to the detuning between the two-qubit, that is, f(ε) = � α2 − 1 4(ε − ωA), (12) where α ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Specifically, the matrix elements of ρA read ρ++ A (t) ≡ ⟨+| ρA(t) |+⟩ = c2 2 |xB|2 {1 − cos[2α(ε − ωA)t]} , ρ+− A (t) ≡ ⟨+| ρA(t) |−⟩ = −icxBy∗ Be− i 2 (ε+ωA)t sin [α(ε − ωA)t] , (13) where c = √ 4α2 − 1/(2α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Here, we assume that ε is subject to a Gaussian distribution and the ensemble- averaged state ⟨ρ(t)⟩ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (3) can be written as ⟨ρ(t)⟩++ ≡ρ++ A (t) =1 2c2|xB|2 � 1 − cos(2αωAt)e−2α2ε2t2� , ⟨ρ(t)⟩+− ≡ρ+− A (t) (14) =1 2cxBy∗ B � e− 1 2 (α+ 1 2 )2ε2t2ei(α− 1 2 )ωAt −e− 1 2 (α− 1 2 )2ε2t2e−i(α+ 1 2 )ωAt� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Here, we have utilized the 2n-order moment identity of a Gaussian distribution with mean zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ε2n = ε2(2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='/(2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=') [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' This result can be considered as the thermalization of the working qubit A in a thermal bath, which is in a thermal equilibrium at temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' When the system A reaches the thermal equilibrium, its probability at the excited state is P+ = e−β∆(1 + e−β∆)−1 [2], with β = 1/kBT, and kB being the Boltzmann constant, where ∆ is the energy-level difference between the two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The dissipation rate Γ(t) = 2α2ε2t is linear with respect to time t, which can be utilized to improve the quantum metrology and thus achieve Zeno limit [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' To simulate this steady-state distribution at arbitrary temperature T, we can effectively tune the coupling strength, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', α, and the initial state of the auxiliary qubit B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', xB and yB, to fulfill that P+ = ρ++ A (t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Notice that for any temperature T, this formula can always be satisfied because 0 ≤ c < 1 and 0 ≤ xB ≤ 1, and thus 0 ≤ ρ++ A (t → ∞) < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Here, as a demonstration, we simulate a process of a single qubit, initialized in the ground state, relaxation to the equilibrium state at a finite temperature T through interaction with the heat reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In this simulation, this process can be controlled by the initial state of the auxiliary qubit, the properties of the noise characterized by ε2 and the coupling strength characterized by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The behaviors of the longitudinal relaxation against the parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', xB, ε2 and α, are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 2(a), we leave ε2 and α unchanged and only vary xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We find that xB does not change the relaxation time but the steady-state population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In contrast, we can observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 2(b), the relaxation time will decrease with the increase of the noise variance ε2, which does not change the steady-state population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Interestingly, the coupling strength between the subsystem A and the auxiliary qubit B determines both the steady-state population and the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 2(c), the greater the coupling strength is, the shorter the relaxation time becomes while the larger the steady- state population reaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' It is worth noting that in a real Markovian open quantum system, the steady-state 4 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6 (a) 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='3 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Comparison of analytical and numerical results for the longitudinal relaxation of a single qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the numerical calculation, we select N = 5000 random samples {ε} of Gaussian distribution with variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6 and expectation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The quantum dynamics of (a) the excited-state population ⟨ρ(t)⟩++, (b) the real part of the coherence Re[⟨ρ(t)⟩+−], when ωA = 4, xB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='9, ε2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6, and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' population is only subject to the temperature of the environment and the energy-level difference of the system [2], and thus is independent of the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' However, in our model, the steady-state population of the subsystem A is also determined by the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Similar discovery has also been observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In order to verify our numerical simulation, the analytical and numerical results of full elements of the density matrix are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We find that when ωA ̸= 0, both the longitudinal and transverse relaxation behaviors demonstrate an oscillatory decay, but the decay of the transverse relaxation is slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 3(b), the interaction between the subsystem A and the auxiliary qubit B will induce the coherence between the ground state and the excited state, since the auxiliary qubit is initially in a superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' However, when the steady state is reached, the coherence of the subsystem A disappears and thus becomes a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Moreover, since the numerical results agree with the analytical results, our numerical simulations are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' To summarize, in this section, we utilize an auxiliary qubit to effectively simulate the longitudinal relaxation process of a single qubit at arbitrary temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' It is found that the initial state xB, frequency variance ε2 of the auxiliary qubit, and the interaction strength between the working qubit and the auxiliary qubit together determine the relaxation time and steady-state population of the longitudinal relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Schematic illustration of simulating sudden death of entanglement in a two-qubit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Qubit A1 and B1 are initialized in the maximum-entangled state and have no interaction with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The random energy level- spacing characterized by εB is used to simulate the transverse relaxation of B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We simulate the longitudinal noise of A1 through the interaction between A1 and the auxiliary qubit A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' FINITE-TIME DISENTANGLEMENT In this section, we simulate the quantum dynamics of two-qubit disentanglement and investigate the effects of longitudinal and transverse relaxation on the disen- tanglement behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The system includes two working qubits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', qubit A1 and B1, where the former interacts with an auxiliary qubit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', qubit A2, as schematically demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Thus, the total Hamiltonian can be written in two parts as ˆHε = ˆHA ε + ˆHB ε , where ˆHA ε =ω0 2 (ωAσA1 z + εAσA2 z ) + f(ω0εA)σA1 + σA2 − + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ˆHB ε =ω0 2 (ωBσB1 z + εBσB1 z ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (15) The composite system composed of A1 and B1 is initialized in the maximum-entangled state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', |ψ(0)⟩A1B1 = (|++⟩A1B1 + |−−⟩A1B1)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We let A1 interact with an auxiliary qubit A2 to mimic the longitudinal relaxation, as depicted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' III, where f(εA) is the coupling strength between A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' For qubit B1, we apply a random energy-level spacing described by εB to simulate the transverse relaxation [9, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Before the ensemble average, we first of all solve the quantum dynamics of each realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The initial state of the three qubits reads ρ(0) = x |ψ1(0)⟩ ⟨ψ1(0)| + y |ψ2(0)⟩ ⟨ψ2(0)| , (16) where |ψ1(0)⟩ = 1 √ 2 |+⟩A2 ⊗ (|++⟩A1B1 + |−−⟩A1B1), |ψ2(0)⟩ = 1 √ 2 |−⟩A2 ⊗ (|++⟩A1B1 + |−−⟩A1B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (17) In other words, the two working qubits A1 and B1 are in a maximum-entangled state, while the auxiliary qubit A2 is in a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' At time t, |ψ1(0)⟩ evolves into 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The critical disentanglement time tc against α and ε2 A for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2, ε2 B = 0 and ωA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' the state |ψ1(t)⟩ = η1 |+ + +⟩A2A1B1 + η2 |− + −⟩A2A1B1 +η3 |+ − −⟩A2A1B1 , (18) where η1 = exp [−i(ωA + εA + ωB + εB)t/2] / √ 2, because |+ + +⟩A2A1B1 is the eigenstate of ˆHε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the invariant subspace spanned by the basis {|− + −⟩A2A1B1 , |+ − −⟩A2A1B1}, the effective Hamilto- nian can be simplified as ˆHε = −1 2(ωB + εB)I + 1 2(ωA − εA)σz + f(εA)σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (19) And thus the evolution operator ˆUε = exp � −i ˆHεt � reads ˆUε = � cos(EAt) + i sin(EAt) EA �εA − ωA 2 σz − f(εA)σx �� × ei(εB+ωB)t/2, (20) where EA = � 1 4(ωA − εA)2 + f(εA)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (21) Since [η2, η3]T = ˆUε � 0, 1/ √ 2 �T , the three coefficients of |ψ1(t)⟩ are explicitly given as η1= 1 √ 2e− i 2 (ωA+εA+ωB+εB)t, η2= −i √ 2e i 2 (εB+ωB)tf(εA)sin EAt EA , η3=e i 2 (εB+ωB)t √ 2 � cos EAt + i sin EAt 2EA (ωA − εA) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (22) Suppose |ψ2(t)⟩ can be expanded as |ψ2(t)⟩ = ξ1 |+ + +⟩A2A1B1 + ξ2 |− + −⟩A2A1B1 +ξ3 |+ − −⟩A2A1B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (23) Following the above steps, we can obtain ξ1= 1 √ 2e i 2 (ωA+εA+ωB+εB)t, ξ2= −i √ 2e− i 2 (εB+ωB)tf(εA)sin EAt EA , ξ3=e− i 2 (εB+ωB)t √ 2 � cos EAt − i sin EAt 2EA (ωA − εA) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (24) Because we have solved the quantum dynamics of the three qubits, by tracing over the auxiliary qubit A2, we can obtain the reduced density matrix of the two working qubits ρA1B1(t) = TrA2 ρ(t) in the basis {|+−⟩A1B1 , |++⟩A1B1 , |−−⟩A1B1 , |−+⟩A1B1} as ρA1B1(t) = � � � a(t) 0 0 0 0 b(t) z(t) 0 0 z∗(t) c(t) 0 0 0 0 d(t) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (25) As in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' III, we define the coupling strength as f(εA) = (εA − ωA) � α2 − 1/4, where α ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The non- vanishing matrix elements are given by a(t) = xc2 2 γ(t)2, b(t) = x 2 + y 2 � 1 − c2γ(t)2� , c(t) = y 2 + x 2 � 1 − c2γ(t)2� , (26) z(t) = x + y 2 ζ(t) � 1 − 2γ(t)2 − iγ(t) 2α � , d(t) = yc2 2 γ(t)2, where ζ(t) = exp � −i 2 (ωA + εA + 2ωB + 2εB)t � , γ(t) = sin(α(ωA − εA)t) and c = √ 4α2 − 1/(2α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In our numerical simulation, we assume that εA and εB are subject to independent Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' After the ensemble average, the non-vanishing matrix elements of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (25) are explicitly given as a(t)=xc2 4 � 1 − cos(2αωAt)e−2α2ε2 At2� , b(t)=x 2 + y 2 � 1 − c2 2 (1 − cos(2αωAt)e−2α2ε2 At2) � , c(t)=y 2 + x 2 � 1 − c2 2 (1 − cos(2αωAt)e−2α2ε2 At2) � , z(t)=1 4e− 1 2 ε2 Bt2e− i 2 (ωA+2ωB)t[eiαωAte− 1 2 (α+ 1 2 )2ε2 At2 ×(1 − 1 2α) + e−iαωAte− 1 2 (α− 1 2 )2ε2 At2(1 + 1 2α)], d(t)=yc2 4 � 1 − cos(2αωAt)e−2α2ε2 At2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (27) To investigate the disentanglement behavior of the composite system composed of A1 and B1 under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='45 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='35 6 log(tewo) 5 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='25 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='15 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='05 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='95 a6 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The concurrence C(t) as a function of time t in the presence of both longitudinal and transverse relaxation for ε2 B = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='5, 2 when x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2, ε2 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='5, α = 1, and ωA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Notice that the transverse relaxation is turned off when ε2 B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The analytic results are shown as solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The dotted lines are generated by N = 300 random samples with εA = εB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' both longitudinal and transverse relaxation, we utilize the concurrence [29] to characterize the entanglement property C(ρA1B1) = max(0, √κ1 − √κ2 − √κ3 − √κ4), where κi’s are the eigenvalues of the matrix G in decreasing order G ≡ ρA1B1 � σA y ⊗ σB y � ρA1B1 ∗ � σA y ⊗ σB y � , (28) where σα y (α = A, B) are the Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' When C = 1, the two working qubits are in the maximum- entangled state, while C = 0, they are disentangled with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Here, we can simplify the concurrence as C(ρA1B1) = 2max � 0, |z| − � ad � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (29) At the beginning, A1 and B1 are initialized in the maximum-entangled state with |z| = 1/2 and √ ad = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the following, we will show two categories of disentanglement in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the first category, the entanglement tends to vanish only when the time approaches infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In the second category, the entanglement decays exactly to zero at a critical disentanglement time tc and remains zero thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We first investigate the influence of the longitudinal relaxation on the entanglement properties when there is no transverse relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ε2 B = 0, in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' In this case, the disentanglement behavior of the system is dominated by the coupling strength α and the noise fluctuation ε2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Obviously, when α = 1/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', the subsystem A1 does not interact with the auxiliary qubit A2, the system will always be in the maximum-entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' When α > 1/2, the non-vanishing noise fluctuation ε2 A will determine whether the entanglement of the system can disappear at a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' When α > 1/2 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='8 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The concurrence C(t) as a function of time t in the presence of both longitudinal and transverse relaxation for ωA = 0, 3, 6, when x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='2, ε2 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='5, ε2 B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='5, and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The analytic results are shown as solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The dotted lines are generated by N = 300 random samples with εA = εB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' and ε2 A = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', the longitudinal relaxation is turned off, |z| and √ ad can be written as |z| = √ 2 4 � 1 + 1 4α2 + � 1 − 1 4α2 � cos(2αωAt), � ad = √xy 4α2 − 1 16α2 [1 − cos(2αωAt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (30) There always exist t = nπ/(αωA) with n ∈ Z, such that |z| = 1/2 and √ ad = 0, and thus the entanglement of the system will not disappear persistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' However, if we turn on the longitudinal relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', α > 1/2 and ε2 A > 0, |z| tends to zero and √ ad = √xy(4α2−1)/(16α2) when the time approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Thus, as long as xy > 0, there exists a finite tc, making the entanglement disappear after time tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The critical disentanglement time tc against ε2 A and α without transverse relaxation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We find that when α > 1/2 and ε2 A > 0, the entanglement will disappear at a finite time and tc decays monotonically and rapidly as α and ε2 A increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Now we consider the case when there is only transverse relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ε2 B > 0 and ε2 A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The evolution of |z| and √ ad can be written as |z| = √ 2 4 e− 1 2 ε2 Bt2 � 1 + 1 4α2 + � 1 − 1 4α2 � cos(2αωAt), � ad = √xy 4α2 − 1 16α2 [1 − cos(2αωAt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (31) Obviously, |z| tends to zero when the time approaches infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' No matter how long the time passes, there always exist 2αωAt = 2kπ with k ∈ Z so that √ ad vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Thus, the concurrence will not constantly 7 remain zero after a finite time but will go to zero at infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We can conclude that when the longitudinal relaxation exists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', α > 1/2 and ε2 A > 0, entanglement will disappear at a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The larger the transverse noise fluctuates, the faster the entanglement disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' And the oscillation frequency of A1 hardly affects the time of disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The dynamics of concurrence against ε2 B and ωA in this case are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 6 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 6, we find that even if ε2 B = 0, the concurrence will remain zero after the critical disentanglement time and decay faster with the increase of ε2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 7, we find that when the noise fluctuation and interaction strength are kept constant, the higher the frequency of subsystem A1 is, the faster the entanglement C oscillates, but the critical disentanglement time is almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' When the system is immune to the longitudinal relaxation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=', ε2 A = 0 or α = 1/2, the entanglement will not die at a finite time but will disappear at infinite time due to the transverse relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' CONCLUSION We have utilized the Hamiltonian-ensemble approach assisted by an auxiliary qubit to simulate the longitudinal relaxation of a single qubit in open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Concretely, many sets of auxiliary qubits with random level spacings are used to simulate the environmental noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' They are prepared in the same initial state and interact with the working qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' The theoretical results show that the simulated dynamics of the working qubit can be described by a real thermalization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We can simulate the equilibrium-state distribution at any temperature, which is determined by the initial state of the auxiliary qubit, noise fluctuation and interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Furthermore, we simulate the dynamics of two-qubit entanglement initialized in the maximum- entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We let the first qubit interact with the auxiliary qubit and relax longitudinally, and let the second qubit relax transversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' We find that if there is not longitudinal relaxation on the first qubit but transverse relaxation on the second qubit, the entanglement of the system will disappear when the time approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' However, if the longitudinal relaxation exists, no matter whether the transverse relaxation is present or not, the entanglement of the two- qubit will disappear after a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' And the larger the noise fluctuation is, the faster the entanglement will decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' ACKNOWLEDGMENTS Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' thanks the financial support from Beijing Natural Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 1202017 and the National Natural Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 11674033, 11505007, and Beijing Normal University under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 2022129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' thanks the financial support from National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 61675028 and National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 12274037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Leggett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Chakravarty, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Dorsey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Fisher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Garg, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zwerger, “Dynamics of the dissipative two-state system,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 59, 1 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Breuer and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Petruccione, The Theory of Open Quantum Systems (Oxford University Presss, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Xiong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Tu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nori, “General non-Markovian dynamics of open quantum systems,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 109, 170402 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Breuer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Laine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Piilo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Vacchini, “Colloquium: Non-Markovian dynamics in open quan- tum systems,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 88, 021002 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' de Vega and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Alonso, “Dynamics of non-Markovian open quantum systems,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 89, 015001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [6] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Kong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Deng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ai, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Long, “Global correlation and local information flows in controllable non-Markovian open quantum dynamics,” npj Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 8, 22 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Buluta and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nori, “Quantum simulators,” Science 326, 108 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Georgescu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ashhab, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nori, “Quantum simulation,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 86, 153 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Kropf, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Gneiting, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Buchleitner, “Effective dynamics of disordered quantum systems,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' X 6, 031023 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Gneiting and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nori, “Disorder-induced dephasing in backscattering-free quantum transport,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 119, 176802 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Gneiting, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Chen, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nori, “Simulating open quantum systems with Hamiltonian ensembles and the nonclassicality of the dynamics,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 120, 030403 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [12] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Tao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Xin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lambert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ruan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Cheng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nori, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Deng, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Long, “Efficient quantum simulation of photosynthetic light harvesting,” npj Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 4, 52 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Tao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Kong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Deng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lambert, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ai, “Efficient quantum simulation of open quantum dynamics at various Hamiltonians and spectral densities,” Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys 16, 51502 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Soare, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ball, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Hayes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Sastrawan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Jarratt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' McLoughlin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Green, and 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Biercuk, “Experimental noise filtering by quantum control,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 10, 825 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Soare, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ball, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Hayes, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Jarratt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Sastrawan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Uys, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Biercuk, “Experimental bath engineering for quantitative studies of quantum control,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' A 89, 042329 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [16] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Feng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Li, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Long, “Optimal experimental dynamical decoupling of both longitudinal and transverse relaxations,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' A 93, 022304 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Yu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Eberly, “Finite-time disentanglement via spontaneous emission,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 93, 140404 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Almeida, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' de Melo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Hor-Meyll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Salles, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Walborn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Souto Ribeiro, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Yu and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Eberly, “Sudden death of entanglement,” Science 323, 598 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [20] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Tao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ai, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Deng, “Longitudinal relaxation of a nitrogen-vacancy center in a spin bath by generalized cluster-correlation expansion method,” Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=') 413, 168063 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Onizhuk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Miao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Blanton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Anderson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Bourassa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Awschalom, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Galli, “Probing the coherence of solid-state qubits at avoided crossings,” PRX Quantum 2, 010311 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Head-Marsden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Flick, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ciccarino, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Narang, “Quantum information and algorithms for correlated quantum matter,” Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 121, 3061 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Yao and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ai, “Optical non-reciprocity in coupled resonators inspired by photosynthetic energy transfer,” arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content='05841 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Goodman, Statistical Optics (Wiley, Hoboken, NJ, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Chin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Huelga, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Plenio, “Quantum metrology in non-Markovian environments,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 109, 233601 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Matsuzaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Benjamin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Fitzsimons, “Magnetic field sensing beyond the standard quantum limit under the effect of decoherence,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' A 84, 012103 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [27] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Long, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' He, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Tang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' D Lin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Nie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Xin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Ai, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lu, “Entanglement-enhanced quantum metrology in colored noise by quantum Zeno effect,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 129, 070502 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Dong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Liu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Sun, “Quantum thermalization with couplings,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' A 76, 044104 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' [29] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} +page_content=' 80, 2245 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfjPjp/content/2301.00413v1.pdf'} diff --git a/eNAzT4oBgHgl3EQfaPxf/content/tmp_files/2301.01365v1.pdf.txt b/eNAzT4oBgHgl3EQfaPxf/content/tmp_files/2301.01365v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..364a114477046b42408382a08f69952d8d34e519 --- /dev/null +++ b/eNAzT4oBgHgl3EQfaPxf/content/tmp_files/2301.01365v1.pdf.txt @@ -0,0 +1,1112 @@ +Primordial black holes, early galaxies, and antimatter +in the Milky Way +Plenary ralk presented at 6th International Conference on Particle Physics and Astrophysics +(ICCPA-2022) +A. D. Dolgova,b +January 5, 2023 +aDepartment of Physics, Novosibirsk State University, +Pirogova st. 2, Novosibirsk, 630090 Russia +bBogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, +Joliot-Curie st. 6, Dubna, Moscow region, 141980 Russia +Abstract +Astronomical observations strongly incompatible with the canonical cosmological +model are reviewed. In particular too early formation of galaxies, as discovered by +HST and JWST, are discussed in detail. Other data revealing highly dense popula- +tion of the very young universe with plethora of other different types of objects are +presented. It is demonstrated that similar or maybe even more pronounced problems +can be seen also in the present day universe. It is argued that all of the above men- +tioned problems can be nicely fixed by assumption that the universe is filled with +primordial black holes in wide mass interval from a fraction of the solar mass up to +supermassive BH. The mechanism of PBH formation presented in 1993 is described. +The predicted by this mechanism log-normal mass spectrum of such PBH is shown +to agree very well with the data. Finally possible rich population of our Galaxy by +antimatter is discussed and new ways of its identification are presented +arXiv:2301.01365v1 [astro-ph.CO] 3 Jan 2023 + +1 +Introduction +Discoveries of the last decades, especially by Hubble Space Telescope (HST) and James +Webb Space Telescope (JWST), created strong confusion among the astronomical commu- +nity. The early universe at redshifts z ∼ 10 and the age of a few hundred million years +was found to be densely populated by galaxies, quasars, gamma-bursters, supernovae, and +contained a very high level of dust. According to the common understanding the available +universe age was by far less than the necessary time for the creation of such rich universe +population. The frustrating feeling came to life that the standard cosmological model, well +supported by the theory and ”experiment”, was on the verge of breaking. +Not less striking and equally puzzling evidence, that does not fit the conventional +frameworks, came from the observations of the contemporary universe. The review of the +problems, we encounter both in the early as well as in the present day universe, at the +stage of art which existed 4 years ago, can be found in ref. [1]. But since that time much +higher amount of controversies was accumulated thanks to active and precise astronomical +observations. +All those surprising phenomena were in fact anticipated thirty years ago according to +our paper [2], where a new mechanism of formation of highly massive primordial black hole +(PBH) was proposed. This mechanism was further elaborated in ref. [3]. A very simple +log-normal mass spectrum of PBH was predicted that, as later verified, very well agrees +with observational data. The abundant PBH population in very wide mass interval can +eliminate the tension between theory and observations, in particular, because supermassive +PBH could seed galaxy formation as it was envisaged in refs. [2,3]. +In addition to high mass PBH formation, the mechanism of refs [2, 3] could lead to +noticeable antimatter population of galaxies. In particular antimatter, including antistars, +may exist in the Milky Way. This exciting predition seems to be confirmed by the recent +studies. +Outline of this talk. +1. Discovery by JWST of the dense population of the early universe with galaxies, which +according to the conventional cosmology cannot be there. +2. More data contradicting traditional cosmology and astrophysics. +3. Anticipated resolution of the problems by PBH pre-suggested in papers [2,3]. +4. Prediction and test of Log-normal mass spectrum of PBHs. The only mass spectrum +confirmed by observatons. +5. Black dark matter. +6. Gravitational waves and PBH. +7. Antimatter in the Galaxy (antistars, anit-nuclej, positrons). +8. Basics of the mechanism of PBH and antimatter creation . +2 +Strong blow to conventional cosmology by JWST +Observatkions of the several recent months made by JWST created almost panic among +traditional cosmologists and astrophysicists. It was discovered that the pretty young uni- +verse with the age 200-300 million years contains plenty of bright galaxies, see e.g. [4]- [12], +1 + +which simply cannot be there according to the accepted faith. +As is stated in the JWST publications: an unexpectedly large density (stellar mass +density ϱ∗ ≳ 106M⊙ Mpc−3) of massive galaxies (stellar masses M∗ ≥ 1010.5M⊙) are dis- +covered at extremely high redshifts z ≳ 10. The following galaxies with record redshifts +and the corresponding age of the universe are observed as reported by: CEERS (Cosmic +Evolution Early Release Science): +z = 14.3 ± 0.4, tU = 264 Myr; z = 16.7, tU = 235 +Myr Such unbelievably early observed galaxies forced the JWST team to some retreat +reported as: ”Bit of panic: Astronomers forced to rethink early JWST findings. +Re- +vised instrument calibrations are bedevilling work on the distant Universe”, according to +https://www.nature.com/articles/d41586-022-... +Several examples of the JWST data with very high redshifts are presented in Fig. 1 +and compared with theoretical expectations by the standard ΛCDM model. In Fig. 2 the +Figure 1: Comparison of some JWST data with theoretical expectation +results of JWST are compared with those by HST for two events for which both telescopes +registered the same objects. An impressive agreement is demonstrated. +According to the canonical theory of large scale structure (LSS) formation the density +contrast ∆ ≡ δϱ/ϱ started to rise at the onset of the matter dominated stage at z = 104. +After that ∆ evolved as the cosmological scale factor. Since initially ∆in ≲ 10−4, by the +present time it may reach unity and after that fast LSS formation takes place (violent +relaxation - strong rising of the gravitational field of the inhomogeneity) leading to the +observed highly inhomogeneous universe at the galactic and galaxy clusters scales. +2 + +Moritz Haslbauer et al, Has JWST already falsified dark-matter-driven galaxy formation? arXiv:2210.14915 +12 +Comparison of the size of the most massive galaxies, obtained +ID 14924 +in models of formation and growth of galaxies based on LCDM +11 +(colored dots) with JWsT observations (black dots with errors) +ID 1514 +10 +GL-z11 +CEERS-1749 +depending on the redshift of the observed galaxies. ++ +GL-z13 +6 +IGN-z11 +Redshift, z +8 +21.8 +15.8 +11.4 +8.1 +10.5 +7 +61 +10.0 +9 +1011.12131415 +5161718 +Redshift, z +9.5 +TNG50-1 +RefL0025N0752 +RefL0050N0752 +TNG100-1 +RecalL0025N0752 +RefL0100N1504 +9.0 +8.5 +8.0 +ACDM & +invariant IMF +7.5. +8.2 +8.4 +8.6 +8.8 +logio(Age/yr)Figure 2 +3 + +XDFH-2395446286 +Rychard J.Bouwens et al, Evolution of the UV LF from +100日 +Z~15 to z~8 Using New JWST NIRCam Medium-Band +Observations over the HUDF/XDF. arXiv:2211.02607 +10 +New coverage +Joint observation of object XDFH-2395446286 and +measuring its redshift z=12 HST and JWST. This is the +fromJWST +most distant galaxy ever discovered by HTS 30 years of +observation. +1 +0 +5 +10 +15 +Wavelength [μm] +Redshift +HST +JWST +Opt +F125W +F140W +F160W +F182M +F210M +F430M +F460M +F480M +Marco Castellano et al, Early results from GLASS-JWST.lll: +Galaxy candidates at z~9-15. arXiv:2207.09436 +Two more examples of galaxies with +z=10.62 and z=12.3 found JWST in +GHZ2 +GHZ1 +a couple of months of observations. +EP(z)E +121618 +26 +0369121618 +28 +上 +28 +个 +30E +30 E +104 +5×104 +104 +5×104 +入oba(A) +入oba(A)In a simple way the process of structure foirmation can be understood as follows. The +velocity of the Hubble runaway at distance r is vH = Hr and the virial velocity in the +gravitational field of the inhomogeneity is +v2 +grav = 4πr3 +rm2 +Pl +δϱ +(1) +Using H2 = (8πϱ)/(3m2 +pl) we find vgrav ≥ vH if ∆ > 1. +The probability of such huge +density fluctuation for the flat spectrum of perturbations is quite low. +There are two effects operating in the same directions. Firstly the available time is +constrained by the universe age, which is essentially equal to tU = 1/H and is quite short. +In addition, a large value of H means expansion is very fast. That suppresses the efficiency +of the structure formation. +3 +Problems preceding JWSP +Similar serious problems are known already for several years. The Hubble space telescope +discovered that the early universe, at z = 6 − 7 was too densely populated with quasars, +alias SMBH, supernovae, gamma-bursters and happened to be very dusty. No understand- +ing is found in conventional cosmology how all these creature were given birth to in such +a short time. Moreover great lots of phenomena in the ∼ 15 billion year old, present day +universe are in strong tension with the conventional cosmological expectations. +HST sees the universe up to z = 6 − 7, but accidentally a galaxy at z ≈ 12 has been +discovered for which both Hubble and Webb are in good agreement, as we have already +mentioned in the previous section. Still, despite the earlier discoveries by HST, only after +publications of JWST data the astronomical establishment became seriously worried. +To summarise: observational data of the last decades present more and more evidence +indicating existence of the objects contradicting conventional astrophysics and cosmology +in the present day and in quite young universe. Rephrasing Marcellus from ”The Tragedy +of Hamlet, Prince of Denmark” we can say ”Something is rotten in the state of Denmark +the Universe”. However, all the problems can be neatly solved if the universe is populated +by primordial black holes. +4 +BH types by formation mechanisms +There three known types of BH depending upon the mechanism of their creation: +1. Astrophysical black holes. +This BHs are created by the collapse of a star which exhausted its nuclear fuel. The +expected masses should start immediately above the neutron star mass, i.e. about +3M⊙, but noticeably below 100M⊙. Instead we observe that the BH mass spectrum +in the galaxy has maximum at M ≈ 8M⊙ with the width ∼ (1−2)M⊙. The result is +somewhat unexpected but an explanations in the conventional astrophysical frame- +works is not excluded. +4 + +Recently LIGO/Virgo discovered BHs with masses close to 100M⊙. +Their astro- +physical origin was considered to be complitely impossible. Now some, though quite +exotic, formation mechanisms have been suggested. +2. Accretion created BHs. +Such BHs are formed by the accretion of matter on the mass excess in galactic centres. +It is known that in any large galaxy at the centre there exists a supermassive black +holes (SMBH) with masses varying from a few millions M⊙ (e,g, Milky Way) up to +almost hundred billions M⊙. However, the conventional accretion mechanisms are +not efficient enough to create such monsters during the universe life-time, tU ≈ 14.6 +Gyr. At least 10-fold longer time is necessary, some references can be found in [1], +to say nothing about SMBH in 10 times younger universe. +3. Primordial black holes (PBH). +PBH are supposed to be formed in the very early universe during pre-stellar epoch, i.e. +prior to star formation .The idea of primordial black holes and a possible mechanism +of their creation was pioneered by Zeldovich and Novikov [13]. According to their +idea, the density contrast in the early universe inside the bubble radius, essentially +equal to the cosmological horizon, might accidentally happen to be large, δϱ/ϱ ≈ 1, +then that piece of volume would be inside its gravitational radius i.e. it became a +PBH, that decoupled from the cosmological expansion. +The mechanism was elaborated later by Hawking [14], and by Carr and Hawking [15]. +5 +BH types by masses +Rather arbitrary black holes are separated into three groups depending on their mass: +1. Supermassive black holes (SMBH): M = (106 − 1010)M⊙. +2. Intermediate mass black holes (IMBH): M = (102 − 105)M⊙. +3. Solar mass black holes: masses from a fraction of M⊙ up to 100M⊙. +4. There can be also very light black holes, not yet observed, with masses in the region +∼ 1020 g; they may be the ”particles” of the cosmological dark matter. +The origin of most of these BHs is unclear, except maybe of the BHs with masses of a few +solar masses, which may be astrophysical. +Extremely unexpected was very high abundance of IMBH which are appearing during last +several years in huge numbers. +The assumption that (almost) all these black holes in the universe are primordial except +possibly the very light ones, strongly reduces or even eliminates the tension between their +observed abundances and possible mechanisms of their formation. +6 +Problems of the contemporary universe. Summary. +1. SMBH in all large galaxies. The universe age is too short for their formation through +the commonly accepted accretion mechanism. +5 + +2. Several SMBH are found in very small galaxies and even in (almost) empty space, +where not only the time duration but also an amount of material is insufficient. An inter- +esting recent observation was made by the Hobby-Eberly Telescope at Texas’s McDonald +Observatory suggesting the presence of a black hole with a mass of about 17 billion M⊙ +equivalent to 14% of the total stellar mass of the galaxy. Usually the mass of the central +BH is about 0.1 % of the galaxy mass. This SMBH was observed by the analysis of the +motions of the stars near the center of the galaxy. +There appeared recently fresh evidence [16] indicating to supermassive BH with the +mass 3 × 106M⊙ in dwarf galaxy Leo 1. Much more new data are presented practically +today [17]. Six dwarf galaxies are identified that have X-ray AGN. They are presumbly +powered by SMBHs of M > 107M⊙. +It is not excluded, that such SMBHs, that are not hosted by a large galaxy, might be +pushed out of large galaxies in the process of galaxy collisions. Such catastrophic event +may even create plenty of wandering single supermassive black holes. However, taking into +account a large number of such exotics, much more natural seems that all SMBH in small +galaxies are primordial. Simply they were unlucky not to acquire their own large galaxy, +since there was not enough matter around to build large galaxies. +3. Problems with the BH mass spectrum in the Galaxy. According to the data the +masses are concentrated in the narrow interval (7.8 ± 1.2)M⊙. +4. Origin and properties of the sources of the observed gravitational waves, encounter +considerable difficulties, if one tries to explain them assuming astrophysical formation of +back hole binaries emitting the observed gravitational radiation. +5. IMBH, with M ∼ (103 − 105)M⊙ are unexpectedly discovered in dwarfs and globular +clusters. Their origin is unclear, if they are not primordial. +6. Invisible Massive Astrophysical Compact Halo Objects (MACHOs), non-luminous +objects with masses ∼ 0.5M⊙ observed through microlensing. It is unknown what are they +and how they were created. +7. Existence of very unusual stars in the Galaxy, among which there are too fast moving +stars and stars with unusual chemistry. Moreover, too old stars, are found. Many of them +look older than the Galaxy and maybe one is even older that the universe (sic!?). +An assumption, that the black holes mentioned in the list above, are primordial elimi- +nates all the problems. The mechanism of PBH formation suggested in papers [2,3] predicts +also existence of the unusual stars mentioned in point 7. +7 +Observations of black holes +The ancient point of view: BH are objects with so strong gravitational field that nothing +can escape it. According to Mitchell (1784): there may be bodies for which the second +cosmic velocity is larger than the speed of light. They do not shine and do not reflect light, +i.e. are absolutely dark, invisible. +However, the truth is quite the opposite, black holes are very well seen. BH can emit all +kind of radiation through the Hawking evaporation (though nobody has yet seen it). The +most powerful sources of radiation in the universe are SMBH - quasars, point-like objects +6 + +shining as a thousands of galaxies. +The methods of the BH observation include: +1. Central mass estimate through analysis of stellar motion around the supposed BH as +e.g. discovery of BH in the center of the Millki Way. +2. Gravitational lensing (MACHO and some other BHs). +3. Electromagnetic radiation from the accreting matter; it is the mechanism of quasar +central engine, but mush smaller BH are also observed that way. +However, all these methods allow only to establish that there is a large mass inside small +volume. We need theory to conclude that there should be a black hole inside. But the +following method is free from this restriction: +4. Registration of gravitational waves from coalescing double systems of black holes. The +data directly show that there are exactly coalescence of BHs. +This is the first test of +General Relativity for strong fields and the first observational proof of existence of the +Schwarzschild solution. +8 +Galaxy seeding by SMBH +We see thar a large amount of observational data are at odds with the conventional model +but nicely agrees with the model of creation of primordial black holes and primordial +stars suggested in refs. [2, 3]. According to the standard approach the SMBH in galactic +centres are formed after galaxies were created by accretion mechanism. In papers. [2,3] the +validity of the opposite scenario was conjectured, namely, SMBHs were formed first and +subsequently seeded galaxy formation. The hypothesis advocated in these works allows to +explain presence of SMBH in all large and several small galaxies accessible to observation +and resolves the problem of very early existence of galaxies observed by HST and JWST. +This statement was recently rediscovered in ref. [18] according to which ”Recent ob- +servations with JWST have identified several bright galaxy candidates at z ≳ 10, some +of which appear unusually massive (up to ∼ 1011 M⊙)... Such early formation of mas- +sive galaxies is difficult to reconcile with standard ΛCDM predictions, ...we show that +the observed massive galaxy candidates can be explained... if structure formation is ac- +celerated by massive (≳ 109 M⊙) primordial black holes that enhance primordial density +fluctuations.” +Very recently, in December, 2022, there appeared another paper on the possibility of +SMBH impact on JWST-galaxy formation [19]. +9 +PBH and inflation +The mechanism suggested in ref. [2, 3] introduced some new features which were later +explored in a series of subsequent works. The proposed there scenarios are heavily based +on the Affleck-Dine [20] model of baryogenesis, that permits to create very interesting +features of the PBH population or some other macroscopic compact objects, see below, +sec. 15 +In paper [2] inflationary mechanism was first implied for PBH formation. It allowed to +7 + +create PBH with huge masses, much larger than those in the previously studied models. A +year later inflationary creation of PBH was explored in ref. [21] soon after in ref. [22], and +two years later in [23]. Nowadays there exploded an avalanche of papers on inflationary +formation of PBH. +Presently inflationary mechanism of PBH production is commonly used. +However, +except for predicting large masses of PBH, the models do not have much predictive power +because the mass spectra of the created PBHs are quite complicated and strongly parameter +dependent. No simple analytic expressions have been presented. The only exception is the +mechanism of refs [2, 3], which predicts extremely simple log-normal mass spectrum of +PBH: +dN +dM = µ2 exp [−γ ln2(M/M0)]. +(2) +The central value mass can be predicted theoretically [24]: M0 ∼ 10M⊙. It is equal to the +horizon mass at QCD phase transition from the free quark-gluon phase to the confinement +phase. To be more precise the horizon mass is approximately equal to 10M⊙ for the cosmic +plasma with vanishingly small chemical potential µ. In our case µ is supposed to be large +of the order of the plasma temperature. In this case the phase tranasion was probably +delayed and the horizon mass could be 2-3 times larger. +An impressive feature of the the log-normal mass spectrum with the predicted value of +M0 is that it is the only known spectrum tested by ”experiment” in very good agreement +with the observed densities of black holes in all mass intervals from the solar mass BH, up +to black holes with intermediates masses, and further up to supermassive black holes. In +partcular, the mechanism developed in [2, 3] allows to explain the presence of SMBH in +all large and several small galaxies accessible to observation. Especially impressive is the +confirmation of the model by the chirp mass binaries measured by LIGO/Virgo which is +discussed in the following section. +10 +Gravitational waves from BH binaries +There is general agreement between several groups that the gravitational waves discovered +by LIGO/Virgo interferometers originated from PBH binaries. We discuss this issue here +following our paper [25]. There are three problems which indicate that the sources of GWs +are most naturally primordial black holes: +1. Origin of heavy BHs (with masses ∼ 30M⊙). To form so heavy BHs, the progenitors +should have M > 100M⊙ and a low metal abundance to avoid too much mass loss during +the evolution. Such heavy stars might be present in young star-forming galaxies but they +are not observed in the necessary amount. Recently there emerged much more striking +problem because of the observation of BH with M ∼ 100M⊙. Formation of such black +holes in the process of stellar collapse was considered to be strictly forbidden. Some exotic +mechanisms might be possibly allowed, such as e.g. BH formation in the process of collapse +of supermassive star heated by dark matter annihilation inside [26]. +On the other hand, primordial black holes with the observed by LIGO masses may be +easily created with sufficient density. +8 + +2. Formation of BH binaries from the original stellar binaries. +Stellar binaries are formed +from common interstellar gas clouds and are quite frequent in galaxies. If BH is created +through stellar collapse, small non-sphericity results in a huge velocity of the BH and the +binary is destroyed. BH formation from PopIII stars and subsequent formation of BH +binaries with tens of M⊙ is estimated to be small. The problem of the binary formation +is simply solved if the observed sources of GWs are the binaries of primordial black holes. +They were at rest in the comoving volume, when inside horizon they are gravitationally +attracted and may loose energy due to dynamical friction in the early universe. The prob- +ability for them to become gravitationally bound is significant. +The conventional astrophysical scenario is not excluded but less natural. +3. Low spins of the coalescing BHs . The low values of the BH spins sae observed in +GW150914 and in almost all (except for three) other events. It strongly constrains as- +trophysical BH formation from close binary systems. +Astrophysical BHs are expected +to have considerable angular momentum, nevertheless the dynamical formation of double +massive low-spin BHs in dense stellar clusters is not excluded, though difficult. On the +other hand, PBH practically do not rotate because vorticity perturbations in the early +universe are vanishingly small. Still, individual PBH forming a binary initially rotating on +elliptic orbit could gain collinear spins about 0.1 - 0.3, rising with the PBH masses and +eccentricity [27,28]. This result is in agreement with the GW170729 LIGO event produced +by the binary with masses 50M⊙ and 30M⊙ and and GW190521. +To summarize: each of the mentioned problems may be solved in the conventional +frameworks but it looks much simpler to assume that the LIGO/Virgo sources are primor- +dial. +11 +Chirp mass distribution +Two rotating gravitationally bound massive bodies are known to emit gravitational waves. +In quasi-stationary inspiral regime, the radius of the orbit and the rotation frequency +are approximately constant and the GW frequency is twice the rotation frequency. The +luminosity of the GW radiation is: +L = 32 +5 m2 +Pl +�Mc ωorb +m2 +Pl +�10/3 +, +(3) +where M1, M2 are the masses of two bodies in the binary system and Mc is the so called +chirp mass: +Mc = +(M1 M2)3/5 +(M1 + M2)1/5 , +(4) +and +ω2 +orb = M1 + M2 +m2 +PlR3 +. +(5) +The available data on the chirp mass distribution of the black holes in the coalescing +binaries in O1-O3 LIGO/Virgo runs are analyzed and compared with theoretical expecta- +tions based on the hypothesis that these black holes are primordial with log-normal mass +9 + +spectrum [29]. The inferred best-fit mass spectrum parameters are: +M0 = 17M⊙ and +γ = 0.9, fall within the theoretically expected range and show excellent agreement with +observations. On the opposite, binary black hole formation based +on massive binary star +evolution require additional adjustments to reproduce the observed chirp mass distribution. +The results are presented in Figs. 3 and 4. +0 +10 +20 +30 +40 +50 +60 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ℳ +KS +M0=15, γ=0.7 +M0=17, γ=0.9 +EDF +F(<ℳ) +Figure 3: Model distribution FPBH(< M) with parameters M0 and γ for two +best Kolmogorov-Smirnov tests. EDF= empirical distribution function. +0 +10 +20 +30 +40 +50 +60 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ℳ +CO a=1 +a=0.1 +BH a=1 +a=0.1 + EDF +F(<ℳ) +Figure 4: Cumulative distributions F(< M) for several astrophysical models +of binary BH coalescences. +So we can conclude that PBHs with log-normal mass spectrum perfectly fit the data, +while astrophysical BHs seem to be disfavoured. +New recent dats on GW observations were analysed by K.A. Postnov in his talk at +XXXIV International Workshop on High Energy Physics ”From Quarks to Galaxies: Elu- +cidating Dark Sides” are depicted in fig. 5. +In this talk is also presented an approximate fitting of the observed chirp-mass distri- +bution in the O1-O3 LVK GW compact binary coalescences (from GWTC-3 catalog) by +10 + +two independent PBH populations with initial log-normal mass distributions M (1) +0 += 5M⊙ +and M (2) +0 += 30M⊙, see fig. 6. +Figure 5 +0 +10 +20 +30 +40 +50 +60 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ℳ +N (< ℳ ) +EDF +Mc 1=5MSun, γ1=1.1, Mc 2=30MSun, γ2=7, fPBH 1=0.37, fPBH 2=0.63 +Figure 6: Approximation of the observed chirp-mass distribution in the O1- +O3 LVK GW compact binary coalescences (from GWTC-3 catalog) by two +independent PBH populations, from K.Postnov talk +In fig. +7 one can see the same approximation but by the simplest model of astro- +physical BH formation from the collapse of the CO-core of a massive star and standard +common-envelope parameter, with taking into account evolution of star-formation rate in +the universe with redshift (Postnov & Kuranov, 2019), plus a population of PBHs with +log-normal initial mass distribution with M (2) +0 += 33M⊙ can be seen in Fig. 7. +The picture looks more complicated than the earlier one described at the beginning of +this section. One possible interpretation is that there are two populations of PBH with +log-normal mass spectrum each but with different values of M0. The model can be modified +11 + +Example: log-normal PBH mass +function GWTC1+GWTC2 +F(M) = Aexp[-yIn?(M /Mo)) +1.0 +F( 107M⊙. Most probably these +dwarfs were seeded by primordial supermassive black holes in accordance with ref [34]. +As argued in ref. [34] IMBHs with masses of a few thousand solar mass, or higher, can +seed formation of globular clusters (GCs) and dwarf galaxies. In the last several years such +IMBH inside GSs are observed confirming this suggestion. For example the BH with the +mass M ∼ 105M⊙ is discovered almost yesterday in the dwarf galaxy SDSS J1521+1404. +The astrophysical origin of IMBH encounter serious problems for all mass values, but +nevertheless huge number of them are discovered with all possible masses. On the contrary, +the described above model of PBH formation excellently resolves all the inconsistencies. +14 +Intermediate summary and antimatter in the Galaxy +The mechanism of PBH formation suggested in refs [2, 3] neatly cure all the problem +related to the observed population of the universe at high redshifts as well as of the present +day universe; The predicted log-normal spectrum of PBH is tested and confirmed by the +observations (the only one existing in the literature). The predicted existence of IMBH in +globular clusters is confirmed. +So the model works great. Thus the seemingly crazy by-product of refs [2, 3], namely +prediction of antimatter in the Galaxy can come true as well. Probably it is indeed the +case. A surprisingly huge flux of cosmic positrons, of He-antinuclei, and possibly even a +population of antistars seem to be observed. +14 + +14.1 +Anti-evidence: cosmic positrons +The observation of intense 0.511 line see refs [35–37], and earlier references therein, presents +a strong proof of abundant positron population in the Galaxy. In the central region of the +Galaxy electron–positron annihilation proceeds at a surprisingly high rate, creating the +flux: +Φ511 keV = 1.07 ± 0.03 · 10−3 photons cm−2 s−1. +(7) +The width of the line is about 3 keV. It proves that the annihilation takes place at rest. The +emission mostly goes from the Galactic bulge and at much lower level from the disk, The +source of 0.511 MeV line in the Galactic bulge. even got the name ”Great Annihihilator”. +Until recently the commonly accepted explanation was that e+ are created in the strong +magnetic fields of pulsars but the recent results of AMS probably exclude this mechanism, +since the spectra of ¯p and e+ at high energies are identical [38, 39]. It means that their +origin is the same and since antiprotons are not created in magnetic fields of pulsars, the +conclusion might follow that positrons are not produced by pulsars as well. However, this +might be true only for positrons with very high energies, about or above tera-electronvolts. +14.2 +Anti-evidence: cosmic antinuclei +The striking result reported by AMS team is the registration of anti-Helium-3 and anti- +Helium-4 nuclei. +Namely in 2018 AMS-02 announced possible observation of six He +3 +and two He +4 [40] [38]. In 2022 already 7 D (E ≲ 15 GeV) and 9 He, (E ∼ 50 GeV) +were observed. Surprisingly high fraction He/He ∼ 10−9 was registered [39]. It is not +excluded that the flux of anti-helium is even much higher because low energy He may +escape registration in AMS. +Secondary production of different antinuclei in cosmic ray was estimated in ref. [41]. +According to this work anti-deuterium could be tmost efficiently produced in the collisions +¯p p or ¯p He which can create the flux ∼ 10−7/m2/s−1/steradian/GeV/neutron), i.e. +5 +orders of magnitude below the observed flux of antiprotons. Antihelium could be created +in the similar reactions and the fluxes of He +3 and He +4, that could be created in cosmic +rays would respectively be 4 and 8 orders of magnitude smaller than the flux of anti-D. +After the AMS announcements of observations of anti-He4 there appeared theoretical +attempts to create anti-He4 through dark matter annihilation. This possibility does not +look natural. Moreover, DM annihilation would presumably create strong cosmic ray flux +of other particles, which is not observed. +In accordance with our model the observed antinuclej can be signatures of primordial +antimatter. However if they are synthesised in the standard big bang anti-nucleosynthesis +(anti-BBN) one would naturally expect the same abundances of light elements as those +created by the canonical BBN. According to the latter the abundances of deuterium and +helium-3 are much smaller then that of helium-4, approximately by 4 orders of magnitude, +while the relative fraction of these antinuclej are approximately equal. There might be +some astrophysical explanation of that or this anomaly is related to the fact that in our +model antismatter is created in bubbles with unusually high baryon-to-photon ratio β. In +the canonical BBN β ∼ 10−9, while in our case it may be as large as unity. However +15 + +150° +120° +90° +60° +30° +0° +-30° +-60° +-90° +-120° +-150° +-75° +-60° +-45° +-30° +-15° +0° +15° +30° +45° +60° +75° +10−12 +10−11 +10−10 +Energy flux, 100 MeV - 100 GeV [erg cm−2 s−1] +Figure 9: Positions and energy flux in the 100 MeV - 100 GeV range of antis- +tar candidates selected in 4FGL-DR2. Galactic coordinates. The background +image shows the Fermi 5-year all-sky photon counts above 1 GeV +if β ∼ 1 there is no primordial D. On the other hand in our scenario the formation of +primordial elements takes place inside non-expanding compact stellar-like objects with +fixed temperature. If the temperature is sufficiently high, this so called BBN may stop +before abundant He formation and ends with almost equal abundances of D and He. One +can see that looking at abundances of light elements at a function of temperature. If it is +so, antistars may have equal amount of D and He. +14.3 +Anti-evidence: antistars in the Galaxy +Last year a sensational announcement has been made [42] about possible discovery of 14 +antistars in our Galaxy, Milky Way. Quoting the authios: ”We identify in the catalog 14 +antistar candidates not associated with any objects belonging to established gamma-ray +source classes and with a spectrum compatible with baryon-antibaryon annihilation”. The +map of the observed anti-sources is presented in fig. 9. +An additional possible method for antistar detection in the Galaxy or in its halo has +been proposed in ref. [43]. In astrophysically plausible cases of the interaction of neutral +atmospheres or winds from antistars with ionised interstellar gas, the hadronic annihilation +will be preceded by the formation of excited p¯p and He¯p atoms. These atoms rapidly +cascade down to low levels prior to annihilation giving rise to a series of narrow lines which +can be associated with the hadronic annihilation gamma-ray emission. The most significant +are L (3p-2p) 1.73 keV line (yield more than 90%) from p¯p atoms, and M (4-3) 4.86 keV +(yield ∼ 60%) and L (3-2) 11.13 keV (yield about 25%) lines from He4¯p atoms. These +lines can be probed in dedicated observations by forthcoming sensitive X-ray spectroscopic +missions XRISM and Athena and in wide-field X-ray surveys like SRG/eROSITA all-sky +survey. +Bounds on the possible density if antistars in the Galaxy were studied in several our +papers [44–46]. It was shown that the restrictions are rather mild and an abundant density +of compact anti-stars in the universe even in the Galaxy does not violate existing obser- +vations. The reason is that the annihilation proceeds on a thin surface layer with a very +short depth by the order equal to the proton mean free path in the dense stellar medium. +16 + +On the other hand if the there were disperse antimatter clouds, the annihilation would be +by far more efficient and if so, the anticlouds did not survive to our time, though they +might have been existed in the early universe +A vey impressive would be star-antistar collision, which may even be a quasi-periodic +process of a star-antistar direct contact, explosion forcing them apart and return to each +other by gravitatinal attraction, etc... +15 +PBH and anti-creation mechanism +The model of PBH and antimatter creation of refs. [2, 3] is essentially based on the su- +persymmetry (SUSY) motivated baryogenesis, proposed by Affleck and Dine (AD) [20], +though the full extend SUSY is not necessary. SUSY predicts existence of scalar field χ (or +several such fields) with non-zero baryon number, B ̸= 0. Another important feature of +the scenario is existence of the flat directions in the self-potential of χ, i.e. the directions +along which the potential does not rise. Simple examples of such quadratic and quartic +potentials with flat directions are the following, for quartic self-interaction +Uλ(χ) = λ|χ|4 (1 − cos 4θ) +(8) +and of the quadratic mass-like term: +Um(χ) = m2|χ|2[[1 − cos(2θ + 2α)] +(9) +where χ = |χ| exp(iθ) and m = |m|eα. If α ̸= 0, C and CP are broken. In GUT SUSY +baryonic number is naturally non-conserved, due non-invariance of U(χ) with respect to +phase rotation, χ → χ exp(iθ)χ. +In the course of inflation χ quantum-fluctuates along flat directions with increasing +amplitude due to quantum instability of massless fields at De Sitter stage [47, 48]. Thus +taking into account that the wave length of the fluciuations exponentially rises, χ could +acquire a large classical value. +When inflation is over and the symmetry maintaining flat directions brakes down, χ +starts to evolve to the equilibrium point, χ = 0, according to the equation of the Newtonian +mechanics with the liquid friction term: +¨χ + 3H ˙χ + U ′(χ) = 0. +(10) +Due to quantum fluctuations orthogonal to the flat directions χ obtains momentum in this +orthogonal direction. This is how baryonic number of χ is generated: +Bχ = ˙θ|χ|2, +(11) +B is analogous to mechanical angular momentum in the two dimensional complex plane +[ℜeχ, ℑm χ]. +Decays of χ into quarks and antiquarks is supposed to conserve baryonic number and +transforms the baryonic number of χ into the baryonic number of quarks. In this process +a huge cosmological baryon asymmetry can be generated, much larger than the observed +17 + +one, β ≈ 10−9. If m ̸= 0, the angular momentum, B, could be generated due to possible +different direction of the quartic and quadratic valleys at low χ. In this case orthogonal +quantum fluctuation are unnecessary. If CP-odd phase α is small but non-vanishing, both +baryonic and antibaryonic domains might be formed with possible dominance of one of +them. Matter and antimatter domains might be created with possible global B ̸= 0. +In the model of ref. [2,3] the AD, scenario of baryogenesis was essentially modified due +to addition of interaction of the Affleck-Dine field χ with the inflaton Φ. The interaction +potential is taken in the form: +U = g|χ|2(Φ − Φ1)2 + λ|χ|4 ln +�|χ|2 +σ2 +� ++ (λ1χ4 + h.c.) + (m2χ2 + h.c.). +(12) +The first term in this expression is the new type of interaction potential between Φ and +χ. Φ1 is a constant, It is the amplitude of the inflaton field, taken by Φ in the process of +inflation. The remaining duration of inflation after Φ passed Φ1 should secure the num- +ber of e-foldings about 30-40 to allow for formation of sufficiently massive PBH. Though +the interaction between χ and Φ looks rather artificial, it is not so. This is the general +renormalizable coupling of two scalar fields. +The second term in eq. (12) is the Coleman-Weinberg potential [49] which arises as a +resukt of one-loop quantum corrections to λ|χ|4 interaction. +The remaining two terms are the toy model ones describing flat directions. +The constants λ1 and m are generally speaking complex. This may lead to C and CP +violation. However the charge symmetry would be broken only if the relative phase of λ1 +and m is nonzero. Coupling of χ to fermions (quarks) can break C and CP as well. +When Φ > Φ1, potential (12) has deep minimum near χ = 0 and χ classically stays +there. In the course of inflation Φ drops down and at some moment reaches Φ1. So the +barrier disappears and the window to the flat direction opens. During this period, when Φ +stays close to Φ1, the field χ started to diffuse away from the old minimum, according to +quantum diffusion equation derived by Starobinsky which we have generalised to a complex +field χ. At some stage for sufficiently large chi the diffusion turns into classical motion. +If the window to the flat direction, when Φ ≈ Φ1 is open only during relatively short +period, cosmologically small but possibly astronomically large bubbles with high β could +be created, occupying a small fraction of the universe volume. Indeed, when Φ passes +the value Φ1 sufficiently far, the old minimum at χ = 0 reappears and χ goes back to +zero. While χ is large it propagates along the flat direction of the quartic potential. When +finally χ becomes small, it starts to feel quadratic potential and in the process of motion +from quartic to quadratic potentials χ acquires a large angular momentum, that is a large +baryonic number. +If the probability of χ to reach a large value is not too big, cosmologically small but +possibly astrophysically large bubbles with high baryon density would be formed while the +rest of the universe has normal β ≈ 6 · 10−10, created by small χ. +Initially large isocurvature perturbations are generated in chemical content of massless +quarks, while, density perturbations stay practically zero. Density perturbations are gen- +erated rather late after the QCD phase transition, when massless quarks turns into heavy +nucleons with masses about 1 GeV, much larger than the temperature of the phase tran- +sition, Tqcd ∼ 100 MeV, The emerging universe looks like a piece of Swiss cheese, where +18 + +Figure 10: Schematic of the effective potential for χ and its evolution +holes are high baryonic density objects occupying a minor fraction of the universe volume. +These Hi-B Bubbles (HBB) mostly turn into primordial black holes The evolution of the +effective potential of χ is presented in Fig. 10. +This mechanism of massive PBH formation quite different from all others. The fun- +dament of PBH creation is build at inflation by making large isocurvature fluctuations at +relatively small scales, with practically vanishing density perturbations, which appeared +only much later. The mass spectrum of PBH reflects the distribution of the bubble by size +during inflation. Its log-normal form is a general feature of diffusion processes. +16 +Summary of the results and conclusion +• A large lot of outstanding problems of the canonical cosmology can be nicely resolved if +the universe is populated by primordial black holes with masses in the interval from the +19 + +Effective potential of x for different values of the inflaton field . The upper blue +curve corresponds to a large value >> Φl which gradually decreases down to Φ : +Dl, red curve. Then the potential returns back to the almost initial shape, as drops +down to zero. The evolution of x in such a potential is similar to a motion of a point- +like particle (shown as a black ball in the figure) in Newtonian mechanics. First, due to +guantum initial fluctuations x left the unstable extremum of the potential at x = O and +"tried" to keep pace with the moving potential minimum and later started to oscillate +around it with decreasing amplitude. The decrease of the oscillation amplitude was +induced by the cosmological expansion. In mechanical analogy the effect of the +expansion is equivalent to the liquid friction term, 3Hx. When D dropped below Φ1 +the potential recovered its original form with the minimum at x = O and x ultimately +returned to zero but before that it could give rise to a large baryon asymmetry +U(I +Φ >>1 +x+3Hx+U'(x) = 0. +=中1solar mass up to supermassive BHs with masses of the order of billion solar masses. +• The inverted mechanism of galaxy formation is proposed when firstly a SMBH was +formed and later it seeded the galaxy formation by gravitational attraction of the sur- +rounding matter. Thus an existence of SMBH in almost empty space can be understood. +• The early galaxies observed by HST and JWST, when the universe was only several +hundred million years old, could be created if seeded by SMBHs. +• Existence of a noticeable number of galaxies with masses that are is too small to allow +for creation the observed SMBHs inside, can be explained if these SMBH are primordial. +• Creation of SMBH in large contemporary galaxies by conventional accretion mechanism +demands time larger than the universe age. The problem disappears if the central SPBH +is primordial. +• Observations of several dwarf galaxies with SMBH in their centres, confirmed our pre- +diction made several years ago. +• The theoretically predicted log-normal mass spectrum of PBH is verified by the chirp +mass distribution of the gravitational waves observed by LIGO/Virgo. The agreement +between observation and theory is impressively good. +• PBHs formed according to our scenario explain the peculiar features of the sources of +GWs observed by LIGO/Virgo, e.g. an existence of BH with M = 100M⊙ +• The density of the intermediate mass black holes (IMBH), M = (102 − 105)M⊙ well +agrees with their primordial origin. Assumption of the astrophysical formation of IMBH +enoounters problems. +• Predicted by the model extremely old stars seem to exist even, one ”older than universe +star” is found; the older age is mimicked by the unusual initial chemistry. The model also +predicts too fast moving stars, which are also observed. +• Natural consequence of the suggested model of PBH creation leads to noticeable popu- +lation of our Galaxy by antimatter. This striking consequence seems to be confirmed by +recent observations. +Acknowledgement +The work was supported by the Russian Federation Ministry of Science and Higher Edu- +cation (project FSUS-2022-0015). +References +[1] A.D. Dolgov, Massive and supermassive black holes in the contemporary and early +Universe and problems in cosmology and astrophysics, Usp. Fiz. Nauk 188 (2018) 2, +121-142, Phys. Usp. 61 (2018) 2, 115-132. +[2] A. Dolgov, J. Silk, Baryon isocurvature fluctuations at small scale and baryonic dark +matter, PRD 47 (1993) 4244-4256. +20 + +[3] A.D. Dolgov, M. Kawasaki, N. Kevlishvili Inhomogeneous baryogenesis, cosmic anti- +matter, and dark matter, Nucl. Phys.B 807 (2009) 229-250, e-Print: 0806.2986 [hep- +ph] +[4] M. Castellano et al, ”Early results from GLASS-JWST. III: Galaxy candidates at +z ∼ (9 − 15)”, arXiv:2207.09436; +[5] S.L. Finkelstein, et al ”Long Time Ago in a Galaxy Far, Far Away: A Candidate +z ∼ 14 Galaxy...” arXiv:2207.12474. +[6] N. J. Adams, et al, ”Discovery and properties of ultra-high redshift galaxies +(9 < z < 12)...” arXiv:2207.11217. +[7] H. Yan et al, ”First Batch of Candidate Galaxies at Redshifts +11 to 20... ” +arXiv:2207.11558. +[8] N. Leethochawalit, et al, Early results from GLASS-JWST. X: Rest-frame UV-optical +properties of galaxies at 7 < z < 9”, arXiv:2207.11135; +[9] C. T. Donnan, et al, ”JWST Geers CEERS-93316, existed just 235 million years after +the Big Bang,” arXive 2207.12356. +[10] Y. Harikane, et al ”Comprehensive Study on Galaxies at z = 9 − 17 Found in the +Early JWST Data: UV Luminosity Functions and Cosmic Star-Formation History at +the Pre-Reionization Epoch” arXiv:2208.01612. +[11] A. Ferrara, A. Pallottini, P. Dayal, On the stunning abundance of super-early, massive +galaxies revealed by JWST, arXiv:2208.00720 +[12] M. Haslbauer, P. Kroupa, A.H. Zonoozi, H. Haghi, Has JWST already falsified dark- +matter-driven galaxy formation?, arXiv:2210.14915 [astro-ph.GA]. +[13] Ya. B. Zeldovich, I.D. Novikov, The Hypothesis of Cores Retarded During Expansion +and the Hot Cosmological Model, Astronomicheskij Zhurnal, 43 (1966) 758: Soviet +Astronomy, 10(4), 602-603 (1967). +[14] S. Hawking, ”Gravitationally collapsed objects of very low mass”. +[15] B. J. Carr and S. W. Hawking, ”Black holes in the early Universe,” Mon. Not. Roy. +Astron. Soc. 168, 399 (1974). +[16] F. Pacucci, A. Loeb, ”Accretion from Winds of Red Giant Branch Stars May Reveal +the Supermassive Black Hole in Leo I”, ApJL 940 L33, 2022. +[17] M. Mezcua, M. Siudek, H. Suh, et al, ”Overmassive black holes in dwarf galaxies out +to z ∼ 0.9 in the VIPERS survey”, arXiv:2212.14057 [astro-ph.GA] +[18] B. Liu, V. Bromm, ”Accelerating early galaxy formation with primordial black holes”, +arXiv:2208.13178, +21 + +[19] M. Volonteri, M Habouzit, M. Colpi, ”What if young z¿9 JWST galaxies hosted +massive black holes?” e-Print: 2212.04710. +[20] I. Affleck, M. Dine, A New Mechanism for Baryogenesis, Nucl. Phys. B 249 (1985) +361-380. +[21] B.J. Carr, J.H. Hilbert, J.E. Lidsey, ”Black hole relics and inflation: Limits on blue +perturbation spectra”, Phys.Rev.D 50 (1994) 4853, astro-ph/9405027 +[22] P. Ivanov, P. Naselsky, I. Novikov, Inflation and primordial black holes as dark matter, +PRD 50 (1994) 7173. +[23] J. Garcia-Bellido(, A.D. Linde, Density perturbations and black hole formation in +hybrid inflation Phys.Rev.D 54 (1996) 6040-6058, e-Print: astro-ph/9605094 [astro- +ph] +[24] A. Dolgov, K. Postnov, Why the mean mass of primordial black hole distribution is +close to 10M⊙”. JCAP 07 (2020) 063. +[25] S. Blinnkov, A. Dolgov, N. Porayko, K. Postnov, ”Solving puzzles of GW150914 by +primordial black holes,” CAP 1611 (2016), 036. +[26] J. Ziegler, K. Freese, ”Filling the Black Hole Mass Gap: Avoiding Pair Instability in +Massive Stars through Addition of Non-Nuclear Energy”, Phys. Rev. D 104, 043015 +(2021), arXiv:2010.00254: +[27] K. Postnov, N. Mitichkin, Spins of primordial binary black holes before coalescence, +JCAP 1906 (2019) no.06, 044, arXiv 1904.00570v5 [astro-ph.HE] +[28] K. Postnov, A. Kuranov, N. Mitichkin, ”Spins of black holes in coalescing compact +binaries”, Physics-Uspekhi vol. 62, No. 11, (2019), arXiv:1907.04218) . +[29] A.D. Dolgov, A.G. Kuranov, N.A. Mitichkin, S. Porey, K.A. Postnov, O.S. Sazhina, +I.V. Simkine On mass distribution of coalescing black holes, JCAP 12 (2020) 017, +e-Print: 2005.00892. +[30] G.F. Chapline, ”Cosmological effects of primordial black holes”. Nature, 253, 251 +(1975). +[31] B.Carr, F. Kuhnel, Primordial black holes as dark matter candidates, SciPost +Phys.Lect.Notes 48 (2022), e-Print: 2110.02821 [astro-ph.CO] +[32] C. Boehm, et al, Eliminating the LIGO bounds on primordial black hole dark matter, +arXiv:2008.10743 +[33] C. Corian`o, P.H. Frampton, Does CMB Distortion Disfavour Intermediate Mass Dark +Matter? arXiv:2012.13821 [astro-ph.GA] +[34] A. Dolgov, K. Postnov, ”Globular Cluster Seeding by Primordial Black Hole Popula- +tion”, JCAP 04 (2017) 036, e-Print: 1702.07621 [astro-ph.CO]. +22 + +[35] G. Weidenspointner et al., ”The sky distribution of positronium annihilation contin- +uum emission measured with SPI/INTEGRAL”, Astron. Astrophys. 450, 1013 (2006); +astro-ph/0601673. +[36] J. Knodlseder et al., ”The sky distribution of positronium annihilation continuum +emission measured with SPI/INTEGRAL”, Astron. Astrophys. 441, 513 (2005); +[arXiv:astro-ph/0506026]. +[37] P. Jean et al., ”Spectral analysis of the Galactic e+e- annihilation emission”, Astron. +Astrophys. 445, 579 (2006). [arXiv:astro-ph/0509298]. +[38] S. Ting, L’Aquila Joint Astroparticle Colloquium, 10th November, 2021. +[39] S. Ting, COSPAR 2022, 16-24 July.Choutko +[40] A. Choutko, AMS-02 Collaboration, AMS Days at La Palma, La Palma, Canary +Islands, Spain, (2018); +S. Ting, Latest Results from the AMS Experiment on the International Space Station. +Colloquium at CERN, May, 2018. +[41] R. Duperray, B. Baret, D. Maurin, et al, ”Flux of light antimatter nuclei near Earth, +induced by cosmic rays in the Galaxy and in the atmosphere”, Phys.Rev.D 71 (2005) +083013 • e-Print: astro-ph/0503544 [astro-ph]. +[42] S. Dupourqu´e, L. Tibaldo, P. von Ballmoos, ”Constraints on the antistar fraction +in the Solar System neighbourhood from the 10-year Fermi Large Area Telescope +gamma-ray source catalog”, Phys Rev D.103.083016 103 (2021) 083016. +[43] A.E. Bondar, S.I. Blinnikov, A.M. Bykov, A.D. Dolgov, K.A. Postnov, ”X-ray sig- +nature of antistars in the Galaxy”, JCAP 03 (2022) 03, 009, e-Print: 2109.12699 +[astro-ph.HE]. +[44] C. Bambi, A.D. Dolgov, +Antimatter in the Milky Way, Nucl.Phys. B 784 (2007) +132-150 e-Print: astro-ph/0702350, +[45] A.D. Dolgov, S.I. Blinnikov, +Stars and Black Holes from the very Early Universe, +Phys.Rev.D 89 (2014) 2, 021301 e-Print: 1309.3395. +[46] S.I. Blinnikov, A.D. Dolgov, K.A. Postnov, Antimatter and antistars in the universe +and in the Galaxy, Phys.Rev.D 92 (2015) 2, 023516 e-print: 1409.5736 +[47] A.D. Linde, Scalar field fluctuations in the expanding universe and the new inflation- +ary universe scenario, Phys. Lett. B116 (1982) 335-339 +[48] A.A. Starobinski, Dynamics of phase transitions in the new inflationary universe sce- +nario and generation of perturbations, Phys.Lett, B117 (1982) 175-178; +[49] S.R. Coleman, E.J. Weinberg, ’Radiative Corrections as the Origin of Spontaneous +Symmetry Breaking”, Phys.Rev.D 7 (1973) 1888-1910. +23 + diff --git a/eNAzT4oBgHgl3EQfaPxf/content/tmp_files/load_file.txt b/eNAzT4oBgHgl3EQfaPxf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e804858e6585287155fe243c22f4b48db4aa6e1 --- /dev/null +++ b/eNAzT4oBgHgl3EQfaPxf/content/tmp_files/load_file.txt @@ -0,0 +1,821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf,len=820 +page_content='Primordial black holes, early galaxies, and antimatter in the Milky Way Plenary ralk presented at 6th International Conference on Particle Physics and Astrophysics (ICCPA-2022) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Dolgova,b January 5, 2023 aDepartment of Physics, Novosibirsk State University, Pirogova st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2, Novosibirsk, 630090 Russia bBogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 6, Dubna, Moscow region, 141980 Russia Abstract Astronomical observations strongly incompatible with the canonical cosmological model are reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In particular too early formation of galaxies, as discovered by HST and JWST, are discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Other data revealing highly dense popula- tion of the very young universe with plethora of other different types of objects are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It is demonstrated that similar or maybe even more pronounced problems can be seen also in the present day universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It is argued that all of the above men- tioned problems can be nicely fixed by assumption that the universe is filled with primordial black holes in wide mass interval from a fraction of the solar mass up to supermassive BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The mechanism of PBH formation presented in 1993 is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The predicted by this mechanism log-normal mass spectrum of such PBH is shown to agree very well with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Finally possible rich population of our Galaxy by antimatter is discussed and new ways of its identification are presented arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='01365v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='CO] 3 Jan 2023 1 Introduction Discoveries of the last decades, especially by Hubble Space Telescope (HST) and James Webb Space Telescope (JWST), created strong confusion among the astronomical commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The early universe at redshifts z ∼ 10 and the age of a few hundred million years was found to be densely populated by galaxies, quasars, gamma-bursters, supernovae, and contained a very high level of dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' According to the common understanding the available universe age was by far less than the necessary time for the creation of such rich universe population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The frustrating feeling came to life that the standard cosmological model, well supported by the theory and ”experiment”, was on the verge of breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Not less striking and equally puzzling evidence, that does not fit the conventional frameworks, came from the observations of the contemporary universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The review of the problems, we encounter both in the early as well as in the present day universe, at the stage of art which existed 4 years ago, can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' But since that time much higher amount of controversies was accumulated thanks to active and precise astronomical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' All those surprising phenomena were in fact anticipated thirty years ago according to our paper [2], where a new mechanism of formation of highly massive primordial black hole (PBH) was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This mechanism was further elaborated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' A very simple log-normal mass spectrum of PBH was predicted that, as later verified, very well agrees with observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The abundant PBH population in very wide mass interval can eliminate the tension between theory and observations, in particular, because supermassive PBH could seed galaxy formation as it was envisaged in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In addition to high mass PBH formation, the mechanism of refs [2, 3] could lead to noticeable antimatter population of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In particular antimatter, including antistars, may exist in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This exciting predition seems to be confirmed by the recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Outline of this talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Discovery by JWST of the dense population of the early universe with galaxies, which according to the conventional cosmology cannot be there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' More data contradicting traditional cosmology and astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Anticipated resolution of the problems by PBH pre-suggested in papers [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Prediction and test of Log-normal mass spectrum of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The only mass spectrum confirmed by observatons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Black dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Gravitational waves and PBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Antimatter in the Galaxy (antistars, anit-nuclej, positrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Basics of the mechanism of PBH and antimatter creation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2 Strong blow to conventional cosmology by JWST Observatkions of the several recent months made by JWST created almost panic among traditional cosmologists and astrophysicists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It was discovered that the pretty young uni- verse with the age 200-300 million years contains plenty of bright galaxies, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [4]- [12], 1 which simply cannot be there according to the accepted faith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' As is stated in the JWST publications: an unexpectedly large density (stellar mass density ϱ∗ ≳ 106M⊙ Mpc−3) of massive galaxies (stellar masses M∗ ≥ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='5M⊙) are dis- covered at extremely high redshifts z ≳ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The following galaxies with record redshifts and the corresponding age of the universe are observed as reported by: CEERS (Cosmic Evolution Early Release Science): z = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='4, tU = 264 Myr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' z = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='7, tU = 235 Myr Such unbelievably early observed galaxies forced the JWST team to some retreat reported as: ”Bit of panic: Astronomers forced to rethink early JWST findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Re- vised instrument calibrations are bedevilling work on the distant Universe”, according to https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='com/articles/d41586-022-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Several examples of the JWST data with very high redshifts are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 1 and compared with theoretical expectations by the standard ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2 the Figure 1: Comparison of some JWST data with theoretical expectation results of JWST are compared with those by HST for two events for which both telescopes registered the same objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' An impressive agreement is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' According to the canonical theory of large scale structure (LSS) formation the density contrast ∆ ≡ δϱ/ϱ started to rise at the onset of the matter dominated stage at z = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' After that ∆ evolved as the cosmological scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Since initially ∆in ≲ 10−4, by the present time it may reach unity and after that fast LSS formation takes place (violent relaxation - strong rising of the gravitational field of the inhomogeneity) leading to the observed highly inhomogeneous universe at the galactic and galaxy clusters scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2 Moritz Haslbauer et al, Has JWST already falsified dark-matter-driven galaxy formation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='14915 12 Comparison of the size of the most massive galaxies, obtained ID 14924 in models of formation and growth of galaxies based on LCDM 11 (colored dots) with JWsT observations (black dots with errors) ID 1514 10 GL-z11 CEERS-1749 depending on the redshift of the observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' + GL-z13 6 IGN-z11 Redshift, z 8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='5 7 61 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 9 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='12131415 5161718 Redshift, z 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='5 TNG50-1 RefL0025N0752 RefL0050N0752 TNG100-1 RecalL0025N0752 RefL0100N1504 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 ACDM & invariant IMF 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 logio(Age/yr)Figure 2 3 XDFH-2395446286 Rychard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='Bouwens et al, Evolution of the UV LF from 100日 Z~15 to z~8 Using New JWST NIRCam Medium-Band Observations over the HUDF/XDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='02607 10 New coverage Joint observation of object XDFH-2395446286 and measuring its redshift z=12 HST and JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This is the fromJWST most distant galaxy ever discovered by HTS 30 years of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 1 0 5 10 15 Wavelength [μm] Redshift HST JWST Opt F125W F140W F160W F182M F210M F430M F460M F480M Marco Castellano et al, Early results from GLASS-JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='lll: Galaxy candidates at z~9-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='09436 Two more examples of galaxies with z=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='62 and z=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='3 found JWST in GHZ2 GHZ1 a couple of months of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' EP(z)E 121618 26 0369121618 28 上 28 个 30E 30 E 104 5×104 104 5×104 入oba(A) 入oba(A)In a simple way the process of structure foirmation can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The velocity of the Hubble runaway at distance r is vH = Hr and the virial velocity in the gravitational field of the inhomogeneity is v2 grav = 4πr3 rm2 Pl δϱ (1) Using H2 = (8πϱ)/(3m2 pl) we find vgrav ≥ vH if ∆ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The probability of such huge density fluctuation for the flat spectrum of perturbations is quite low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' There are two effects operating in the same directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Firstly the available time is constrained by the universe age, which is essentially equal to tU = 1/H and is quite short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In addition, a large value of H means expansion is very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' That suppresses the efficiency of the structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3 Problems preceding JWSP Similar serious problems are known already for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The Hubble space telescope discovered that the early universe, at z = 6 − 7 was too densely populated with quasars, alias SMBH, supernovae, gamma-bursters and happened to be very dusty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' No understand- ing is found in conventional cosmology how all these creature were given birth to in such a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Moreover great lots of phenomena in the ∼ 15 billion year old, present day universe are in strong tension with the conventional cosmological expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' HST sees the universe up to z = 6 − 7, but accidentally a galaxy at z ≈ 12 has been discovered for which both Hubble and Webb are in good agreement, as we have already mentioned in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Still, despite the earlier discoveries by HST, only after publications of JWST data the astronomical establishment became seriously worried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' To summarise: observational data of the last decades present more and more evidence indicating existence of the objects contradicting conventional astrophysics and cosmology in the present day and in quite young universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Rephrasing Marcellus from ”The Tragedy of Hamlet, Prince of Denmark” we can say ”Something is rotten in the state of Denmark the Universe”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' However, all the problems can be neatly solved if the universe is populated by primordial black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 4 BH types by formation mechanisms There three known types of BH depending upon the mechanism of their creation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Astrophysical black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This BHs are created by the collapse of a star which exhausted its nuclear fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The expected masses should start immediately above the neutron star mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' about 3M⊙, but noticeably below 100M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Instead we observe that the BH mass spectrum in the galaxy has maximum at M ≈ 8M⊙ with the width ∼ (1−2)M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The result is somewhat unexpected but an explanations in the conventional astrophysical frame- works is not excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 4 Recently LIGO/Virgo discovered BHs with masses close to 100M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Their astro- physical origin was considered to be complitely impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Now some, though quite exotic, formation mechanisms have been suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Accretion created BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Such BHs are formed by the accretion of matter on the mass excess in galactic centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It is known that in any large galaxy at the centre there exists a supermassive black holes (SMBH) with masses varying from a few millions M⊙ (e,g, Milky Way) up to almost hundred billions M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' However, the conventional accretion mechanisms are not efficient enough to create such monsters during the universe life-time, tU ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='6 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' At least 10-fold longer time is necessary, some references can be found in [1], to say nothing about SMBH in 10 times younger universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Primordial black holes (PBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' PBH are supposed to be formed in the very early universe during pre-stellar epoch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' prior to star formation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='The idea of primordial black holes and a possible mechanism of their creation was pioneered by Zeldovich and Novikov [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' According to their idea, the density contrast in the early universe inside the bubble radius, essentially equal to the cosmological horizon, might accidentally happen to be large, δϱ/ϱ ≈ 1, then that piece of volume would be inside its gravitational radius i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' it became a PBH, that decoupled from the cosmological expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The mechanism was elaborated later by Hawking [14], and by Carr and Hawking [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 5 BH types by masses Rather arbitrary black holes are separated into three groups depending on their mass: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Supermassive black holes (SMBH): M = (106 − 1010)M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Intermediate mass black holes (IMBH): M = (102 − 105)M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Solar mass black holes: masses from a fraction of M⊙ up to 100M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' There can be also very light black holes, not yet observed, with masses in the region ∼ 1020 g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' they may be the ”particles” of the cosmological dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The origin of most of these BHs is unclear, except maybe of the BHs with masses of a few solar masses, which may be astrophysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Extremely unexpected was very high abundance of IMBH which are appearing during last several years in huge numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The assumption that (almost) all these black holes in the universe are primordial except possibly the very light ones, strongly reduces or even eliminates the tension between their observed abundances and possible mechanisms of their formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 6 Problems of the contemporary universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' SMBH in all large galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The universe age is too short for their formation through the commonly accepted accretion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Several SMBH are found in very small galaxies and even in (almost) empty space, where not only the time duration but also an amount of material is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' An inter- esting recent observation was made by the Hobby-Eberly Telescope at Texas’s McDonald Observatory suggesting the presence of a black hole with a mass of about 17 billion M⊙ equivalent to 14% of the total stellar mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Usually the mass of the central BH is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='1 % of the galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This SMBH was observed by the analysis of the motions of the stars near the center of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' There appeared recently fresh evidence [16] indicating to supermassive BH with the mass 3 × 106M⊙ in dwarf galaxy Leo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Much more new data are presented practically today [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Six dwarf galaxies are identified that have X-ray AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' They are presumbly powered by SMBHs of M > 107M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It is not excluded, that such SMBHs, that are not hosted by a large galaxy, might be pushed out of large galaxies in the process of galaxy collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Such catastrophic event may even create plenty of wandering single supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' However, taking into account a large number of such exotics, much more natural seems that all SMBH in small galaxies are primordial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Simply they were unlucky not to acquire their own large galaxy, since there was not enough matter around to build large galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Problems with the BH mass spectrum in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' According to the data the masses are concentrated in the narrow interval (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='2)M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Origin and properties of the sources of the observed gravitational waves, encounter considerable difficulties, if one tries to explain them assuming astrophysical formation of back hole binaries emitting the observed gravitational radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' IMBH, with M ∼ (103 − 105)M⊙ are unexpectedly discovered in dwarfs and globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Their origin is unclear, if they are not primordial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Invisible Massive Astrophysical Compact Halo Objects (MACHOs), non-luminous objects with masses ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='5M⊙ observed through microlensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It is unknown what are they and how they were created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Existence of very unusual stars in the Galaxy, among which there are too fast moving stars and stars with unusual chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Moreover, too old stars, are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Many of them look older than the Galaxy and maybe one is even older that the universe (sic!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' An assumption, that the black holes mentioned in the list above, are primordial elimi- nates all the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The mechanism of PBH formation suggested in papers [2,3] predicts also existence of the unusual stars mentioned in point 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 7 Observations of black holes The ancient point of view: BH are objects with so strong gravitational field that nothing can escape it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' According to Mitchell (1784): there may be bodies for which the second cosmic velocity is larger than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' They do not shine and do not reflect light, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' are absolutely dark, invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' However, the truth is quite the opposite, black holes are very well seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' BH can emit all kind of radiation through the Hawking evaporation (though nobody has yet seen it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The most powerful sources of radiation in the universe are SMBH - quasars, point-like objects 6 shining as a thousands of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The methods of the BH observation include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Central mass estimate through analysis of stellar motion around the supposed BH as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' discovery of BH in the center of the Millki Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Gravitational lensing (MACHO and some other BHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Electromagnetic radiation from the accreting matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' it is the mechanism of quasar central engine, but mush smaller BH are also observed that way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' However, all these methods allow only to establish that there is a large mass inside small volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' We need theory to conclude that there should be a black hole inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' But the following method is free from this restriction: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Registration of gravitational waves from coalescing double systems of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The data directly show that there are exactly coalescence of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This is the first test of General Relativity for strong fields and the first observational proof of existence of the Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 8 Galaxy seeding by SMBH We see thar a large amount of observational data are at odds with the conventional model but nicely agrees with the model of creation of primordial black holes and primordial stars suggested in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' According to the standard approach the SMBH in galactic centres are formed after galaxies were created by accretion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [2,3] the validity of the opposite scenario was conjectured, namely, SMBHs were formed first and subsequently seeded galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The hypothesis advocated in these works allows to explain presence of SMBH in all large and several small galaxies accessible to observation and resolves the problem of very early existence of galaxies observed by HST and JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This statement was recently rediscovered in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [18] according to which ”Recent ob- servations with JWST have identified several bright galaxy candidates at z ≳ 10, some of which appear unusually massive (up to ∼ 1011 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Such early formation of mas- sive galaxies is difficult to reconcile with standard ΛCDM predictions, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='we show that the observed massive galaxy candidates can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' if structure formation is ac- celerated by massive (≳ 109 M⊙) primordial black holes that enhance primordial density fluctuations.” Very recently, in December, 2022, there appeared another paper on the possibility of SMBH impact on JWST-galaxy formation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 9 PBH and inflation The mechanism suggested in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [2, 3] introduced some new features which were later explored in a series of subsequent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The proposed there scenarios are heavily based on the Affleck-Dine [20] model of baryogenesis, that permits to create very interesting features of the PBH population or some other macroscopic compact objects, see below, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 15 In paper [2] inflationary mechanism was first implied for PBH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It allowed to 7 create PBH with huge masses, much larger than those in the previously studied models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' A year later inflationary creation of PBH was explored in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [21] soon after in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' [22], and two years later in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Nowadays there exploded an avalanche of papers on inflationary formation of PBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Presently inflationary mechanism of PBH production is commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' However, except for predicting large masses of PBH, the models do not have much predictive power because the mass spectra of the created PBHs are quite complicated and strongly parameter dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' No simple analytic expressions have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The only exception is the mechanism of refs [2, 3], which predicts extremely simple log-normal mass spectrum of PBH: dN dM = µ2 exp [−γ ln2(M/M0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' (2) The central value mass can be predicted theoretically [24]: M0 ∼ 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It is equal to the horizon mass at QCD phase transition from the free quark-gluon phase to the confinement phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' To be more precise the horizon mass is approximately equal to 10M⊙ for the cosmic plasma with vanishingly small chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In our case µ is supposed to be large of the order of the plasma temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In this case the phase tranasion was probably delayed and the horizon mass could be 2-3 times larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' An impressive feature of the the log-normal mass spectrum with the predicted value of M0 is that it is the only known spectrum tested by ”experiment” in very good agreement with the observed densities of black holes in all mass intervals from the solar mass BH, up to black holes with intermediates masses, and further up to supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In partcular, the mechanism developed in [2, 3] allows to explain the presence of SMBH in all large and several small galaxies accessible to observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Especially impressive is the confirmation of the model by the chirp mass binaries measured by LIGO/Virgo which is discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 10 Gravitational waves from BH binaries There is general agreement between several groups that the gravitational waves discovered by LIGO/Virgo interferometers originated from PBH binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' We discuss this issue here following our paper [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' There are three problems which indicate that the sources of GWs are most naturally primordial black holes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Origin of heavy BHs (with masses ∼ 30M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' To form so heavy BHs, the progenitors should have M > 100M⊙ and a low metal abundance to avoid too much mass loss during the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Such heavy stars might be present in young star-forming galaxies but they are not observed in the necessary amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Recently there emerged much more striking problem because of the observation of BH with M ∼ 100M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Formation of such black holes in the process of stellar collapse was considered to be strictly forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Some exotic mechanisms might be possibly allowed, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' BH formation in the process of collapse of supermassive star heated by dark matter annihilation inside [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' On the other hand, primordial black holes with the observed by LIGO masses may be easily created with sufficient density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Formation of BH binaries from the original stellar binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Stellar binaries are formed from common interstellar gas clouds and are quite frequent in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' If BH is created through stellar collapse, small non-sphericity results in a huge velocity of the BH and the binary is destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' BH formation from PopIII stars and subsequent formation of BH binaries with tens of M⊙ is estimated to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The problem of the binary formation is simply solved if the observed sources of GWs are the binaries of primordial black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' They were at rest in the comoving volume, when inside horizon they are gravitationally attracted and may loose energy due to dynamical friction in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The prob- ability for them to become gravitationally bound is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The conventional astrophysical scenario is not excluded but less natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Low spins of the coalescing BHs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The low values of the BH spins sae observed in GW150914 and in almost all (except for three) other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' It strongly constrains as- trophysical BH formation from close binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Astrophysical BHs are expected to have considerable angular momentum, nevertheless the dynamical formation of double massive low-spin BHs in dense stellar clusters is not excluded, though difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' On the other hand, PBH practically do not rotate because vorticity perturbations in the early universe are vanishingly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Still, individual PBH forming a binary initially rotating on elliptic orbit could gain collinear spins about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='3, rising with the PBH masses and eccentricity [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' This result is in agreement with the GW170729 LIGO event produced by the binary with masses 50M⊙ and 30M⊙ and and GW190521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' To summarize: each of the mentioned problems may be solved in the conventional frameworks but it looks much simpler to assume that the LIGO/Virgo sources are primor- dial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 11 Chirp mass distribution Two rotating gravitationally bound massive bodies are known to emit gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In quasi-stationary inspiral regime, the radius of the orbit and the rotation frequency are approximately constant and the GW frequency is twice the rotation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The luminosity of the GW radiation is: L = 32 5 m2 Pl �Mc ωorb m2 Pl �10/3 , (3) where M1, M2 are the masses of two bodies in the binary system and Mc is the so called chirp mass: Mc = (M1 M2)3/5 (M1 + M2)1/5 , (4) and ω2 orb = M1 + M2 m2 PlR3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' (5) The available data on the chirp mass distribution of the black holes in the coalescing binaries in O1-O3 LIGO/Virgo runs are analyzed and compared with theoretical expecta- tions based on the hypothesis that these black holes are primordial with log-normal mass 9 spectrum [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The inferred best-fit mass spectrum parameters are: M0 = 17M⊙ and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='9, fall within the theoretically expected range and show excellent agreement with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' On the opposite, binary black hole formation based on massive binary star evolution require additional adjustments to reproduce the observed chirp mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The results are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 ℳ KS M0=15, γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='7 M0=17, γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='9 EDF F(<ℳ) Figure 3: Model distribution FPBH(< M) with parameters M0 and γ for two best Kolmogorov-Smirnov tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' EDF= empirical distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 ℳ CO a=1 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='1 BH a=1 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='1 EDF F(<ℳ) Figure 4: Cumulative distributions F(< M) for several astrophysical models of binary BH coalescences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' So we can conclude that PBHs with log-normal mass spectrum perfectly fit the data, while astrophysical BHs seem to be disfavoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' New recent dats on GW observations were analysed by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Postnov in his talk at XXXIV International Workshop on High Energy Physics ”From Quarks to Galaxies: Elu- cidating Dark Sides” are depicted in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' In this talk is also presented an approximate fitting of the observed chirp-mass distri- bution in the O1-O3 LVK GW compact binary coalescences (from GWTC-3 catalog) by 10 two independent PBH populations with initial log-normal mass distributions M (1) 0 = 5M⊙ and M (2) 0 = 30M⊙, see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' Figure 5 0 10 20 30 40 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 ℳ N (< ℳ ) EDF Mc 1=5MSun, γ1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='1, Mc 2=30MSun, γ2=7, fPBH 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='37, fPBH 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='63 Figure 6: Approximation of the observed chirp-mass distribution in the O1- O3 LVK GW compact binary coalescences (from GWTC-3 catalog) by two independent PBH populations, from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='Postnov talk In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 7 one can see the same approximation but by the simplest model of astro- physical BH formation from the collapse of the CO-core of a massive star and standard common-envelope parameter, with taking into account evolution of star-formation rate in the universe with redshift (Postnov & Kuranov, 2019), plus a population of PBHs with log-normal initial mass distribution with M (2) 0 = 33M⊙ can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The picture looks more complicated than the earlier one described at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' One possible interpretation is that there are two populations of PBH with log-normal mass spectrum each but with different values of M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' The model can be modified 11 Example: log-normal PBH mass function GWTC1+GWTC2 F(M) = Aexp[-yIn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content=' (M /Mo)) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfaPxf/content/2301.01365v1.pdf'} +page_content='0 F( 0 through the use of a first-order method proposed in [26], where L is the AL +function of (1) defined as +L(x, y, λx, λy; ρ) = F(x, y)+ 1 +2ρ +� +∥[λx + ρc(x)]+∥2 − ∥λx∥2� +− 1 +2ρ +� +∥[λy + ρd(x, y)]+∥2 − ∥λy∥2� +. +(3) +The Lagrangian multiplier estimate is then updated by λk+1 +x += ΠB+ +Λ (λk +x+ρkc(xk+1)) and λk+1 +y += +[λk +y + ρkd(xk+1, yk+1)]+ for some Λ > 0, where ΠB+ +Λ(·) and [·]+ are defined in Section 1.1. +The main contributions of this paper are summarized below. +• We propose a first-order AL method for solving problem (1). To the best of our knowledge, +this is the first yet implementable method for solving (1). +• We show that under some suitable assumptions, our first-order AL method enjoys an iter- +ation complexity of O(log ε−1) and an operation complexity of O(ε−4 log ε−1), measured +by the amount of evaluations of ∇f, ∇c, ∇d and proximal operator of p and q, for finding +an ε-KKT solution of (1). +The rest of this paper is organized as follows. In Subsection 1.1, we introduce some notation +and terminology. In Section 2, we propose a first-order AL method for solving problem (1). In +2 + +Section 3, we present complexity results for the proposed method. In Section 4, we provide the +proof of the main result. +1.1 +Notation and terminology +The following notation will be used throughout this paper. Let Rn denote the Euclidean space +of dimension n and Rn ++ denote the nonnegative orthant in Rn. The standard inner product, +l1-norm and Euclidean norm are denoted by ⟨·, ·⟩, ∥ · ∥1 and ∥ · ∥, respectively. For any Λ > 0, +let B+ +Λ = {x ≥ 0 : ∥x∥ ≤ Λ}, whose dimension is clear from the context. For any v ∈ Rn, let v+ +denote the nonnegative part of v, that is, (v+)i = max{vi, 0} for all i. Given a point x and a +closed set S in Rn, let dist(x, S) = minx′∈S ∥x′ − x∥, ΠS(x) denote the Euclidean projection of +x onto S, and IS denote the indicator function associated with S. +A function or mapping φ is said to be Lφ-Lipschitz continuous on a set S if ∥φ(x)−φ(x′)∥ ≤ +Lφ∥x−x′∥ for all x, x′ ∈ S. In addition, it is said to be L∇φ-smooth on S if ∥∇φ(x)−∇φ(x′)∥ ≤ +L∇φ∥x − x′∥ for all x, x′ ∈ S. For a closed convex function p : Rn → R ∪ {∞},2 the proximal +operator associated with p is denoted by proxp, that is, +proxp(x) = arg min +x′∈Rn +�1 +2∥x′ − x∥2 + p(x′) +� +∀x ∈ Rn. +Given that evaluation of proxγp(x) is often as cheap as proxp(x), we count the evaluation of +proxγp(x) as one evaluation of proximal operator of p for any γ > 0 and x ∈ Rn. +For a lower semicontinuous function φ : Rn → R ∪ {∞}, its domain is the set dom φ := +{x|φ(x) < ∞}. The upper subderivative of φ at x ∈ dom φ in a direction d ∈ Rn is defined by +φ′(x; d) = lim sup +x′ φ→x, t↓0 +inf +d′→d +φ(x′ + td′) − φ(x′) +t +, +where t ↓ 0 means both t > 0 and t → 0, and x′ +φ→ x means both x′ → x and φ(x′) → φ(x). +The subdifferential of φ at x ∈ dom φ is the set +∂φ(x) = {s ∈ Rn��sTd ≤ φ′(x; d) ∀d ∈ Rn}. +We use ∂xiφ(x) to denote the subdifferential with respect to xi. +In addition, for an upper +semicontinuous function φ, its subdifferential is defined as ∂φ = −∂(−φ). If φ is locally Lipschitz +continuous, the above definition of subdifferential coincides with the Clarke subdifferential. +Besides, if φ is convex, it coincides with the ordinary subdifferential for convex functions. Also, +if φ is continuously differentiable at x , we simply have ∂φ(x) = {∇φ(x)}, where ∇φ(x) is the +gradient of φ at x. In addition, it is not hard to verify that ∂(φ1 + φ2)(x) = ∇φ1(x) + ∂φ2(x) if +φ1 is continuously differentiable at x and φ2 is lower or upper semicontinuous at x. See [7, 46] +for more details. +Finally, we introduce an (approximate) stationary point (e.g., see [9, 10, 21]) for a general +minimax problem +min +x max +y +Ψ(x, y), +(4) +where Ψ(·, y) : Rn → R∪{∞} is a lower semicontinuous function, and Ψ(x, ·) : Rm → R∪{−∞} +is an upper semicontinuous function. +Definition 1. A point (x, y) is said to be a stationary point of the minimax problem (4) if +0 ∈ ∂xΨ(x, y), +0 ∈ ∂yΨ(x, y). +In addition, for any ǫ > 0, a point (xǫ, yǫ) is said to be an ǫ-stationary point of the minimax +problem (4) if +dist (0, ∂xΨ(xǫ, yǫ)) ≤ ǫ, +dist (0, ∂yΨ(xǫ, yǫ)) ≤ ǫ. +2For convenience, ∞ stands for +∞. +3 + +2 +A first-order augmented Lagrangian method for problem (1) +In this section we propose a first-order augmented Lagrangian (FAL) method for problem (1). +One standard approach for solving constrained nonlinear program is to solve a sequence +of unconstrained nonlinear program problems, which are typically penalty or augmented La- +grangian subproblems (e.g., see [32]). In a similar spirit, we next propose an FAL method in +Algorithm 1 for solving (1). In particular, at each iteration, the FAL method finds an approxi- +mate stationary point of an AL subproblem in the form of +min +x max +y +L(x, y, λx, λy; ρ) +(5) +for some ρ > 0, λx ∈ R˜n ++ and λy ∈ R ˜m ++, where L is the AL function associated with problem +(1) defined in (3). In view of Assumption 1, one can observe that L enjoys the following nice +properties. +• For any given ρ > 0, λx ∈ R˜n ++ and λy ∈ R ˜m ++, L is the sum of smooth function f(x, y) + +� +∥[λx + ρc(x)]+∥2 − ∥λx∥2� +/(2ρ)− +� +∥[λy + ρd(x, y)]+∥2 − ∥λy∥2� +/(2ρ) with Lipschitz con- +tinuous gradient and possibly nonsmooth function p(x) − q(y) with exactly computable +proximal operator. +• L is nonconvex in x but concave in y. +Thanks to such a nice structure of L, an approximate stationary point of the AL subproblem +(5) can be found by Algorithm 3 (see Appendix A), which is a first-order method proposed in +[26, Algorithm 2]) for solving nonconvex-concave minimax problems. +Before presenting an FAL method for (1), we let +Lx(x, y, λx; ρ) := F(x, y) + 1 +2ρ +� +∥[λx + ρc(x)]+∥2 − ∥λx∥2� +, +(6) +chi := max{∥c(x)∥ +��x ∈ X}, +dhi := max{∥d(x, y)∥ +��(x, y) ∈ X × Y}, +(7) +and make one additional assumption on problem (1). +Assumption 2. For any given η ∈ (0, 1], an η-approximately feasible point zη of problem (1), +namely zη ∈ X satisfying ∥[c(zη)]+∥ ≤ η, can be found. +Remark 1. A very similar assumption as Assumption 2 was considered in [5, 17, 27, 48]. One +example of the problem instances satisfying Assumption 2 arises when the error bound condition +∥[c(x)]+∥ = O(dist(0, ∂(∥[c(x)]+∥2 +IX (x))))ν) holds on a level set of ∥[c(x)]+∥ for some ν > 0 +(e.g., see [24, 36]). Indeed, one can find the above zη by applying a projected gradient method +to the problem minx∈X ∥[c(x)]+∥2. +We are now ready to present an FAL method for solving problem (1). +4 + +Algorithm 1 A first-order augmented Lagrangian method for problem (1) +Input: ε, τ ∈ (0, 1), ǫ0 ∈ (τε, 1], ǫk = ǫ0τ k, ρk = ǫ−1 +k , Λ > 0, λ0 +x ∈ B+ +Λ, λ0 +y ∈ R ˜m ++, (x0, y0) ∈ +X × Y, and xnf ∈ X with ∥[c(xnf)]+∥ ≤ √ε. +1: for k = 0, 1, . . . do +2: +Set +xk +init = +� xk, +if Lx(xk, yk, λk +x; ρk) ≤ Lx(xnf, yk, λk +x; ρk), +xnf, +otherwise. +(8) +3: +Call Algorithm 3 (see Appendix A) with ǫ ← ǫk, ǫ0 ← ǫk/(2√ρk), (x0, y0) ← (xk +init, yk) +and L∇h ← Lk to find an ǫk-stationary point (xk+1, yk+1) of +min +x max +y +L(x, y, λk +x, λk +y; ρk) +(9) +where +Lk = L∇f + ρkL2 +c + ρkchiL∇c + ∥λk +x∥L∇c + ρkL2 +d + ρkdhiL∇d + ∥λk +y∥L∇d. +(10) +4: +Set λk+1 +x += ΠB+ +Λ(λk +x + ρkc(xk+1)) and λk+1 +y += [λk +y + ρkd(xk+1, yk+1)]+. +5: +Terminate the algorithm and output (xk+1, yk+1) if ǫk ≤ ε. +6: end for +Remark 2. +(i) xnf is an √ε-approximately feasible point of problem (1), where the subscript +“nf” stands for “nearly feasible”. It follows from Assumption 2 that xnf can be found in +advance. +(ii) λk+1 +x +results from projecting onto a nonnegative Euclidean ball the standard Lagrangian +multiplier estimate ˜λk+1 +x +obtained by the classical scheme ˜λk+1 +x += [λk +x + ρkc(xk+1)]+. It is +called a safeguarded Lagrangian multiplier in the relevant literature [2, 20, 3], which has +been shown to enjoy many practical and theoretical advantages (see [2] for discussions). +(iii) In view of Theorem 2 (see Appendix A), one can see that an ǫk-stationary point of (9) +can be successfully found in step 3 of Algorithm 1 by applying Algorithm 3 to problem (9) +and thus Algorithm 1 is well-defined. +3 +Complexity results of Algorithm 1 +In this section we establish iteration and operation complexity results for Algorithm 1. Before +proceeding, we make one additional assumption that a generalized Mangasarian-Fromowitz +constraint qualification holds for the minimization part of (1) and a uniform Slater’s condition +holds for the maximization part of (1). +Assumption 3. +(i) There exist some constants δc, θa, θf > 0 such that for each x ∈ F(θf) +there exists some vx ∈ Rn satisfying ∥vx∥ = 1 and vT +x ∇ci(x) ≤ −δc for all i ∈ A(x; θa), +where +F(θf) = {x ∈ X +��∥[c(x)]+∥ ≤ θf}, +A(x; θa) = {i|ci(x) ≥ −θa, 1 ≤ i ≤ ˜n}. +(11) +(ii) For each x ∈ X, there exists some ˆyx ∈ Y such that di(x, ˆyx) < 0 for all i = 1, 2, . . . , ˜m, +and moreover, δd := inf{−di(x, ˆyx)|x ∈ X, i = 1, 2, . . . , ˜m} > 0.3 +3The latter part of this assumption can be weakened to the one that the pointwise Slater’s condition holds for +5 + +In order to characterize the approximate solution found by Algorithm 1, we next introduce +a terminology called an ε-KKT solution of problem (1). +One can observe from Lemma 1(iii) that problem (1) is equivalent to +min +x,λy +� +max +y +F(x, y) − ⟨λy, d(x, y)⟩ + IR ˜ +m ++ (λy) +��c(x) ≤ 0 +� +. +By this, one can further see that problem (1) is equivalent to +min +x,λy max +λx +� +max +y {F(x, y) − ⟨λy, d(x, y)⟩ + IR ˜ +m ++ (λy)} + ⟨λx, c(x)⟩ − IR˜n ++(λx) +� +, +which is a nonconvex-concave minimax problem +min +x,λy max +y,λx +� +F(x, y) + ⟨λx, c(x)⟩ − ⟨λy, d(x, y)⟩ − IR˜n ++(λx) + IR ˜ +m ++ (λy) +� +. +(12) +It then follows from Definition 1 (see also [9, Theorem 3]) that (x, y, λx, λy) ∈ Rn×Rm×R˜n ++×R ˜m ++ +is a stationary point of problem (12) if +0 ∈ ∂xF(x, y) + ∇c(x)λx − ∇xd(x, y)λy, +(13) +0 ∈ ∂yF(x, y) − ∇yd(x, y)λy, +(14) +c(x) ≤ 0, +⟨λx, c(x)⟩ = 0, +(15) +d(x, y) ≤ 0, +⟨λy, d(x, y)⟩ = 0. +(16) +Based on this observation and the equivalence of (1) and (12), we introduce an (approximate) +KKT solution of problem (1) below. +Definition 2. The pair (x, y) is said to be a KKT solution of problem (1) if there exists +(λx, λy) ∈ R˜n ++ × R ˜m ++ such that the conditions (13)-(16) hold. In addition, for any ε > 0, (x, y) +is said to be an ε-KKT point of problem (1) if there exists (λx, λy) ∈ R˜n ++ × R ˜m ++ such that +dist(0, ∂xF(x, y) + ∇c(x)λx − ∇xd(x, y)λy) ≤ ε, +dist(0, ∂yF(x, y) − ∇yd(x, y)λy) ≤ ε, +∥[c(x)]+∥ ≤ ε, +|⟨λx, c(x)⟩| ≤ ε, +∥[d(x, y)]+∥ ≤ ε, +|⟨λy, d(x, y)⟩| ≤ ε. +To study complexity of Algorithm 1, we define +f ∗(x) := max{F(x, y)|d(x, y) ≤ 0}, +(17) +f ∗ +low := inf{f ∗(x)|x ∈ X}, +(18) +Dx := max{∥u − v∥ +��u, v ∈ X}, +Dy := max{∥u − v∥ +��u, v ∈ Y}, +(19) +Fhi := max{F(x, y)|(x, y) ∈ X × Y}, +Flow := min{F(x, y)|(x, y) ∈ X × Y}, +(20) +r := 2δ−1 +d (ǫ0 + LF)Dy, +(21) +K := ⌈(log ε − log ǫ0)/ log τ⌉+ , +K := {0, 1, . . . , K + 1}, +(22) +where LF and δd are given in Assumptions 1 and 3, and ǫ0, ε, and τ are some input parameters +of Algorithm 1. For convenience, we define K − 1 = {k − 1|k ∈ K}. One can observe from +Assumption 1 that Dx, Dy, Fhi and Flow are finite. Besides, as will be shown in Lemma 1, f ∗ +low +is also finite. +We are now ready to present an iteration and operation complexity of Algorithm 1 for finding +an O(ε)-KKT solution of problem (1), whose proof is deferred to Section 4. +the constraint on y in (1), that is, there exists ˆyx ∈ Y such that d(x, ˆyx) < 0 for each x ∈ X . Indeed, if δd > 0, +Assumption 3(ii) holds. Otherwise, one can solve the perturbed counterpart of (1) with d(x, y) being replaced +by d(x, y) − ǫ for some suitable ǫ > 0 instead, which satisfies Assumption 3(ii). +6 + +Theorem 1. Suppose that Assumptions 1, 2 and 3 hold. Let {(xk, yk, λk +x, λk +y)}k∈K be generated +by Algorithm 1, chi, dhi, f ∗ +low, Dx, Dy, Fhi, Flow and K be defined in (7), (18), (19), (20) and +(22), LF, L∇f, L∇d, L∇c, Lc, L∇d, Ld and δd be given in Assumption 1, ε, ǫ0, τ, Λ and λ0 +y be +given in Algorithm 1, and +�L = L∇f + L2 +c + chiL∇c + ΛL∇c + L2 +d + dhiL∇d + L∇d +� +∥λ0y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +, +(23) +ˆα = min +� +1, +� +4/(Dy �L) +� +, +ˆδ = (2 + ˆα−1)�LD2 +x + max{1/Dy, �L/4}D2 +y, +(24) +� +M = 16 max +� +1/(2L2 +c), 4/(ˆαL2 +c) +� � +(3�L + 1/(2Dy))2/ min{L2 +c, 1/(2Dy)} + 3�L + 1/(2Dy) +�2 +× +� +ˆδ + 2ˆα−1� +Fhi − Flow + Λ2 +2 + 3 +2∥λ0 +y∥2 + 3(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ρkd2 +hi + Dy +4 + �LD2 +x +�� +, +(25) +�T = +� +16 +� +LF Dy + Fhi − f ∗ +low + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ ++ Λ2 +2 + Dy +4 +� +�L ++ 8(1 + 4D2 +y �L2) +� ++ +, +(26) +˜λK+1 +x += [λK +x + c(xK+1)/(ǫ0τ K)]+. +(27) +Suppose that +ε−1 ≥ max +� +1, θ−1 +a Λ, θ−2 +f +� +2LF Dy + 2Fhi − 2f ∗ +low + 2Λ + τ −1 + ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ǫ0Dy +2 ++ L−2 +c ++ 4D2 +y �L + Λ2� +, 4∥λ0 +y∥2 +δ2 +dτ ++ 8(Fhi − f ∗ +low + Dyǫ0) +δ2 +dτ(1 − τ) +� +. +(28) +Then the following statements hold. +(i) Algorithm 1 terminates after K+1 outer iterations and outputs an approximate stationary +point (xK+1, yK+1) of (1) satisfying +dist(0, ∂xF(xK+1, yK+1) + ∇c(xK+1)˜λK+1 +x +− ∇xd(xK+1, yK+1)λK+1 +y +) ≤ ε, +(29) +dist +� +0, ∂yF(xK+1, yK+1) − ∇yd(xK+1, yK+1)λK+1 +y +� +≤ ε, +(30) +∥[c(xK+1)]+∥ ≤ εδ−1 +c +� +LF + 2Ldδ−1 +d (ǫ0 + LF )Dy + ǫ0 +� +, +(31) +|⟨˜λK+1 +x +, c(xK+1)⟩| ≤ εδ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF)Dy + ǫ0) +× max{δ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF)Dy + ǫ0), Λ}, +(32) +∥[d(xK+1, yK+1)]+∥ ≤ 2εδ−1 +d (ǫ0 + LF )Dy, +(33) +|⟨λK+1 +y +, d(xK+1, yK+1)⟩| ≤ 2εδ−1 +d (ǫ0 + LF)Dy max{2δ−1 +d (ǫ0 + LF )Dy, ∥λ0 +y∥} +(34) +(ii) The total number of evaluations of ∇f, ∇c, ∇d and proximal operator of p and q performed +in Algorithm 1 is at most N, respectively, where +N = +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy �L +� +�T(1 − τ 4)−1 +× (τε)−4 � +28K log(1/τ) + 28 log(1/ǫ0) + 2(log � +M)+ + 2 + 2 log(2 �T) +� +. +(35) +7 + +Remark 3. One can observe from Theorem 1 that Algorithm 1 enjoys an iteration complexity +of O(log ε−1) and an operation complexity of O(ε−4 log ε−1), measured by the amount of eval- +uations of ∇f, ∇c, ∇d and proximal operator of p and q, for finding an O(ε)-KKT solution +(xK+1, yK+1) of (1) such that +dist +� +∂xF(xK+1, yK+1) + ∇c(xK+1)˜λx − ∇xd(xK+1, yK+1)λK+1 +y +� +≤ ε, +dist +� +∂yF(xK+1, yK+1) − ∇yd(xK+1, yK+1)λK+1 +y +� +≤ ε, +∥[c(xK+1)]+∥ = O(ε), +|⟨˜λK+1 +x +, c(xK+1)⟩| = O(ε), +∥[d(xK+1, yK+1)]+∥ = O(ε), +|⟨λK+1 +y +, d(xK+1, yK+1)⟩| = O(ε). +where ˜λK+1 +x +∈ R˜n ++ is defined in (27) and λK+1 +y +∈ R ˜m ++ is given in Algorithm 1. +4 +Proof of the main result +In this section, we provide a proof of our main result presented in Section 2, which is particularly +Theorem 1. Before proceeding, let +Ly(x, y, λy; ρ) = F(x, y) − 1 +2ρ +� +∥[λy + ρd(x, y)]+∥2 − ∥λy∥2� +. +(36) +In view of (3), (17) and (36), one can observe that +f ∗(x) ≤ max +y +Ly(x, y, λy; ρ) +∀x ∈ X, λy ∈ R ˜m ++, ρ > 0, +(37) +which will be frequently used later. +We next establish several lemmas that will be used to prove Theorem 1 subsequently. +Lemma 1. Suppose that Assumptions 1 and 3 hold. Let f ∗, f ∗ +low, Dy, r, LF and δd be given +in (17), (18), (19), (21) and Assumption 1, respectively. Then the following statements hold. +(i) ∥λ∗ +y∥ ≤ δ−1 +d LF Dy and λ∗ +y ∈ B+ +r for all λ∗ +y ∈ Λ∗(x) and x ∈ X, where Λ∗(x) denotes the +set of optimal Lagrangian multipliers of problem (17) for any x ∈ X. +(ii) The function f ∗ is Lipschitz continuous on X and f ∗ +low is finite. +(iii) It holds that +f ∗(x) = min +λy max +y +F(x, y) − ⟨λy, d(x, y)⟩ + IR ˜ +m ++ (λy) +∀x ∈ X, +(38) +where IR ˜ +m ++ (·) is the indicator function associated with R ˜m ++. +Proof. (i) Let x ∈ X and λ∗ +y ∈ Λ∗(x) be arbitrarily chosen, and let y∗ ∈ Y be such that (y∗, λ∗ +y) +is a pair of primal-dual optimal solutions of (17). It then follows that +y∗ ∈ Argmax +y +F(x, y) − ⟨λ∗ +y, d(x, y)⟩, +⟨λ∗ +y, d(x, y∗)⟩ = 0, +d(x, y∗) ≤ 0, +λ∗ +y ≥ 0. +The first relation above yields +F(x, y∗) − ⟨λ∗ +y, d(x, y∗)⟩ ≥ F(x, ˆyx) − ⟨λ∗ +y, d(x, ˆyx)⟩, +where ˆyx is given in Assumption 3(ii). By this and ⟨λ∗ +y, d(x, y∗)⟩ = 0, one has +⟨λ∗ +y, −d(x, ˆyx)⟩ ≤ F(x, y∗) − F(x, ˆyx), +8 + +which together with (19), λ∗ +y ≥ 0 and Assumption 1 implies that +δd∥λ∗ +y∥1 ≤ ⟨λ∗ +y, −d(x, ˆyx)⟩ ≤ F(x, y∗) − F(x, ˆyx) ≤ LF∥y∗ − ˆyx∥ ≤ LFDy, +(39) +where the first inequality is due to Assumption 3(ii), and the third inequality follows from +(19) and LF -Lipschitz continuity of F (see Assumption 1(i)). Using (21) and (39), we have +∥λ∗ +y∥ ≤ ∥λ∗ +y∥1 ≤ δ−1 +d LF Dy and hence λ∗ +y ∈ B+ +r due to (21). +(ii) Recall from Assumption 1 that F(x, ·) and di(x, ·), i = 1, . . . , l, are convex for any given +x ∈ X. Using this, (17), (21) and the first statement of this lemma, we observe that +f ∗(x) = max +y +min +λ∈B+ +r +F(x, y) − ⟨λ, d(x, y)⟩ +∀x ∈ X. +(40) +Notice from Assumption 1 that F and d are Lipschitz continuous on their domain. Then it is +not hard to observe that min{F(x, y)+⟨λ, d(x, y)⟩|λ ∈ B+ +r } is a Lipschitz continuous function of +(x, y) on its domain. By this and (40), one can easily verify that f ∗ is Lipschitz continuous on X. +In addition, the finiteness of f ∗ +low follows from (18), the continuity of ˜f ∗, and the compactness +of X. +(iii) One can observe from (17) that for all x ∈ X, +f ∗(x) = max +y +min +λy F(x, y)−⟨λy, d(x, y)⟩+IR ˜ +m ++ (λy) ≤ min +λy max +y +F(x, y)−⟨λy, d(x, y)⟩+IR ˜ +m ++ (λy), +where the inequality follows from the weak duality. In addition, it follows from Assumption 1 +that the domain of F(x, ·) is compact for all x ∈ X. By this, (40) and the strong duality, one +has +f ∗(x) = min +λ∈B+ +r +max +y +F(x, y) − ⟨λ, d(x, y)⟩ +∀x ∈ X, +which together with the above inequality implies that (38) holds. +Lemma 2. Suppose that Assumptions 1 and 3 hold. Let {λk +y}k∈K be generated by Algorithm 1, +f ∗ +low, Dy, and Fhi be defined in (18), (19) and (20), and ǫ0, τ, and ρk be given in Algorithm 1. +Then we have +ρ−1 +k ∥λk +y∥2 ≤ ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +∀0 ≤ k ∈ K − 1. +(41) +Proof. One can observe from (18), (20) and Algorithm 1 that Fhi ≥ f ∗ +low and ρ0 ≥ 1 > τ > 0, +which imply that (41) holds for k = 0. It remains to show that (41) holds for all 1 ≤ k ∈ K − 1. +Since (xt+1, yt+1) is an ǫt-stationary point of (9) for all 0 ≤ t ∈ K − 1, it follows from +Definition 1 that there exists some u ∈ ∂yL(xt+1, yt+1, λt +x, λt +y; ρt, ρt) with ∥u∥ ≤ ǫt. Notice +from (3) and (36) that ∂yL(xt+1, yt+1, λt +x, λt +y; ρt, ρt) = ∂yLy(xt+1, yt+1, λt +y; ρt). +Hence, u ∈ +∂yLy(xt+1, yt+1, λt +y; ρt). Also, observe from (1), (36) and Assumption 1 that Ly(xt+1, ·, λt +y; ρt) +is concave. Using this, (19), u ∈ ∂yLy(xt+1, yt+1, λt +y; ρt) and ∥u∥ ≤ ǫt, we obtain +Ly(xt+1, y, λt +y; ρt) ≤ Ly(xt+1, yt+1, λt +y; ρt) + ⟨u, y − yt+1⟩ +≤ Ly(xt+1, yt+1, λt +y; ρt) + Dyǫt +∀y ∈ Y, +which implies that +max +y +Ly(xt+1, y, λt +y; ρt) ≤ Ly(xt+1, yt+1, λt +y; ρt) + Dyǫt. +(42) +By this, (36) and (37), one has +f ∗(xt+1) +(37) +≤ max +y +Ly(xt+1, y, λt +y; ρt) +(36)(42) +≤ +F(xt+1, yt+1) − 1 +2ρt +� +∥[λt +y + ρtd(xt+1, yt+1)]+∥2 − ∥λt +y∥2� ++ Dyǫt += F(xt+1, yt+1) − 1 +2ρt +� +∥λt+1 +y +∥2 − ∥λt +y∥2� ++ Dyǫt, +9 + +where the equality follows from the relation λt+1 +y += [λt +y + ρtd(xt+1, yt+1)]+ (see Algorithm 1). +Using the above inequality, (18), (20) and ǫt ≤ ǫ0 (see Algorithm 1), we have +∥λt+1 +y +∥2 − ∥λt +y∥2 ≤ 2ρk(F(xt+1, yt+1) − f ∗(xt+1) + Dyǫt) ≤ 2ρt(Fhi − f ∗ +low + Dyǫ0). +Summing up this inequality for t = 0, . . . , k − 1 with 1 ≤ k ∈ K − 1 yields +∥λk +y∥2 ≤ ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +k−1 +� +t=0 +ρt. +(43) +Recall from Algorithm 1 that ρt = ǫ−1 +t += (ǫ0τ t)−1. Then we have �k−1 +t=0 ρt ≤ ρk−1/(1 − τ). +Using this, (43) and ρk > ρk−1 ≥ 1 (see Algorithm 1), we obtain that for all 1 ≤ k ∈ K − 1, +ρ−1 +k ∥λk +y∥2 ≤ ρ−1 +k +� +∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0)ρk−1 +1 − τ +� +≤ ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +. +Hence, the conclusion holds as desired. +Lemma 3. Suppose that Assumptions 1 and 3 hold. Let f ∗ +low, Dy and Fhi be defined in (18), +(19) and (20), LF and δd be given in Assumptions 1 and 3, and ǫ0, τ, ǫk and ρk be given in +Algorithm 1. Suppose that (xk+1, yk+1, λk+1 +y +) is generated by Algorithm 1 for some 0 ≤ k ∈ K−1 +with +ρk ≥ 4∥λ0 +y∥2 +δ2 +d ++ 8(Fhi − f ∗ +low + Dyǫ0) +δ2 +d(1 − τ) +. +(44) +Then we have +∥[d(xk+1, yk+1)]+∥ ≤ ρ−1 +k ∥λk+1 +y +∥ ≤ 2ρ−1 +k δ−1 +d (ǫ0 + LF )Dy. +(45) +Proof. Suppose that (xk+1, yk+1, λk+1 +y +) is generated by Algorithm 1 for some 0 ≤ k ∈ K − 1 +with ρk satisfying (44). Since (xk+1, yk+1) is an ǫk-stationary point of (9), it follows from (3) +and Definition 1 that +dist +� +0, ∂yF(xk+1, yk+1) − ∇yd(xk+1, yk+1)[λk +y + ρkd(xk+1, yk+1)]+ +� +≤ ǫk. +Besides, notice from Algorithm 1 that λk+1 +y += [λk +y +ρkd(xk+1, yk+1)]+. Hence, there exists some +u ∈ ∂yF(xk+1, yk+1) such that +∥u − ∇yd(xk+1, yk+1)λk+1 +y +∥ ≤ ǫk. +(46) +By Assumption 3(ii), there exists some ˆyk+1 ∈ Y such that −di(xk+1, ˆyk+1) ≥ δd for all i. Notice +that ⟨λk+1 +y +, λk +y + ρkd(xk+1, yk+1)⟩ = ∥[λk +y + ρkd(xk+1, yk+1)]+∥2 ≥ 0, which implies that +− ⟨λk+1 +y +, ρ−1 +k λk +y⟩ ≤ ⟨λk+1 +y +, d(xk+1, yk+1)⟩. +(47) +Using these and (46), we have +F(xk+1, ˆyk+1) − F(xk+1, yk+1) + δd∥λk+1 +y +∥1 − ρ−1 +k ⟨λk+1 +y +, λk +y⟩ +≤ F(xk+1, ˆyk+1) − F(xk+1, yk+1) − ⟨λk+1 +y +, ρ−1 +k λk +y + d(xk+1, ˆyk+1)⟩ +(47) +≤ F(xk+1, ˆyk+1) − F(xk+1, yk+1) + ⟨λk+1 +y +, d(xk+1, yk+1) − d(xk+1, ˆyk+1))⟩ +≤ ⟨u, ˆyk+1 − yk+1⟩ + ⟨∇yd(xk+1, yk+1)λk+1 +y +, yk+1 − ˆyk+1⟩ += ⟨u − ∇yd(xk+1, yk+1)λk+1 +y +, yk+1 − ˆyk+1⟩ ≤ Dyǫk, +(48) +10 + +where the first inequality is due to λk+1 +y +≥ 0 and −di(xk+1, ˆyk+1) ≥ δd for all i, the third +inequality follows from u ∈ ∂yF(xk+1, yk+1), λk+1 +y +≥ 0, the concavity of F(xk+1, ·) and the +convexity of di(xk+1, ·), and the last inequality is due to (19) and (46). +In view of (19), (48) and the Lipschitz continuity of F (see Assumption 1), one has +Dyǫk + LFDy +(19) +≥ Dyǫk + LF∥ˆyk+1 − yk+1∥ ≥ Dyǫk − F(xk+1, ˆyk+1) + F(xk+1, yk+1) +(48) +≥ δd∥λk+1 +y +∥1 − ρ−1 +k ⟨λk+1 +y +, λk +y⟩ ≥ (δd − ρ−1 +k ∥λk +y∥)∥λk+1 +y +∥, +(49) +where the second inequality follows from LF-Lipschitz continuity of F, and the last inequality +is due to ∥λk+1 +y +∥1 ≥ ∥λk+1 +y +∥. In addition, it follows from (41) and (44) that +δd − ρ−1 +k ∥λk +y∥ +(41) +≥ δd − +� +ρ−1 +k +� +∥λ0y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +� +(44) +≥ 1 +2δd, +which together with (49) yields +1 +2δd∥λk+1 +y +∥ ≤ (δd − ρ−1 +k ∥λk +y∥)∥λk+1 +y +∥ +(49) +≤ Dyǫk + LF Dy. +The conclusion then follows from this, ǫk ≤ ǫ0, and the relations +∥[d(xk+1, yk+1)]+∥ ≤ ρ−1 +k ∥[λk +y + ρkd(xk+1, yk+1)]+∥ = ρ−1 +k ∥λk+1 +y +∥. +Lemma 4. Suppose that Assumptions 1 and 3 hold. Let f ∗ +low, Dy and Flow be defined in (18), +(19) and (20), LF and δd be given in Assumptions 1 and 3, ǫ0, τ, ǫk, ρk and λ0 +y be given +in Algorithm 1. Suppose that (xk+1, yk+1, λk+1 +x +, λk+1 +y +) is generated by Algorithm 1 for some +0 ≤ k ∈ K − 1 with +ρk ≥ 4∥λ0 +y∥2 +δ2 +dτ ++ 8(Fhi − f ∗ +low + Dyǫ0) +δ2 +dτ(1 − τ) +. +(50) +Let +˜λk+1 +x += [λk +x + ρkc(xk+1)]+. +(51) +Then we have +dist(0, ∂xF(xk+1, yk+1) + ∇c(xk+1)˜λk+1 +x +− ∇xd(xk+1, yk+1)λk+1 +y +) ≤ ǫk, +(52) +dist +� +0, ∂yF(xk+1, yk+1) − ∇yd(xk+1, yk+1)λk+1 +y +� +≤ ǫk, +(53) +∥[d(xk+1, yk+1)]+∥ ≤ 2ρ−1 +k δ−1 +d (ǫ0 + LF )Dy, +(54) +|⟨λk+1 +y +, d(xk+1, yk+1)⟩| ≤ 2ρ−1 +k δ−1 +d (ǫ0 + LF )Dy max{∥λ0 +y∥, 2δ−1 +d (ǫ0 + LF)Dy}. +(55) +Proof. Suppose that (xk+1, yk+1, λk+1 +x +, λk+1 +y +) is generated by Algorithm 1 for some 0 ≤ k ∈ K−1 +with ρk satisfying (50). Since (xk+1, yk+1) is an ǫk-stationary point of (9), it then follows from +Definition 1 that +dist +� +0, ∂xL(xk+1, yk+1, λk +x, λk +y; ρk) +� +≤ ǫk, dist +� +0, ∂yL(xk+1, yk+1, λk +x, λk +y; ρk) +� +≤ ǫk. +(56) +Observe from Algorithm 1 that λk+1 +y += [λk +y + ρkd(xk+1, yk+1)]+. In view of this, (3) and (51), +one has +∂xL(xk+1, yk+1, λk +x, λk +y; ρk) = ∂xF(xk+1, yk+1) + ∇c(xk+1)[λk +x + ρkc(xk+1)]+ +− ∇xd(xk+1, yk+1)[λk +y + ρkd(xk+1, yk+1)]+ += ∂xF(xk+1, yk+1) + ∇c(xk+1)˜λk+1 +x +− ∇xd(xk+1, yk+1)λk+1 +y +, +∂yL(xk+1, yk+1, λk +x, λk +y; ρk) = ∂yF(xk+1, yk+1) − ∇yd(xk+1, yk+1)λk+1 +y +. +11 + +These relations together with (56) imply that (52) and (53) hold. +Notice from Algorithm 1 that 0 < τ < 1, which together with (50) implies that (44) holds +for ρk. It then follows that (45) holds, which immediately yields (54) and +∥λk+1 +y +∥ ≤ 2δ−1 +d (ǫ0 + LF )Dy. +(57) +Claim that +∥λk +y∥ ≤ max{∥λ0 +y∥, 2δ−1 +d (ǫ0 + LF)Dy}. +(58) +Indeed, (58) clearly holds if k = 0. We now assume that k > 0. Notice from Algorithm 1 that +ρk−1 = τρk, which together with (50) implies that (44) holds with k replaced by k − 1. By this +and Lemma 3 with k replaced by k − 1, one can conclude that ∥λk +y∥ ≤ 2δ−1 +d (ǫ0 + LF)Dy and +hence (58) holds. +We next show that (55) holds. Indeed, by λk+1 +y +≥ 0, (47), (54), (57) and (58), one has +⟨λk+1 +y +, d(xk+1, yk+1)⟩ +≤ ⟨λk+1 +y +, [d(xk+1, yk+1)]+⟩ ≤ ∥λk+1 +y +∥∥[d(xk+1, yk+1)]+∥ +(54)(57) +≤ +4ρ−1 +k δ−2 +d (ǫ0 + LF)2D2 +y, +⟨λk+1 +y +, d(xk+1, yk+1)⟩ +(47) +≥ ⟨λk+1 +y +, −ρ−1 +k λk +y⟩ ≥ −ρ−1 +k ∥λk+1 +y +∥∥λk +y∥ +≥ −2ρ−1 +k δ−1 +d (ǫ0 + LF)Dy max{∥λ0 +y∥, 2δ−1 +d (ǫ0 + LF)Dy}. +These relations imply that (55) holds. +Lemma 5. Suppose that Assumptions 1, 2 and 3 hold. Let {(λk +x, λk +y)}k∈K be generated by Algo- +rithm 1, L, f ∗ +low, Dy and Fhi be defined in (3), (18), (19) and (20), LF be given in Assumption +1, and ǫ0, τ, ρk, Λ and xk +init be given in Algorithm 1. Then for all 0 ≤ k ∈ K − 1, we have +max +y +L(xk +init, y, λk +x, λk +y; ρk) ≤ LFDy + Fhi + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ +. +(59) +Proof. In view of (6), (8), (20) and ∥λk +x∥ ≤ Λ (see Algorithm 1), one has +Lx(xk +init, yk, λk +x; ρk) +(8) +≤ Lx(xnf, yk, λk +x; ρk) +(6) += F(xnf, yk) + +1 +2ρk +� +∥[λk +x + ρkc(xnf)]+∥2 − ∥λk +x∥2� +≤ F(xnf, yk) + +1 +2ρk +� +(∥λk +x∥ + ρk∥[c(xnf )]+∥)2 − ∥λk +x∥2� += F(xnf, yk) + ∥λk +x∥∥[c(xnf )]+∥ + 1 +2ρk∥[c(xnf)]+∥2 +(20) +≤ Fhi + Λ∥[c(xnf)]+∥ + 1 +2ρk∥[c(xnf)]+∥2. +(60) +In addition, one can observe from Algorithm 1 that ǫk > τε for all 0 ≤ k ∈ K − 1. By this and +the choice of ρk in Algorithm 1, we obtain that ρk = ǫ−1 +k +< τ −1ε−1 for all 0 ≤ k ∈ K−1. It then +follows from this, (3), (6), (19), (41), (60), ∥[c(xnf)]+∥ ≤ √ε ≤ 1, and the Lipschitz continuity +12 + +of F that +max +y +L(xk +init, y, λk +x, λk +y; ρk) +(3)(6) += +max +y +� +Lx(xk +init, y, λk +x; ρk) − +1 +2ρk +� +∥[λk +y + ρkd(xk +init, y)]+∥2 − ∥λk +y∥2�� +≤ max +y +� +Lx(xk +init, y, λk +x; ρk) + +1 +2ρk +∥λk +y∥2 +� +(6) += max +y +� +F(xk +init, y) − F(xk +init, yk) + Lx(xk +init, yk, λk +x; ρk) + +1 +2ρk +∥λk +y∥2 +� +≤ max +y∈Y LF∥y − yk∥ + Lx(xk +init, yk, λk +x; ρk) + +1 +2ρk +∥λk +y∥2 +≤ LF Dy + Fhi + Λ∥[c(xnf)]+∥ + 1 +2ρk∥[c(xnf)]+∥2 + 1 +2∥λ0 +y∥2 + Fhi − f ∗ +low + Dyǫ0 +1 − τ +≤ LF Dy + Fhi + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ +, +where the second inequality follows from LF-Lipschitz continuity of F (see Assumption 1(i)), +the third inequality follows from (19), (41) and (60), and the last inequality follows from ρk < +τ −1ε−1 and ∥[c(xnf )]+∥ ≤ √ε ≤ 1. +Lemma 6. Suppose that Assumptions 1, 2 and 3 hold. Let Lk, f ∗ +low, Dx, Dy, Fhi and Flow be +defined in (10), (18), (19) and (20), LF be given in Assumption 1, ǫ0, τ, ǫk, ρk, Λ and λ0 +y be +given in Algorithm 1, and +αk = min +� +1, +� +4ǫk/(DyLk) +� +, +(61) +δk = (2 + α−1 +k )LkD2 +x + max {ǫk/Dy, αkLk/4} D2 +y, +(62) +Mk = +16 max {1/(2Lk), min {Dy/ǫk, 4/(αkLk)}} ρk +[(3Lk + ǫk/(2Dy))2/ min{Lk, ǫk/(2Dy)} + 3Lk + ǫk/(2Dy)]−2 ǫ2 +k +× +� +δk + 2α−1 +k +� +Fhi − Flow ++ Λ2 +2ρk ++ 3 +2∥λ0 +y∥2 + 3(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ρkd2 +hi + ǫkDy +4 ++ LkD2 +x +�� +(63) +Tk = +� +16 +� +LF Dy + Fhi − f ∗ +low + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ ++ Λ2 +2ρk ++ ǫkDy +4 +� +Lkǫ−2 +k ++ 8(1 + 4D2 +yL2 +kǫ−2 +k )ρ−1 +k +− 1 +� ++ +, +(64) +Nk = +�� +96 +√ +2 +� +1 + (24Lk + 4ǫk/Dy) L−1 +k +�� ++ 2 +� +max +� +2, +� +DyLkǫ−1 +k +� +× ((Tk + 1)(log Mk)+ + Tk + 1 + 2Tk log(Tk + 1)) . +(65) +Then for all 0 ≤ k ∈ K − 1, Algorithm 1 finds an ǫk-stationary point (xk+1, yk+1) of problem +(9) that satisfies +max +y +L(xk+1, y, λk +x, λk +y; ρk) ≤ LFDy + Fhi + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ ++ ǫkDy +4 ++ +1 +2ρk +� +L−1 +k ǫ2 +k + 4D2 +yLk +� +. +(66) +Moreover, the total number of evaluations of ∇f, ∇c, ∇d and proximal operator of p and q +performed in iteration k of Algorithm 1 is no more than Nk, respectively. +13 + +Proof. Observe from (1) and (3) that problem (9) can be viewed as +min +x max +y {h(x, y) + p(x) − q(y)}, +where +h(x, y) = f(x, y) + +1 +2ρk +� +∥[λk +x + ρkc(x)]+∥2 − ∥λk +x∥2� +− +1 +2ρk +� +∥[λk +y + ρkd(x, y)]+∥2 − ∥λk +y∥2� +. +Notice that +∇xh(x, y) = ∇xf(x, y) + ∇c(x)[λk +x + ρkc(x)]+ + ∇xd(x, y)[λk +y + ρkd(x, y)]+, +∇yh(x, y) = ∇yf(x, y) + ∇yd(x, y)[λk +y + ρkd(x, y)]+. +It follows from Assumption 1(iii) that +∥∇c(x)∥ ≤ Lc, +∥∇d(x, y)∥ ≤ Ld +∀(x, y) ∈ X × Y. +In view of the above relations, (7) and Assumption 1, one can observe that ∇c(x)[λk +x +ρkc(x)]+ +is (ρkL2 +c + ρkchiL∇c + ∥λk +x∥L∇c)-Lipschitz continuous on X, and ∇d(x, y)[λk +y + ρkd(x, y)]+ is +(ρkL2 +d + ρkdhiL∇d + ∥λk +y∥L∇d)-Lipschitz continuous on X × Y. Using these and the fact that +∇f(x, y) is L∇f-Lipschitz continuous on X × Y, we can see that h(x, y) is Lk-smooth on X × Y +for all 0 ≤ k ∈ K − 1, where Lk is given in (10). +Consequently, it follows from Theorem +2 that Algorithm 3 can be suitably applied to problem (9) for finding an ǫk-stationary point +(xk+1, yk+1) of it. +In addition, by (3), (18), (36), (37) and ∥λk +x∥ ≤ Λ (see Algorithm 1), one has +min +x max +y +L(x, y, λk +x, λk +y; ρk) +(3)(36) += +min +x max +y +� +Ly(x, y, λk +y; ρk) + +1 +2ρk +� +∥[λk +x + ρkc(x)]+∥2 − ∥λk +x∥2�� +(37) +≥ min +x +� +f ∗(x) + +1 +2ρk +� +∥[λk +x + ρkc(x)]+∥2 − ∥λk +x∥2�� (18) +≥ f ∗ +low − +1 +2ρk +∥λk +x∥2 ≥ f ∗ +low − Λ2 +2ρk +. +(67) +Let (x∗, y∗) be an optimal solution of (1). It then follows that c(x∗) ≤ 0. Using this, (3), (20) +and (41), we obtain that +min +x max +y +L(x, y, λk +x, λk +y; ρk) ≤ max +y +L(x∗, y, λk +x, λk +y; ρk) +(3) += max +y +� +F(x∗, y) + +1 +2ρk +� +∥[λk +x + ρkc(x∗)]+∥2 − ∥λk +x∥2� +− +1 +2ρk +� +∥[λk +y + ρkd(x∗, y)]+∥2 − ∥λk +y∥2�� +≤ max +y +� +F(x∗, y) − +1 +2ρk +� +∥[λk +y + ρkd(x∗, y)]+∥2 − ∥λk +y∥2�� +(20) +≤ Fhi + +1 +2ρk +∥λk +y∥2 (41) +≤ Fhi + 1 +2∥λ0 +y∥2 + Fhi − f ∗ +low + Dyǫ0 +1 − τ +, +(68) +where the second inequality is due to c(x∗) ≤ 0. Moreover, it follows from this, (3), (7), (20), +(41), λk +y ∈ R ˜m ++ and ∥λk +x∥ ≤ Λ that +min +(x,y)∈X×Y L(x, y, λk +x, λk +y; ρk) +(3) +≥ +min +(x,y)∈X×Y +� +F(x, y) − +1 +2ρk +∥λk +x∥2 − +1 +2ρk +∥[λk +y + ρkd(x, y)]+∥2 +� +≥ +min +(x,y)∈X×Y +� +F(x, y) − +1 +2ρk +∥λk +x∥2 − +1 +2ρk +� +∥λk +y∥ + ρk∥[d(x, y)]+∥ +�2� +≥ +min +(x,y)∈X×Y +� +F(x, y) − +1 +2ρk +∥λk +x∥2 − ρ−1 +k ∥λk +y∥2 − ρk∥[d(x, y)]+∥2 +� +≥ Flow − Λ2 +2ρk +− ∥λ0 +y∥2 − 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +− ρkd2 +hi, +(69) +14 + +where the second inequality is due to λk +y ∈ R ˜m ++ and the last inequality is due to (7), (20), (41) +and ∥λk +x∥ ≤ Λ. +To complete the rest of the proof, let +H(x, y) = L(x, y, λk +x, λk +y; ρk), +H∗ = min +x max +y +L(x, y, λk +x, λk +y; ρk), +(70) +Hlow = +min +(x,y)∈X×Y L(x, y, λk +x, λk +y; ρk). +(71) +In view of these, (59), (67), (68), (69), we obtain that +max +y +H(xk +init, y) +(59) +≤ LFDy + Fhi + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ +, +f ∗ +low − Λ2 +2ρk +(67) +≤ H∗ (68) +≤ Fhi + 1 +2∥λ0 +y∥2 + Fhi − f ∗ +low + Dyǫ0 +1 − τ +, +Hlow +(69) +≥ Flow − Λ2 +2ρk +− ∥λ0 +y∥2 − 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +− ρkd2 +hi. +Using these and Theorem 2 (see Appendix A) with x0 = xk +init, Dp = Dx, Dq = Dy, ǫ = ǫk, +ǫ0 = ǫk/(2√ρk), L∇h = Lk, α = αk, δ = δk, and H, H∗, Hlow given in (70) and (71), we +can conclude that Algorithm 3 performs at most Nk evaluations of ∇f, ∇c, ∇d and proximal +operator of p and q for finding an ǫk-stationary point of problem (9) satisfying (66). +Lemma 7. Suppose that Assumptions 1, 2 and 3 hold. Let f ∗ +low, Dy, Fhi and �L be defined in +(18), (19), (20) and (23), LF , Lc, δc, θf and θa be given in Assumptions 1 and 3, and ǫ0, τ, +ρk, Λ and λ0 +y be given in Algorithm 1. Suppose that (xk+1, λk+1 +x +) is generated by Algorithm 1 +for some 0 ≤ k ∈ K − 1 with +ρk ≥ max +� +θ−1 +a Λ, θ−2 +f +� +2LF Dy + 2Fhi − 2f ∗ +low + 2Λ + τ −1 + ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ǫ0Dy +2 ++ L−2 +c ++ 4D2 +y �L + Λ2� +, 4∥λ0 +y∥2 +δ2 +dτ ++ 8(Fhi − f ∗ +low + Dyǫ0) +δ2 +dτ(1 − τ) +� +. +(72) +Let +˜λk+1 +x += [λk +x + ρkc(xk+1)]+. +(73) +Then we have +∥[c(xk+1)]+∥ ≤ ρ−1 +k δ−1 +c +� +LF + 2Ldδ−1 +d (ǫ0 + LF )Dy + ǫ0 +� +, +(74) +|⟨˜λk+1 +x +, c(xk+1)⟩| ≤ ρ−1 +k δ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF)Dy + ǫ0) max{δ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF)Dy + ǫ0), Λ}. +(75) +Proof. One can observe from (3), (18), (36) and (37) that +max +y +L(xk+1, y, λk +x, λk +y; ρk) = +max +y +Ly(xk+1, y, λk +y; ρk) + +1 +2ρk +� +∥[λk +x + ρkc(xk+1)]+∥2 − ∥λk +x∥2� +(37) +≥ f ∗(xk+1) + +1 +2ρk +� +∥[λk +x + ρkc(xk+1)]+∥2 − ∥λk +x∥2� +(18) +≥ +f ∗ +low + +1 +2ρk +� +∥[λk +x + ρkc(xk+1)]+∥2 − ∥λk +x∥2� +. +15 + +By this inequality, (66) and ∥λk +x∥ ≤ Λ, one has +∥[λk +x + ρkc(xk+1)]+∥2 ≤ 2ρk max +y +L(xk+1, y, λk +x, λk +y; ρk) − 2ρkf ∗ +low + ∥λk +x∥2 +≤ 2ρk max +y +L(xk+1, y, λk +x, λk +y; ρk) − 2ρkf ∗ +low + Λ2 +(66) +≤ 2ρkLF Dy + 2ρkFhi + 2ρkΛ + ρk(τ −1 + ∥λ0 +y∥2) + 2ρk(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ρkǫkDy +2 ++ L−1 +k ǫ2 +k + 4D2 +yLk − 2ρkf ∗ +low + Λ2. +This together with ρ2 +k∥[c(xk+1)]+∥2 ≤ ∥[λk +x + ρkc(xk+1)]+∥2 implies that +∥[c(xk+1)]+∥2 ≤ ρ−1 +k +� +2LF Dy + 2Fhi − 2f ∗ +low + 2Λ + τ −1 + ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ǫkDy +2 +� ++ ρ−2 +k +� +L−1 +k ǫ2 +k + 4D2 +yLk + Λ2� +. +(76) +In addition, we observe from (10), (23), (41), ρk ≥ 1 and ∥λk +x∥ ≤ Λ that for all 0 ≤ k ≤ K, +ρkL2 +c ≤ Lk = L∇f + ρkL2 +c + ρkchiL∇c + ∥λk +x∥L∇c + ρkL2 +d + ρkdhiL∇d + ∥λk +y∥L∇d +≤ L∇f + ρkL2 +c + ρkchiL∇c + ΛL∇c + ρkL2 +d + ρkdhiL∇d ++ L∇d +� +ρk +� +∥λ0y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ +� +≤ ρk �L. +(77) +Using this relation, (72), (76), ρk ≥ 1 and ǫk ≤ ǫ0, we have +∥[c(xk+1)]+∥2 ≤ ρ−1 +k +� +2LF Dy + 2Fhi − f ∗ +low + 2Λ + τ −1 + ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ǫkDy +2 +� ++ ρ−2 +k +� +(ρkL2 +c)−1ǫ2 +k + 4ρkD2 +y�L + Λ2� +≤ ρ−1 +k +� +2LF Dy + 2Fhi − f ∗ +low + 2Λ + τ −1 + ∥λ0 +y∥2 + 2(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ǫ0Dy +2 +� ++ ρ−1 +k +� +L−2 +c ++ 4D2 +y �L + Λ2� (72) +≤ θ2 +f, +which together with (11) implies that xk+1 ∈ F(θf). +It follows from xk+1 ∈ F(θf) and Assumption 3(i) that there exists some vx such that +∥vx∥ = 1 and vT +x ∇ci(xk+1) ≤ −δc for all i ∈ A(xk+1; θa), where A(xk+1; θa) is defined in +(11). Let ¯ +A(xk+1; θa) = {1, 2, . . . , ˜n}\A(xk+1; θa). Notice from (11) that ci(xk+1) < −θa for all +i ∈ ¯ +A(xk+1; θa). In addition, observe from (72) that ρk ≥ θ−1 +a Λ. Using these and ∥λk +x∥ ≤ Λ, we +obtain that (λk +x + ρkc(xk+1))i ≤ Λ − ρkθa ≤ 0 for all i ∈ ¯ +A(xk+1; θa). By this and the fact that +vT +x ∇ci(xk+1) ≤ −δc for all i ∈ A(xk+1; θa), one has +vT +x ∇c(xk+1)˜λk+1 +x +(73) += vT +x ∇c(xk+1)[λk +x + ρkc(xk+1)]+ = +˜n +� +i=1 +vT +x ∇ci(xk+1)([λk +x + ρkc(xk+1)]+)i += +� +i∈A(xk+1;θa) +vT +x ∇ci(xk+1)([λk +x + ρkc(xk+1)]+)i + +� +i∈ ¯ +A(xk+1;θa) +vT +x ∇ci(xk+1)([λk +x + ρkc(xk+1)]+)i +≤ −δc +� +i∈A(xk+1;θa) +([λk +x + ρkc(xk+1)]+)i = −δc +˜n +� +i=1 +([λk +x + ρkc(xk+1)]+)i +(73) += −δc∥˜λk+1 +x +∥1. +(78) +Since (xk+1, yk+1) is an ǫk-stationary point of (9), it follows from (3) and (56) that there +exists some s ∈ ∂xF(xk+1, yk+1) such that +∥s + ∇c(xk+1)[λk +x + ρkc(xk+1)]+ − ∇xd(xk+1, yk+1)[λk +y + ρkd(xk+1, yk+1)]+∥ ≤ ǫk, +16 + +which along with (73) and λk+1 +y += [λk +y + ρxd(xk+1, yk+1)]+ implies that +∥s + ∇c(xk+1)˜λk+1 +x +− ∇xd(xk+1, yk+1)λk+1 +y +∥ ≤ ǫk. +By this, (78) and ∥vx∥ = 1, one has +ǫk ≥ ∥s + ∇c(xk+1)˜λk+1 +x +− ∇xd(xk+1, yk+1)λk+1 +y +∥ · ∥vx∥ +≥ ⟨s + ∇c(xk+1)˜λk+1 +x +− ∇xd(xk+1, yk+1)λk+1 +y +, −vx⟩ += −⟨s − ∇xd(xk+1, yk+1)λk+1 +y +, vx⟩ − vT +x ∇c(xk+1)˜λk+1 +x +(78) +≥ − +� +∥s∥ + ∥∇xd(xk+1, yk+1)∥∥λk+1 +y +∥ +� +∥vx∥ + δc∥˜λk+1 +x +∥1. +≥ −LF − Ld∥λk+1 +y +∥ + δc∥˜λk+1 +x +∥1, +where the last inequality is due to ∥vx∥ = 1 and Assumptions 1(i) and 1(iii). Notice from (72) +that (44) holds. It then follows from (45) that ∥λk+1 +y +∥ ≤ 2δ−1 +d (ǫ0 + LF )Dy, which together with +the above inequality and ǫk ≤ ǫ0 yields +∥˜λk+1 +x +∥ ≤ ∥˜λk+1 +x +∥1 ≤ δ−1 +c (LF + Ld∥λk+1 +y +∥ + ǫk) ≤ δ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF )Dy + ǫ0). +(79) +By this and (73), one can observe that +∥[c(xk+1)]+∥ ≤ ρ−1 +k ∥[λk +x + ρkc(xk+1)]+∥ = ρ−1 +k ∥˜λk+1 +x +∥ ≤ ρ−1 +k δ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF )Dy + ǫ0). +Hence, (74) holds as desired. +We next show that (75) holds. Indeed, by ˜λk+1 +x +≥ 0, (74) and (79), one has +⟨˜λk+1 +x +, c(xk+1)⟩ +≤ ⟨˜λk+1 +x +, [c(xk+1)]+⟩ ≤ ∥˜λk+1 +x +∥∥[c(xk+1)]+∥ +(74)(79) +≤ +ρ−1 +k δ−2 +c (LF + 2Ldδ−1 +d (ǫ0 + LF )Dy + ǫ0)2. +(80) +Using a similar argument as for the proof of (47), we have +−⟨˜λk+1 +x +, ρ−1 +k λk +x⟩ ≤ ⟨˜λk+1 +x +, c(xk+1)⟩, +which along with ∥λk +x∥ ≤ Λ and (79) yields +⟨˜λk+1 +x +, c(xk+1)⟩ ≥ −ρ−1 +k ∥˜λk+1 +x +∥∥λk +x∥ ≥ −ρ−1 +k δ−1 +c (LF + 2Ldδ−1 +d (ǫ0 + LF )Dy + ǫ0)Λ. +The relation (75) then follows from this and (80). +We are now ready to prove Theorem 1. +Proof of Theorem 1. (i) Observe from the definition of K in (22) and ǫk = ǫ0τ k that K is +the smallest nonnegative integer such that ǫK ≤ ε. Hence, Algorithm 1 terminates and outputs +(xK+1, yK+1) after K + 1 outer iterations. It follows from these and ρk = ǫ−1 +k +that ǫK ≤ ε and +ρK ≥ ε−1. By this and (28), one can see that (50) and (72) holds for k = K. It then follows +from Lemmas 4 and 7 that (29)-(34) hold. +(ii) Let K and N be given in (22) and (35). Recall from Lemma 6 that the number of +evaluations of ∇f, ∇c, ∇d, proximal operator of p and q performed by Algorithm 3 at iteration +k of Algorithm 1 is at most Nk, where Nk is given in (65). By this and statement (i) of this +theorem, one can observe that the total number of evaluations of ∇f, ∇c, ∇d, proximal operator +of p and q performed in Algorithm 1 is no more than �K +k=0 Nk, respectively. As a result, to +prove statement (ii) of this theorem, it suffices to show that �K +k=0 Nk ≤ N. Recall from (77) +17 + +and Algorithm 1 that ρkL2 +c ≤ Lk ≤ ρk �L and ρk ≥ 1 ≥ ǫk. Using these, (24), (25), (26), (61), +(62), (63) and (64), we obtain that +1 ≥ αk ≥ min +� +1, +� +4ǫk/(ρkDy �L) +� +≥ ǫ1/2 +k +ρ−1/2 +k +ˆα, +(81) +δk ≤ (2 + ǫ−1/2 +k +ρ1/2 +k +ˆα−1)ρk �LD2 +x + max{1/Dy, ρk �L/4}D2 +y ≤ ǫ−1/2 +k +ρ3/2 +k +ˆδ, +(82) +Mk ≤ +16 max +� +1/(2ρkL2 +c), 4/(ǫ1/2 +k +ρ−1/2 +k +ˆαρkL2 +c) +� +ρk +� +(3ρk �L + 1/(2Dy))2/ min{ρkL2c, ǫk/(2Dy)} + 3ρk �L + 1/(2Dy) +�−2 +ǫ2 +k +× +� +ǫ−1/2 +k +ρ3/2 +k +ˆδ ++ 2ǫ−1/2 +k +ρ1/2 +k +ˆα−1� +Fhi − Flow + Λ2 +2 + 3 +2∥λ0 +y∥2 + 3(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ ρkd2 +hi ++ Dy +4 + ρk �LD2 +x +�� +(83) +≤ +16ǫ−1/2 +k +ρ−1/2 +k +max +� +1/(2L2 +c), 4/(ˆαL2 +c) +� +ρk +ǫ2 +kρ−4 +k +� +(3�L + 1/(2Dy))2/ min{L2c, 1/(2Dy)} + 3�L + 1/(2Dy) +�−2 +ǫ2 +k +× (ǫ−1/2 +k +ρ3/2 +k +) +� +ˆδ + 2ˆα−1 +× +� +Fhi − Flow + Λ2 +2 + 3 +2∥λ0 +y∥2 + 3(Fhi − f ∗ +low + Dyǫ0) +1 − τ ++ d2 +hi + Dy +4 + �LD2 +x +�� +≤ ǫ−5 +k ρ6 +k � +M, +Tk ≤ +� +16 +� +LF Dy + Fhi − f ∗ +low + Λ + 1 +2(τ −1 + ∥λ0 +y∥2) + Fhi − f ∗ +low + Dyǫ0 +1 − τ ++ Λ2 +2 + Dy +4 +� +ǫ−2 +k ρk �L ++ 8(1 + 4D2 +yρ2 +k �L2ǫ−2 +k )ρ−1 +k +− 1 +� ++ +≤ ǫ−2 +k ρk �T, +where (83) follows from (24), (25), (26), (81), (82), ρkL2 +c ≤ Lk ≤ ρk �L, and ρk ≥ 1 ≥ ǫk. By the +above inequalities, (65), (77), �T ≥ 1 and ρk ≥ 1 ≥ ǫk, one has +K +� +k=0 +Nk ≤ +K +� +k=0 +�� +96 +√ +2 +� +1 + +� +24ρk �L + 4/Dy +� +/(ρkL2 +c) +�� ++ 2 +� +max +� +2, +� +Dyρk �Lǫ−1 +k +� +× +� +(ǫ−2 +k ρk �T + 1)(log(ǫ−5 +k ρ6 +k � +M))+ + ǫ−2 +k ρk �T + 1 + 2ǫ−2 +k ρk �T log(ǫ−2 +k ρk �T + 1) +� +≤ +K +� +k=0 +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy�L +� +ǫ−1/2 +k +ρ1/2 +k +× ǫ−2 +k ρk +� +( �T + 1)(log(ǫ−5 +k ρ6 +k � +M))+ + �T + 1 + 2 �T log(ǫ−2 +k ρk �T + 1) +� +≤ +K +� +k=0 +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy�L +� +× ǫ−5/2 +k +ρ3/2 +k +�T +� +2(log(ǫ−5 +k ρ6 +k � +M))+ + 2 + 2 log(2ǫ−2 +k ρk �T) +� +≤ +K +� +k=0 +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy�L +� +�T +× ǫ−5/2 +k +ρ3/2 +k +� +14 log ρk − 14 log ǫk + 2(log � +M)+ + 2 + 2 log(2 �T) +� +, +(84) +By the definition of K in (22), one has τ K ≥ τε/ǫ0. +Also, notice from Algorithm 1 that +18 + +ρk = ǫ−1 +k += (ǫ0τ k)−1. It then follows from these, (35) and (84) that +K +� +k=0 +Nk ≤ +K +� +k=0 +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy �L +� +�T +× ǫ−4 +k +� +28 log(1/ǫk) + 2(log � +M)+ + 2 + 2 log(2 �T ) +� += +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy �L +� +�T +× +K +� +k=0 +ǫ−4 +0 τ −4k � +28k log(1/τ) + 28 log(1/ǫ0) + 2(log � +M)+ + 2 + 2 log(2 �T) +� +≤ +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy �L +� +�T +× +K +� +k=0 +ǫ−4 +0 τ −4k � +28K log(1/τ) + 28 log(1/ǫ0) + 2(log � +M)+ + 2 + 2 log(2 �T ) +� +≤ +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy �L +� +�Tǫ−4 +0 +× τ −4K(1 − τ 4)−1 � +28K log(1/τ) + 28 log(1/ǫ0) + 2(log � +M)+ + 2 + 2 log(2 �T ) +� +≤ +�� +96 +√ +2 +� +1 + +� +24�L + 4/Dy +� +/L2 +c +�� ++ 2 +� +max +� +2, +� +Dy �L +� +�Tǫ−4 +0 (1 − τ 4)−1 +× (τε/ǫ0)−4 � +28K log(1/τ) + 28 log(1/ǫ0) + 2(log � +M)+ + 2 + 2 log(2 �T ) +� (35) += N, +where the second last inequality is due to �K +k=0 τ −4k ≤ τ −4K/(1 − τ 4), and the last inequality +is due to τ K ≥ τε/ǫ0. Hence, statement (ii) of this theorem holds as desired. +References +[1] K. Antonakopoulos, E. V. Belmega, and P. Mertikopoulos. Adaptive extra-gradient meth- +ods for min-max optimization and games. In The International Conference on Learning +Representations, 2021. +[2] E. G. Birgin and J. M. Mart´ınez. Practical Augmented Lagrangian Methods for Constrained +Optimization. SIAM, 2014. +[3] E. G. Birgin and J. M. Mart´ınez. Complexity and performance of an augmented Lagrangian +algorithm. Optim. Methods and Softw., 35(5):885–920, 2020. +[4] N. Cesa-Bianchi and G. Lugosi. Prediction, learning, and games. Cambridge University +Press, 2006. +[5] X. Chen, L. Guo, Z. Lu, and J. J. Ye. An augmented Lagrangian method for non-Lipschitz +nonconvex programming. SIAM J. Numer. Anal., 55(1):168–193, 2017. +[6] Z. Chen, Y. Zhou, T. Xu, and Y. Liang. Proximal gradient descent-ascent: variable con- +vergence under K�L geometry. arXiv preprint arXiv:2102.04653, 2021. +[7] F. H. Clarke. Optimization and nonsmooth analysis. SIAM, 1990. +[8] B. Dai, A. Shaw, L. Li, L. Xiao, N. He, Z. Liu, J. Chen, and L. Song. SBEED: Convergent +reinforcement learning with nonlinear function approximation. In International Conference +on Machine Learning, pages 1125–1134, 2018. +19 + +[9] Y.-H. Dai, J. Wang, and L. Zhang. +Optimality conditions and numerical algorithms +for a class of linearly constrained minimax optimization problems. +arXiv preprint +arXiv:2204.09185, 2022. +[10] Y.-H. Dai and L. Zhang. Optimality conditions for constrained minimax optimization. +arXiv preprint arXiv:2004.09730, 2020. +[11] S. S. Du, J. Chen, L. Li, L. Xiao, and D. Zhou. Stochastic variance reduction methods +for policy evaluation. In International Conference on Machine Learning, pages 1049–1058, +2017. +[12] J. Duchi and H. Namkoong. Variance-based regularization with convex objectives. Journal +of Machine Learning Research, 20(1):2450–2504, 2019. +[13] G. Gidel, H. Berard, G. Vignoud, P. Vincent, and S. Lacoste-Julien. A variational inequality +perspective on generative adversarial networks. In International Conference on Learning +Representations, 2019. +[14] D. Goktas and A. Greenwald. Convex-concave min-max stackelberg games. Advances in +Neural Information Processing Systems, 34:2991–3003, 2021. +[15] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, +and Y. Bengio. Generative adversarial nets. In Advances in Neural Information Processing +Systems, pages 2672–2680, 2014. +[16] I. J. Goodfellow, J. Shlens, and C. Szegedy. Explaining and harnessing adversarial exam- +ples. In International Conference on Learning Representations, 2015. +[17] G. N. Grapiglia and Y. Yuan. On the complexity of an augmented Lagrangian method for +nonconvex optimization. IMA J. Numer. Anal., 41(2):1508–1530, 2021. +[18] Z. Guo, Z. Yuan, Y. Yan, and T. Yang. +Fast objective & duality gap convergence for +nonconvex-strongly-concave min-max problems. arXiv preprint arXiv:2006.06889, 2020. +[19] F. Huang, S. Gao, J. Pei, and H. Huang. Accelerated zeroth-order momentum methods +from mini to minimax optimization. arXiv preprint arXiv:2008.08170, 3, 2020. +[20] C. Kanzow and D. Steck. An example comparing the standard and safeguarded augmented +Lagrangian methods. Oper. Res. Lett., 45(6):598–603, 2017. +[21] W. Kong and R. D. Monteiro. An accelerated inexact proximal point method for solving +nonconvex-concave min-max problems. SIAM Journal on Optimization, 31(4):2558–2585, +2021. +[22] T. Lin, C. Jin, and M. Jordan. On gradient descent ascent for nonconvex-concave minimax +problems. In International Conference on Machine Learning, pages 6083–6093, 2020. +[23] T. Lin, C. Jin, and M. I. Jordan. Near-optimal algorithms for minimax optimization. In +Conference on Learning Theory, pages 2738–2779. PMLR, 2020. +[24] S. Lu. A single-loop gradient descent and perturbed ascent algorithm for nonconvex func- +tional constrained optimization. In International Conference on Machine Learning, pages +14315–14357, 2022. +[25] S. Lu, I. Tsaknakis, M. Hong, and Y. Chen. Hybrid block successive approximation for +one-sided non-convex min-max problems: algorithms and applications. IEEE Transactions +on Signal Processing, 68:3676–3691, 2020. +20 + +[26] Z. Lu and S. Mei. First-order penalty methods for bilevel optimization. arXiv preprint +arXiv:2301.01716, 2023. +[27] Z. Lu and Y. Zhang. An augmented Lagrangian approach for sparse principal component +analysis. Math. Program., 135(1-2):149–193, 2012. +[28] L. Luo, H. Ye, Z. Huang, and T. Zhang. Stochastic recursive gradient descent ascent for +stochastic nonconvex-strongly-concave minimax problems. Advances in Neural Information +Processing Systems, 33:20566–20577, 2020. +[29] A. Madry, A. Makelov, L. Schmidt, D. Tsipras, and A. Vladu. Towards deep learning +models resistant to adversarial attacks. In International Conference on Learning Repre- +sentations, 2018. +[30] G. Mateos, J. A. Bazerque, and G. B. Giannakis. Distributed sparse linear regression. +IEEE Transactions on Signal Processing, 58:5262–5276, 2010. +[31] O. Nachum, Y. Chow, B. Dai, and L. Li. DualDICE: Behavior-agnostic estimation of dis- +counted stationary distribution corrections. In Advances in Neural Information Processing +Systems, pages 2315–2325, 2019. +[32] J. Nocedal and S. J. Wright. Numerical optimization. Springer, 1999. +[33] M. Nouiehed, M. Sanjabi, T. Huang, J. D. Lee, and M. Razaviyayn. Solving a class of non- +convex min-max games using iterative first order methods. Advances in Neural Information +Processing Systems, 32, 2019. +[34] S. Qiu, Z. Yang, X. Wei, J. Ye, and Z. Wang. Single-timescale stochastic nonconvex-concave +optimization for smooth nonlinear td learning. arXiv preprint arXiv:2008.10103, 2020. +[35] A. Rakhlin and K. Sridharan. +Optimization, learning, and games with predictable se- +quences. In Advances in Neural Information Processing Systems, pages 3066–3074, 2013. +[36] M. F. Sahin, A. Eftekhari, A. Alacaoglu, F. Latorre, and V. Cevher. An inexact augmented +Lagrangian framework for nonconvex optimization with nonlinear constraints. Advances +in Neural Information Processing Systems, 32, 2019. +[37] M. Sanjabi, J. Ba, M. Razaviyayn, and J. D. Lee. On the convergence and robustness of +training gans with regularized optimal transport. Advances in Neural Information Process- +ing Systems, 31, 2018. +[38] S. Shafieezadeh-Abadeh, P. M. Esfahani, and D. Kuhn. Distributionally robust logistic +regression. In Advances in Neural Information Processing Systems, page 1576–1584, 2015. +[39] J. Shamma. Cooperative Control of Distributed Multi-Agent Systems. Wiley-Interscience, +2008. +[40] A. Sinha, H. Namkoong, and J. C. Duchi. Certifying some distributional robustness with +principled adversarial training. In International Conference on Learning Representations, +2018. +[41] J. Song, H. Ren, D. Sadigh, and S. Ermon. Multi-agent generative adversarial imitation +learning. Advances in neural information processing systems, 31, 2018. +[42] V. Syrgkanis, A. Agarwal, H. Luo, and R. E. Schapire. Fast convergence of regularized +learning in games. In Advances in Neural Information Processing Systems, page 2989–2997, +2015. +21 + +[43] B. Taskar, S. Lacoste-Julien, and M. Jordan. Structured prediction via the extragradient +method. In Advances in Neural Information Processing Systems, page 1345–1352, 2006. +[44] I. Tsaknakis, M. Hong, and S. Zhang. Minimax problems with coupled linear constraints: +computational complexity, duality and solution methods. arXiv preprint arXiv:2110.11210, +2021. +[45] J. Wang, T. Zhang, S. Liu, P.-Y. Chen, J. Xu, M. Fardad, and B. Li. Adversarial at- +tack generation empowered by min-max optimization. In Advances in Neural Information +Processing Systems, 2021. +[46] D. Ward and J. M. Borwein. Nonsmooth calculus in finite dimensions. SIAM Journal on +control and optimization, 25(5):1312–1340, 1987. +[47] W. Xian, F. Huang, Y. Zhang, and H. Huang. A faster decentralized algorithm for non- +convex minimax problems. Advances in Neural Information Processing Systems, 34, 2021. +[48] Y. Xie and S. J. Wright. Complexity of proximal augmented Lagrangian for nonconvex +optimization with nonlinear equality constraints. J. Sci. Comput., 86(3):1–30, 2021. +[49] H. Xu, C. Caramanis, and S. Mannor. Robustness and regularization of support vector +machines. Journal of Machine Learning Research, 10:1485–1510, 2009. +[50] L. Xu, J. Neufeld, B. Larson, and D. Schuurmans. Maximum margin clustering. In Advances +in Neural Information Processing Systems, page 1537–1544, 2005. +[51] T. Xu, Z. Wang, Y. Liang, and H. V. Poor. Gradient free minimax optimization: Variance +reduction and faster convergence. arXiv preprint arXiv:2006.09361, 2020. +[52] Z. Xu, H. Zhang, Y. Xu, and G. Lan. A unified single-loop alternating gradient projec- +tion algorithm for nonconvex-concave and convex-nonconcave minimax problems. arXiv +preprint arXiv:2006.02032, 2020. +[53] H. Zhang, J. Wang, Z. Xu, and Y.-H. Dai. +Primal dual alternating proximal gradient +algorithms for nonsmooth nonconvex minimax problems with coupled linear constraints. +arXiv preprint arXiv:2212.04672, 2022. +[54] J. Zhang, P. Xiao, R. Sun, and Z. Luo. A single-loop smoothed gradient descent-ascent +algorithm for nonconvex-concave min-max problems. Advances in Neural Information Pro- +cessing Systems, 33:7377–7389, 2020. +A +A first-order method for nonconvex-concave minimax prob- +lem +In this part we present a first-order method proposed in [26, Algorithm 2] for finding an ǫ- +stationary point of the nonconvex-concave minimax problem +H∗ = min +x max +y +{H(x, y) := h(x, y) + p(x) − q(y)} , +(85) +which has at least one optimal solution and satisfies the following assumptions. +Assumption 4. +(i) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper convex functions +and continuous on dom p and dom q, respectively, and moreover, dom p and dom q are +compact. +(ii) The proximal operator associated with p and q can be exactly evaluated. +22 + +(iii) h is L∇h-smooth on dom p × dom q, and moreover, h(x, ·) is concave for any x ∈ dom p. +For ease of presentation, we define +Dp = max{∥u − v∥ +��u, v ∈ dom p}, +Dq = max{∥u − v∥ +��u, v ∈ dom q}, +(86) +Hlow = min{H(x, y)|(x, y) ∈ dom p × dom q}. +(87) +Given an iterate (xk, yk), the first-order method [26, Algorithm 2] finds the next iterate +(xk+1, yk+1) by applying a modified optimal first-order method [26, Algorithm 1] to the strongly- +convex-strongly-concave minimax problem +min +x max +y +� +hk(x, y) = h(x, y) − ǫ∥y − y0∥2/(4Dq) + L∇h∥x − xk∥2� +. +(88) +For ease reference, we next present the modified optimal first-order method [26, Algorithm +1] in Algorithm 2 below for solving the strongly-convex-strongly-concave minimax problem +min +x max +y +�¯h(x, y) + p(x) − q(y) +� +, +(89) +where ¯h(x, y) is σx-strongly-convex-σy-strongly-concave and L∇¯h-smooth on dom p × dom q for +some σx, σy > 0. In Algorithm 2, the functions ˆh, ak +x and ak +y are defined as follows: +ˆh(x, y) = ¯h(x, y) − σx∥x∥2/2 + σy∥y∥2/2, +ak +x(x, y) = ∇xˆh(x, y) + σx(x − σ−1 +x zk +g)/2, +ak +y(x, y) = −∇yˆh(x, y) + σyy + σx(y − yk +g)/8, +where yk +g and zk +g are generated at iteration k of Algorithm 2 below. +23 + +Algorithm 2 A modified optimal first-order method for problem (89) +Input: τ +> 0, ¯z0 = z0 +f +∈ −σxdom p,4 ¯y0 = y0 +f +∈ dom q, (z0, y0) = (¯z0, ¯y0), ¯α = +min +� +1, +� +8σy/σx +� +, ηz += σx/2, ηy += min {1/(2σy), 4/(¯ασx)}, βt = 2/(t + 3), ζ += +� +2 +√ +5(1 + 8L∇¯h/σx) +�−1, γx = γy = 8σ−1 +x , and ˆζ = min{σx, σy}/L2 +∇¯h. +1: for k = 0, 1, 2, . . . do +2: +(zk +g , yk +g) = ¯α(zk, yk) + (1 − ¯α)(zk +f, yk +f). +3: +(xk,−1, yk,−1) = (−σ−1 +x zk +g, yk +g). +4: +xk,0 = proxζγxp(xk,−1 − ζγxak +x(xk,−1, yk,−1)). +5: +yk,0 = proxζγyq(yk,−1 − ζγyak +y(xk,−1, yk,−1)). +6: +bk,0 +x += +1 +ζγx (xk,−1 − ζγxak +x(xk,−1, yk,−1) − xk,0). +7: +bk,0 +y += +1 +ζγy (yk,−1 − ζγyak +y(xk,−1, yk,−1) − yk,0). +8: +t = 0. +9: +while +γx∥ak +x(xk,t, yk,t)+bk,t +x ∥2+γy∥ak +y(xk,t, yk,t)+bk,t +y ∥2 > γ−1 +x ∥xk,t−xk,−1∥2+γ−1 +y ∥yk,t−yk,−1∥2 +do +10: +xk,t+1/2 = xk,t + βt(xk,0 − xk,t) − ζγx(ak +x(xk,t, yk,t) + bk,t +x ). +11: +yk,t+1/2 = yk,t + βt(yk,0 − yk,t) − ζγy(ak +y(xk,t, yk,t) + bk,t +y ). +12: +xk,t+1 = proxζγxp(xk,t + βt(xk,0 − xk,t) − ζγxak +x(xk,t+1/2, yk,t+1/2)). +13: +yk,t+1 = proxζγyq(yk,t + βt(yk,0 − yk,t) − ζγyak +y(xk,t+1/2, yk,t+1/2)). +14: +bk,t+1 +x += +1 +ζγx (xk,t + βt(xk,0 − xk,t) − ζγxak +x(xk,t+1/2, yk,t+1/2) − xk,t+1). +15: +bk,t+1 +y += +1 +ζγy (yk,t + βt(yk,0 − yk,t) − ζγyak +y(xk,t+1/2, yk,t+1/2) − yk,t+1). +16: +t ← t + 1. +17: +end while +18: +(xk+1 +f +, yk+1 +f +) = (xk,t, yk,t). +19: +(zk+1 +f +, wk+1 +f +) = (∇xˆh(xk+1 +f +, yk+1 +f +) + bk,t +x , −∇yˆh(xk+1 +f +, yk+1 +f +) + bk,t +y ). +20: +zk+1 = zk + ηzσ−1 +x (zk+1 +f +− zk) − ηz(xk+1 +f ++ σ−1 +x zk+1 +f +). +21: +yk+1 = yk + ηyσy(yk+1 +f +− yk) − ηy(wk+1 +f ++ σyyk+1 +f +). +22: +xk+1 = −σ−1 +x zk+1. +23: +˜xk+1 = proxˆζp(xk+1 − ˆζ∇x¯h(xk+1, yk+1)). +24: +˜yk+1 = proxˆζq(yk+1 + ˆζ∇y¯h(xk+1, yk+1)). +25: +Terminate the algorithm and output (˜xk+1, ˜yk+1) if +∥ˆζ−1(xk+1 − ˜xk+1, ˜yk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(˜xk+1, ˜yk+1))∥ ≤ τ. +26: end for +We are now ready to present the first-order method [26, Algorithm 2] for finding an ǫ- +stationary point of (85) in Algorithm 3 below. +4For convenience, −σxdom p stands for the set {−σxu|u ∈ dom p}. +24 + +Algorithm 3 A first-order method for problem (85) +Input: ǫ > 0, ǫ0 ∈ (0, ǫ/2], (ˆx0, ˆy0) ∈ dom p × dom q, (x0, y0) = (ˆx0, ˆy0), and ǫk = ǫ0/(k + 1). +1: for k = 0, 1, 2, . . . do +2: +Call Algorithm 2 with ¯h ← hk, τ ← ǫk, σx ← L∇h, σy ← ǫ/(2Dq), L∇¯h ← 3L∇h+ǫ/(2Dq), +¯z0 = z0 +f ← −σxxk, ¯y0 = y0 +f ← yk, and denote its output by (xk+1, yk+1), where hk is +given in (88). +3: +Terminate the algorithm and output (xǫ, yǫ) = (xk+1, yk+1) if +∥xk+1 − xk∥ ≤ ǫ/(4L∇h). +4: end for +The following theorem presents the iteration complexity of Algorithm 3, whose proof is given +in [26, Theorem 2]. +Theorem 2 (Complexity of Algorithm 3). Suppose that Assumption 4 holds. Let H∗, H +Dp, Dq, and Hlow be defined in (85), (86) and (87), L∇h be given in Assumption 4, ǫ, ǫ0 and +x0 be given in Algorithm 3, and +α = min +� +1, +� +4ǫ/(DqL∇h) +� +, +δ = (2 + α−1)L∇hD2 +p + max {ǫ/Dq, αL∇h/4} D2 +q, +K = +� +16(max +y +H(x0, y) − H∗ + ǫDq/4)L∇hǫ−2 + 32ǫ2 +0(1 + 4D2 +qL2 +∇hǫ−2)ǫ−2 − 1 +� ++ +, +N = +�� +96 +√ +2 +� +1 + (24L∇h + 4ǫ/Dq) L−1 +∇h +�� ++ 2 +� � +2, +� +DqL∇hǫ−1 +� +× +� +(K + 1) +� +log +4 max +� +1 +2L∇h , min +� +Dq +ǫ , +4 +αL∇h +�� � +δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2 +p) +� +[(3L∇h + ǫ/(2Dq))2/ min{L∇h, ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 +0 +� ++ ++ K + 1 + 2K log(K + 1) +� +. +Then Algorithm 3 terminates and outputs an ǫ-stationary point (xǫ, yǫ) of (85) in at most K +1 +outer iterations that satisfies +max +y +H(xǫ, y) ≤ max +y +H(ˆx0, y) + ǫDq/4 + 2ǫ2 +0 +� +L−1 +∇h + 4D2 +qL∇hǫ−2� +. +Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed +in Algorithm 3 is no more than N, respectively. +25 + diff --git a/etA0T4oBgHgl3EQfHP_m/content/tmp_files/load_file.txt b/etA0T4oBgHgl3EQfHP_m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c389a197f006859632c5d1d0bfc94ffe014cfd82 --- /dev/null +++ b/etA0T4oBgHgl3EQfHP_m/content/tmp_files/load_file.txt @@ -0,0 +1,1009 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf,len=1008 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='02060v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='OC] 5 Jan 2023 A first-order augmented Lagrangian method for constrained minimax optimization Zhaosong Lu ∗ Sanyou Mei ∗ January 5, 2023 Abstract In this paper we study a class of constrained minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In particular, we pro- pose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first- order method recently developed in [26] by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Under some suitable assumptions, an operation complexity of O(ε−4 log ε−1), measured by its fundamental operations, is es- tablished for the first-order augmented Lagrangian method for finding an ε-KKT solution of the constrained minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Keywords: minimax optimization, augmented Lagrangian method, first-order method, oper- ation complexity Mathematics Subject Classification: 90C26, 90C30, 90C47, 90C99, 65K05 1 Introduction In this paper, we consider a constrained minimax problem F ∗ = min c(x)≤0 max d(x,y)≤0{F(x, y) := f(x, y) + p(x) − q(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (1) Assume that problem (1) has at least one optimal solution and the following additional assump- tions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) F is LF-Lipschitz continuous on X × Y, f is L∇f-smooth on X × Y, and f(x, ·) is concave for any given x ∈ X, where X := dom p and Y := dom q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='1 (ii) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper closed convex functions, and the proximal operator of p and q can be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (iii) c : Rn → R˜n is L∇c-smooth and Lc-Lipschitz continuous on X, d : Rn × Rm → R ˜m is L∇d-smooth and Ld-Lipschitz continuous on X ×Y, and di(x, ·) is convex for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (iv) The sets X and Y (namely, dom p and dom q) are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In the recent years, the minimax problem of a simpler form min x∈X max y∈Y f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' y), (2) ∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email: zhaosong@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='edu, mei00035@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' This work was partially supported by NSF Award IIS-2211491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1The definition of LF -Lipschitz continuity of F and L∇f-smoothness of f is given in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1 where X and Y are a closed set, has received tremendous amount of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Indeed, it has found broad applications in many areas, such as adversarial training [16, 29, 40, 45], generative adversarial networks [13, 15, 37], reinforcement learning [8, 11, 31, 34, 41], computational game [1, 35, 42], distributed computing [30, 39], prediction and regression [4, 43, 49, 50], and distri- butionally robust optimization [12, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Numerous methods have been developed for solving (2) with X and Y being a simple closed convex set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', see [6, 18, 19, 22, 23, 25, 28, 33, 47, 51, 52, 54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' There have also been several studies on some other special cases of problem (1) recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In particular, two first-order methods, called max-oracle gradient-descent and nested gradient descent/ascent methods, were proposed in [14] for solving (1) with c(x) ≡ 0 and p and q being the indicator function of simple compact convex sets X and Y respectively, under the assumption that the function V (x) = maxy∈Y {f(x, y) : d(x, y) ≤ 0} is convex and moreover an optimal Lagrangian multiplier associated with the constraint d(x, y) ≤ 0 can be computed for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, a multiplier gradient descent method was proposed in [44] for solving (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and p and q being the indicator function of a simple compact convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Also, a proximal gradient multi-step ascent decent method was developed in [9] for (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and f(x, y) = g(x) + xT Ay − h(y), under the assumption that f(x, y) − q(y) is strongly concave in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Besides, primal dual alternating proximal gradient methods were proposed in [53] for (1) with c(x) ≡ 0, d(x, y) being an affine mapping, and {f(x, y) being strongly concave in y or [q(y) ≡ 0 and f(x, y) being a linear function in y]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For these methods, an iteration complexity for finding an approximate stationary point of the aforementioned special minimax problem was established in [9, 14, 53], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Yet, their operation complexity, measured by the amount of fundamental operations such as evaluations of gradient of f and proximal operator of p and q, was not studied in these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' There was no algorithmic development for (1) prior to our work, though optimality condi- tions of (1) were recently studied in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In this paper, we propose a first-order augmented Lagrangian (AL) method for solving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Specifically, given an iterate (xk, yk) and a Lagrangian multiplier estimate (λk x, λk y) at the kth iteration, the next iterate (xk+1, yk+1) is obtained by finding an approximate stationary point of the AL subproblem min x max y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) for some ρk > 0 through the use of a first-order method proposed in [26], where L is the AL function of (1) defined as L(x, y, λx, λy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρ) = F(x, y)+ 1 2ρ � ∥[λx + ρc(x)]+∥2 − ∥λx∥2� − 1 2ρ � ∥[λy + ρd(x, y)]+∥2 − ∥λy∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (3) The Lagrangian multiplier estimate is then updated by λk+1 x = ΠB+ Λ (λk x+ρkc(xk+1)) and λk+1 y = [λk y + ρkd(xk+1, yk+1)]+ for some Λ > 0, where ΠB+ Λ(·) and [·]+ are defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The main contributions of this paper are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We propose a first-order AL method for solving problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' To the best of our knowledge, this is the first yet implementable method for solving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We show that under some suitable assumptions, our first-order AL method enjoys an iter- ation complexity of O(log ε−1) and an operation complexity of O(ε−4 log ε−1), measured by the amount of evaluations of ∇f, ∇c, ∇d and proximal operator of p and q, for finding an ε-KKT solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='1, we introduce some notation and terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Section 2, we propose a first-order AL method for solving problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In 2 Section 3, we present complexity results for the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Section 4, we provide the proof of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='1 Notation and terminology The following notation will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let Rn denote the Euclidean space of dimension n and Rn + denote the nonnegative orthant in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The standard inner product, l1-norm and Euclidean norm are denoted by ⟨·, ·⟩, ∥ · ∥1 and ∥ · ∥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For any Λ > 0, let B+ Λ = {x ≥ 0 : ∥x∥ ≤ Λ}, whose dimension is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For any v ∈ Rn, let v+ denote the nonnegative part of v, that is, (v+)i = max{vi, 0} for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Given a point x and a closed set S in Rn, let dist(x, S) = minx′∈S ∥x′ − x∥, ΠS(x) denote the Euclidean projection of x onto S, and IS denote the indicator function associated with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A function or mapping φ is said to be Lφ-Lipschitz continuous on a set S if ∥φ(x)−φ(x′)∥ ≤ Lφ∥x−x′∥ for all x, x′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, it is said to be L∇φ-smooth on S if ∥∇φ(x)−∇φ(x′)∥ ≤ L∇φ∥x − x′∥ for all x, x′ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For a closed convex function p : Rn → R ∪ {∞},2 the proximal operator associated with p is denoted by proxp, that is, proxp(x) = arg min x′∈Rn �1 2∥x′ − x∥2 + p(x′) � ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Given that evaluation of proxγp(x) is often as cheap as proxp(x), we count the evaluation of proxγp(x) as one evaluation of proximal operator of p for any γ > 0 and x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For a lower semicontinuous function φ : Rn → R ∪ {∞}, its domain is the set dom φ := {x|φ(x) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The upper subderivative of φ at x ∈ dom φ in a direction d ∈ Rn is defined by φ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' d) = lim sup x′ φ→x, t↓0 inf d′→d φ(x′ + td′) − φ(x′) t , where t ↓ 0 means both t > 0 and t → 0, and x′ φ→ x means both x′ → x and φ(x′) → φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The subdifferential of φ at x ∈ dom φ is the set ∂φ(x) = {s ∈ Rn��sTd ≤ φ′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' d) ∀d ∈ Rn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We use ∂xiφ(x) to denote the subdifferential with respect to xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, for an upper semicontinuous function φ, its subdifferential is defined as ∂φ = −∂(−φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' If φ is locally Lipschitz continuous, the above definition of subdifferential coincides with the Clarke subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Besides, if φ is convex, it coincides with the ordinary subdifferential for convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Also, if φ is continuously differentiable at x , we simply have ∂φ(x) = {∇φ(x)}, where ∇φ(x) is the gradient of φ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, it is not hard to verify that ∂(φ1 + φ2)(x) = ∇φ1(x) + ∂φ2(x) if φ1 is continuously differentiable at x and φ2 is lower or upper semicontinuous at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' See [7, 46] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Finally, we introduce an (approximate) stationary point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', see [9, 10, 21]) for a general minimax problem min x max y Ψ(x, y), (4) where Ψ(·, y) : Rn → R∪{∞} is a lower semicontinuous function, and Ψ(x, ·) : Rm → R∪{−∞} is an upper semicontinuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A point (x, y) is said to be a stationary point of the minimax problem (4) if 0 ∈ ∂xΨ(x, y), 0 ∈ ∂yΨ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, for any ǫ > 0, a point (xǫ, yǫ) is said to be an ǫ-stationary point of the minimax problem (4) if dist (0, ∂xΨ(xǫ, yǫ)) ≤ ǫ, dist (0, ∂yΨ(xǫ, yǫ)) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 2For convenience, ∞ stands for +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 3 2 A first-order augmented Lagrangian method for problem (1) In this section we propose a first-order augmented Lagrangian (FAL) method for problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One standard approach for solving constrained nonlinear program is to solve a sequence of unconstrained nonlinear program problems, which are typically penalty or augmented La- grangian subproblems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', see [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In a similar spirit, we next propose an FAL method in Algorithm 1 for solving (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In particular, at each iteration, the FAL method finds an approxi- mate stationary point of an AL subproblem in the form of min x max y L(x, y, λx, λy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρ) (5) for some ρ > 0, λx ∈ R˜n + and λy ∈ R ˜m +, where L is the AL function associated with problem (1) defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In view of Assumption 1, one can observe that L enjoys the following nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For any given ρ > 0, λx ∈ R˜n + and λy ∈ R ˜m +, L is the sum of smooth function f(x, y) + � ∥[λx + ρc(x)]+∥2 − ∥λx∥2� /(2ρ)− � ∥[λy + ρd(x, y)]+∥2 − ∥λy∥2� /(2ρ) with Lipschitz con- tinuous gradient and possibly nonsmooth function p(x) − q(y) with exactly computable proximal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' L is nonconvex in x but concave in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Thanks to such a nice structure of L, an approximate stationary point of the AL subproblem (5) can be found by Algorithm 3 (see Appendix A), which is a first-order method proposed in [26, Algorithm 2]) for solving nonconvex-concave minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Before presenting an FAL method for (1), we let Lx(x, y, λx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρ) := F(x, y) + 1 2ρ � ∥[λx + ρc(x)]+∥2 − ∥λx∥2� , (6) chi := max{∥c(x)∥ ��x ∈ X}, dhi := max{∥d(x, y)∥ ��(x, y) ∈ X × Y}, (7) and make one additional assumption on problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For any given η ∈ (0, 1], an η-approximately feasible point zη of problem (1), namely zη ∈ X satisfying ∥[c(zη)]+∥ ≤ η, can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A very similar assumption as Assumption 2 was considered in [5, 17, 27, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One example of the problem instances satisfying Assumption 2 arises when the error bound condition ∥[c(x)]+∥ = O(dist(0, ∂(∥[c(x)]+∥2 +IX (x))))ν) holds on a level set of ∥[c(x)]+∥ for some ν > 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', see [24, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Indeed, one can find the above zη by applying a projected gradient method to the problem minx∈X ∥[c(x)]+∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We are now ready to present an FAL method for solving problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4 Algorithm 1 A first-order augmented Lagrangian method for problem (1) Input: ε, τ ∈ (0, 1), ǫ0 ∈ (τε, 1], ǫk = ǫ0τ k, ρk = ǫ−1 k , Λ > 0, λ0 x ∈ B+ Λ, λ0 y ∈ R ˜m +, (x0, y0) ∈ X × Y, and xnf ∈ X with ∥[c(xnf)]+∥ ≤ √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1: for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' do 2: Set xk init = � xk, if Lx(xk, yk, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) ≤ Lx(xnf, yk, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk), xnf, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (8) 3: Call Algorithm 3 (see Appendix A) with ǫ ← ǫk, ǫ0 ← ǫk/(2√ρk), (x0, y0) ← (xk init, yk) and L∇h ← Lk to find an ǫk-stationary point (xk+1, yk+1) of min x max y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (9) where Lk = L∇f + ρkL2 c + ρkchiL∇c + ∥λk x∥L∇c + ρkL2 d + ρkdhiL∇d + ∥λk y∥L∇d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (10) 4: Set λk+1 x = ΠB+ Λ(λk x + ρkc(xk+1)) and λk+1 y = [λk y + ρkd(xk+1, yk+1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 5: Terminate the algorithm and output (xk+1, yk+1) if ǫk ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 6: end for Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) xnf is an √ε-approximately feasible point of problem (1), where the subscript “nf” stands for “nearly feasible”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It follows from Assumption 2 that xnf can be found in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (ii) λk+1 x results from projecting onto a nonnegative Euclidean ball the standard Lagrangian multiplier estimate ˜λk+1 x obtained by the classical scheme ˜λk+1 x = [λk x + ρkc(xk+1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It is called a safeguarded Lagrangian multiplier in the relevant literature [2, 20, 3], which has been shown to enjoy many practical and theoretical advantages (see [2] for discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (iii) In view of Theorem 2 (see Appendix A), one can see that an ǫk-stationary point of (9) can be successfully found in step 3 of Algorithm 1 by applying Algorithm 3 to problem (9) and thus Algorithm 1 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 3 Complexity results of Algorithm 1 In this section we establish iteration and operation complexity results for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Before proceeding, we make one additional assumption that a generalized Mangasarian-Fromowitz constraint qualification holds for the minimization part of (1) and a uniform Slater’s condition holds for the maximization part of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) There exist some constants δc, θa, θf > 0 such that for each x ∈ F(θf) there exists some vx ∈ Rn satisfying ∥vx∥ = 1 and vT x ∇ci(x) ≤ −δc for all i ∈ A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa), where F(θf) = {x ∈ X ��∥[c(x)]+∥ ≤ θf}, A(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa) = {i|ci(x) ≥ −θa, 1 ≤ i ≤ ˜n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (11) (ii) For each x ∈ X, there exists some ˆyx ∈ Y such that di(x, ˆyx) < 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' , ˜m, and moreover, δd := inf{−di(x, ˆyx)|x ∈ X, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' , ˜m} > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='3 3The latter part of this assumption can be weakened to the one that the pointwise Slater’s condition holds for 5 In order to characterize the approximate solution found by Algorithm 1, we next introduce a terminology called an ε-KKT solution of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One can observe from Lemma 1(iii) that problem (1) is equivalent to min x,λy � max y F(x, y) − ⟨λy, d(x, y)⟩ + IR ˜ m + (λy) ��c(x) ≤ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this, one can further see that problem (1) is equivalent to min x,λy max λx � max y {F(x, y) − ⟨λy, d(x, y)⟩ + IR ˜ m + (λy)} + ⟨λx, c(x)⟩ − IR˜n +(λx) � , which is a nonconvex-concave minimax problem min x,λy max y,λx � F(x, y) + ⟨λx, c(x)⟩ − ⟨λy, d(x, y)⟩ − IR˜n +(λx) + IR ˜ m + (λy) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (12) It then follows from Definition 1 (see also [9, Theorem 3]) that (x, y, λx, λy) ∈ Rn×Rm×R˜n +×R ˜m + is a stationary point of problem (12) if 0 ∈ ∂xF(x, y) + ∇c(x)λx − ∇xd(x, y)λy, (13) 0 ∈ ∂yF(x, y) − ∇yd(x, y)λy, (14) c(x) ≤ 0, ⟨λx, c(x)⟩ = 0, (15) d(x, y) ≤ 0, ⟨λy, d(x, y)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (16) Based on this observation and the equivalence of (1) and (12), we introduce an (approximate) KKT solution of problem (1) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The pair (x, y) is said to be a KKT solution of problem (1) if there exists (λx, λy) ∈ R˜n + × R ˜m + such that the conditions (13)-(16) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, for any ε > 0, (x, y) is said to be an ε-KKT point of problem (1) if there exists (λx, λy) ∈ R˜n + × R ˜m + such that dist(0, ∂xF(x, y) + ∇c(x)λx − ∇xd(x, y)λy) ≤ ε, dist(0, ∂yF(x, y) − ∇yd(x, y)λy) ≤ ε, ∥[c(x)]+∥ ≤ ε, |⟨λx, c(x)⟩| ≤ ε, ∥[d(x, y)]+∥ ≤ ε, |⟨λy, d(x, y)⟩| ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' To study complexity of Algorithm 1, we define f ∗(x) := max{F(x, y)|d(x, y) ≤ 0}, (17) f ∗ low := inf{f ∗(x)|x ∈ X}, (18) Dx := max{∥u − v∥ ��u, v ∈ X}, Dy := max{∥u − v∥ ��u, v ∈ Y}, (19) Fhi := max{F(x, y)|(x, y) ∈ X × Y}, Flow := min{F(x, y)|(x, y) ∈ X × Y}, (20) r := 2δ−1 d (ǫ0 + LF)Dy, (21) K := ⌈(log ε − log ǫ0)/ log τ⌉+ , K := {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' , K + 1}, (22) where LF and δd are given in Assumptions 1 and 3, and ǫ0, ε, and τ are some input parameters of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For convenience, we define K − 1 = {k − 1|k ∈ K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One can observe from Assumption 1 that Dx, Dy, Fhi and Flow are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Besides, as will be shown in Lemma 1, f ∗ low is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We are now ready to present an iteration and operation complexity of Algorithm 1 for finding an O(ε)-KKT solution of problem (1), whose proof is deferred to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' the constraint on y in (1), that is, there exists ˆyx ∈ Y such that d(x, ˆyx) < 0 for each x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Indeed, if δd > 0, Assumption 3(ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Otherwise, one can solve the perturbed counterpart of (1) with d(x, y) being replaced by d(x, y) − ǫ for some suitable ǫ > 0 instead, which satisfies Assumption 3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 6 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1, 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let {(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' λk x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' λk y)}k∈K be generated by Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' chi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' dhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' f ∗ low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Fhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Flow and K be defined in (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (19),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (20) and (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' LF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' L∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' L∇d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' L∇c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' L∇d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ld and δd be given in Assumption 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Λ and λ0 y be given in Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and �L = L∇f + L2 c + chiL∇c + ΛL∇c + L2 d + dhiL∇d + L∇d � ∥λ0y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (23) ˆα = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � 4/(Dy �L) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆδ = (2 + ˆα−1)�LD2 x + max{1/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' �L/4}D2 y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (24) � M = 16 max � 1/(2L2 c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4/(ˆαL2 c) � � (3�L + 1/(2Dy))2/ min{L2 c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1/(2Dy)} + 3�L + 1/(2Dy) �2 × � ˆδ + 2ˆα−1� Fhi − Flow + Λ2 2 + 3 2∥λ0 y∥2 + 3(Fhi − f ∗ low + Dyǫ0) 1 − τ + ρkd2 hi + Dy 4 + �LD2 x �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (25) �T = � 16 � LF Dy + Fhi − f ∗ low + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ + Λ2 2 + Dy 4 � �L + 8(1 + 4D2 y �L2) � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (26) ˜λK+1 x = [λK x + c(xK+1)/(ǫ0τ K)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (27) Suppose that ε−1 ≥ max � 1, θ−1 a Λ, θ−2 f � 2LF Dy + 2Fhi − 2f ∗ low + 2Λ + τ −1 + ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ + ǫ0Dy 2 + L−2 c + 4D2 y �L + Λ2� , 4∥λ0 y∥2 δ2 dτ + 8(Fhi − f ∗ low + Dyǫ0) δ2 dτ(1 − τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (28) Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) Algorithm 1 terminates after K+1 outer iterations and outputs an approximate stationary point (xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1) of (1) satisfying dist(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∂xF(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1) + ∇c(xK+1)˜λK+1 x − ∇xd(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)λK+1 y ) ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (29) dist � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∂yF(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1) − ∇yd(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)λK+1 y � ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (30) ∥[c(xK+1)]+∥ ≤ εδ−1 c � LF + 2Ldδ−1 d (ǫ0 + LF )Dy + ǫ0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (31) |⟨˜λK+1 x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' c(xK+1)⟩| ≤ εδ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF)Dy + ǫ0) × max{δ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF)Dy + ǫ0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Λ},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (32) ∥[d(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)]+∥ ≤ 2εδ−1 d (ǫ0 + LF )Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (33) |⟨λK+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' d(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)⟩| ≤ 2εδ−1 d (ǫ0 + LF)Dy max{2δ−1 d (ǫ0 + LF )Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∥λ0 y∥} (34) (ii) The total number of evaluations of ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∇c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∇d and proximal operator of p and q performed in Algorithm 1 is at most N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' where N = �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy �L � �T(1 − τ 4)−1 × (τε)−4 � 28K log(1/τ) + 28 log(1/ǫ0) + 2(log � M)+ + 2 + 2 log(2 �T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (35) 7 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One can observe from Theorem 1 that Algorithm 1 enjoys an iteration complexity of O(log ε−1) and an operation complexity of O(ε−4 log ε−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' measured by the amount of eval- uations of ∇f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∇c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∇d and proximal operator of p and q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' for finding an O(ε)-KKT solution (xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1) of (1) such that dist � ∂xF(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1) + ∇c(xK+1)˜λx − ∇xd(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)λK+1 y � ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' dist � ∂yF(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1) − ∇yd(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)λK+1 y � ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∥[c(xK+1)]+∥ = O(ε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' |⟨˜λK+1 x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' c(xK+1)⟩| = O(ε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ∥[d(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)]+∥ = O(ε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' |⟨λK+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' d(xK+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yK+1)⟩| = O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' where ˜λK+1 x ∈ R˜n + is defined in (27) and λK+1 y ∈ R ˜m + is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4 Proof of the main result In this section, we provide a proof of our main result presented in Section 2, which is particularly Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Before proceeding, let Ly(x, y, λy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρ) = F(x, y) − 1 2ρ � ∥[λy + ρd(x, y)]+∥2 − ∥λy∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (36) In view of (3), (17) and (36), one can observe that f ∗(x) ≤ max y Ly(x, y, λy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρ) ∀x ∈ X, λy ∈ R ˜m +, ρ > 0, (37) which will be frequently used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We next establish several lemmas that will be used to prove Theorem 1 subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let f ∗, f ∗ low, Dy, r, LF and δd be given in (17), (18), (19), (21) and Assumption 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) ∥λ∗ y∥ ≤ δ−1 d LF Dy and λ∗ y ∈ B+ r for all λ∗ y ∈ Λ∗(x) and x ∈ X, where Λ∗(x) denotes the set of optimal Lagrangian multipliers of problem (17) for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (ii) The function f ∗ is Lipschitz continuous on X and f ∗ low is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (iii) It holds that f ∗(x) = min λy max y F(x, y) − ⟨λy, d(x, y)⟩ + IR ˜ m + (λy) ∀x ∈ X, (38) where IR ˜ m + (·) is the indicator function associated with R ˜m +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) Let x ∈ X and λ∗ y ∈ Λ∗(x) be arbitrarily chosen, and let y∗ ∈ Y be such that (y∗, λ∗ y) is a pair of primal-dual optimal solutions of (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows that y∗ ∈ Argmax y F(x, y) − ⟨λ∗ y, d(x, y)⟩, ⟨λ∗ y, d(x, y∗)⟩ = 0, d(x, y∗) ≤ 0, λ∗ y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The first relation above yields F(x, y∗) − ⟨λ∗ y, d(x, y∗)⟩ ≥ F(x, ˆyx) − ⟨λ∗ y, d(x, ˆyx)⟩, where ˆyx is given in Assumption 3(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and ⟨λ∗ y, d(x, y∗)⟩ = 0, one has ⟨λ∗ y, −d(x, ˆyx)⟩ ≤ F(x, y∗) − F(x, ˆyx), 8 which together with (19), λ∗ y ≥ 0 and Assumption 1 implies that δd∥λ∗ y∥1 ≤ ⟨λ∗ y, −d(x, ˆyx)⟩ ≤ F(x, y∗) − F(x, ˆyx) ≤ LF∥y∗ − ˆyx∥ ≤ LFDy, (39) where the first inequality is due to Assumption 3(ii), and the third inequality follows from (19) and LF -Lipschitz continuity of F (see Assumption 1(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using (21) and (39), we have ∥λ∗ y∥ ≤ ∥λ∗ y∥1 ≤ δ−1 d LF Dy and hence λ∗ y ∈ B+ r due to (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (ii) Recall from Assumption 1 that F(x, ·) and di(x, ·), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' , l, are convex for any given x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using this, (17), (21) and the first statement of this lemma, we observe that f ∗(x) = max y min λ∈B+ r F(x, y) − ⟨λ, d(x, y)⟩ ∀x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (40) Notice from Assumption 1 that F and d are Lipschitz continuous on their domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Then it is not hard to observe that min{F(x, y)+⟨λ, d(x, y)⟩|λ ∈ B+ r } is a Lipschitz continuous function of (x, y) on its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and (40), one can easily verify that f ∗ is Lipschitz continuous on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, the finiteness of f ∗ low follows from (18), the continuity of ˜f ∗, and the compactness of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (iii) One can observe from (17) that for all x ∈ X, f ∗(x) = max y min λy F(x, y)−⟨λy, d(x, y)⟩+IR ˜ m + (λy) ≤ min λy max y F(x, y)−⟨λy, d(x, y)⟩+IR ˜ m + (λy), where the inequality follows from the weak duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, it follows from Assumption 1 that the domain of F(x, ·) is compact for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this, (40) and the strong duality, one has f ∗(x) = min λ∈B+ r max y F(x, y) − ⟨λ, d(x, y)⟩ ∀x ∈ X, which together with the above inequality implies that (38) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let {λk y}k∈K be generated by Algorithm 1, f ∗ low, Dy, and Fhi be defined in (18), (19) and (20), and ǫ0, τ, and ρk be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Then we have ρ−1 k ∥λk y∥2 ≤ ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ ∀0 ≤ k ∈ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (41) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One can observe from (18), (20) and Algorithm 1 that Fhi ≥ f ∗ low and ρ0 ≥ 1 > τ > 0, which imply that (41) holds for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It remains to show that (41) holds for all 1 ≤ k ∈ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Since (xt+1, yt+1) is an ǫt-stationary point of (9) for all 0 ≤ t ∈ K − 1, it follows from Definition 1 that there exists some u ∈ ∂yL(xt+1, yt+1, λt x, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt, ρt) with ∥u∥ ≤ ǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice from (3) and (36) that ∂yL(xt+1, yt+1, λt x, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt, ρt) = ∂yLy(xt+1, yt+1, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hence, u ∈ ∂yLy(xt+1, yt+1, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Also, observe from (1), (36) and Assumption 1 that Ly(xt+1, ·, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) is concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using this, (19), u ∈ ∂yLy(xt+1, yt+1, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) and ∥u∥ ≤ ǫt, we obtain Ly(xt+1, y, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) ≤ Ly(xt+1, yt+1, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) + ⟨u, y − yt+1⟩ ≤ Ly(xt+1, yt+1, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) + Dyǫt ∀y ∈ Y, which implies that max y Ly(xt+1, y, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) ≤ Ly(xt+1, yt+1, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) + Dyǫt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (42) By this, (36) and (37), one has f ∗(xt+1) (37) ≤ max y Ly(xt+1, y, λt y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρt) (36)(42) ≤ F(xt+1, yt+1) − 1 2ρt � ∥[λt y + ρtd(xt+1, yt+1)]+∥2 − ∥λt y∥2� + Dyǫt = F(xt+1, yt+1) − 1 2ρt � ∥λt+1 y ∥2 − ∥λt y∥2� + Dyǫt, 9 where the equality follows from the relation λt+1 y = [λt y + ρtd(xt+1, yt+1)]+ (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using the above inequality, (18), (20) and ǫt ≤ ǫ0 (see Algorithm 1), we have ∥λt+1 y ∥2 − ∥λt y∥2 ≤ 2ρk(F(xt+1, yt+1) − f ∗(xt+1) + Dyǫt) ≤ 2ρt(Fhi − f ∗ low + Dyǫ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Summing up this inequality for t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' , k − 1 with 1 ≤ k ∈ K − 1 yields ∥λk y∥2 ≤ ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) k−1 � t=0 ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (43) Recall from Algorithm 1 that ρt = ǫ−1 t = (ǫ0τ t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Then we have �k−1 t=0 ρt ≤ ρk−1/(1 − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using this, (43) and ρk > ρk−1 ≥ 1 (see Algorithm 1), we obtain that for all 1 ≤ k ∈ K − 1, ρ−1 k ∥λk y∥2 ≤ ρ−1 k � ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0)ρk−1 1 − τ � ≤ ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hence, the conclusion holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let f ∗ low, Dy and Fhi be defined in (18), (19) and (20), LF and δd be given in Assumptions 1 and 3, and ǫ0, τ, ǫk and ρk be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that (xk+1, yk+1, λk+1 y ) is generated by Algorithm 1 for some 0 ≤ k ∈ K−1 with ρk ≥ 4∥λ0 y∥2 δ2 d + 8(Fhi − f ∗ low + Dyǫ0) δ2 d(1 − τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (44) Then we have ∥[d(xk+1, yk+1)]+∥ ≤ ρ−1 k ∥λk+1 y ∥ ≤ 2ρ−1 k δ−1 d (ǫ0 + LF )Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (45) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that (xk+1, yk+1, λk+1 y ) is generated by Algorithm 1 for some 0 ≤ k ∈ K − 1 with ρk satisfying (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Since (xk+1, yk+1) is an ǫk-stationary point of (9), it follows from (3) and Definition 1 that dist � 0, ∂yF(xk+1, yk+1) − ∇yd(xk+1, yk+1)[λk y + ρkd(xk+1, yk+1)]+ � ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Besides, notice from Algorithm 1 that λk+1 y = [λk y +ρkd(xk+1, yk+1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hence, there exists some u ∈ ∂yF(xk+1, yk+1) such that ∥u − ∇yd(xk+1, yk+1)λk+1 y ∥ ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (46) By Assumption 3(ii), there exists some ˆyk+1 ∈ Y such that −di(xk+1, ˆyk+1) ≥ δd for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice that ⟨λk+1 y , λk y + ρkd(xk+1, yk+1)⟩ = ∥[λk y + ρkd(xk+1, yk+1)]+∥2 ≥ 0, which implies that − ⟨λk+1 y , ρ−1 k λk y⟩ ≤ ⟨λk+1 y , d(xk+1, yk+1)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (47) Using these and (46),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' we have F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1) − F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1) + δd∥λk+1 y ∥1 − ρ−1 k ⟨λk+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' λk y⟩ ≤ F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1) − F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1) − ⟨λk+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρ−1 k λk y + d(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1)⟩ (47) ≤ F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1) − F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1) + ⟨λk+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' d(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1) − d(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1))⟩ ≤ ⟨u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1 − yk+1⟩ + ⟨∇yd(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1)λk+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1 − ˆyk+1⟩ = ⟨u − ∇yd(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1)λk+1 y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1 − ˆyk+1⟩ ≤ Dyǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (48) 10 where the first inequality is due to λk+1 y ≥ 0 and −di(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ˆyk+1) ≥ δd for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' the third inequality follows from u ∈ ∂yF(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' yk+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' λk+1 y ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' the concavity of F(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ·) and the convexity of di(xk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and the last inequality is due to (19) and (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In view of (19), (48) and the Lipschitz continuity of F (see Assumption 1), one has Dyǫk + LFDy (19) ≥ Dyǫk + LF∥ˆyk+1 − yk+1∥ ≥ Dyǫk − F(xk+1, ˆyk+1) + F(xk+1, yk+1) (48) ≥ δd∥λk+1 y ∥1 − ρ−1 k ⟨λk+1 y , λk y⟩ ≥ (δd − ρ−1 k ∥λk y∥)∥λk+1 y ∥, (49) where the second inequality follows from LF-Lipschitz continuity of F, and the last inequality is due to ∥λk+1 y ∥1 ≥ ∥λk+1 y ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, it follows from (41) and (44) that δd − ρ−1 k ∥λk y∥ (41) ≥ δd − � ρ−1 k � ∥λ0y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ � (44) ≥ 1 2δd, which together with (49) yields 1 2δd∥λk+1 y ∥ ≤ (δd − ρ−1 k ∥λk y∥)∥λk+1 y ∥ (49) ≤ Dyǫk + LF Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The conclusion then follows from this, ǫk ≤ ǫ0, and the relations ∥[d(xk+1, yk+1)]+∥ ≤ ρ−1 k ∥[λk y + ρkd(xk+1, yk+1)]+∥ = ρ−1 k ∥λk+1 y ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let f ∗ low, Dy and Flow be defined in (18), (19) and (20), LF and δd be given in Assumptions 1 and 3, ǫ0, τ, ǫk, ρk and λ0 y be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that (xk+1, yk+1, λk+1 x , λk+1 y ) is generated by Algorithm 1 for some 0 ≤ k ∈ K − 1 with ρk ≥ 4∥λ0 y∥2 δ2 dτ + 8(Fhi − f ∗ low + Dyǫ0) δ2 dτ(1 − τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (50) Let ˜λk+1 x = [λk x + ρkc(xk+1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (51) Then we have dist(0, ∂xF(xk+1, yk+1) + ∇c(xk+1)˜λk+1 x − ∇xd(xk+1, yk+1)λk+1 y ) ≤ ǫk, (52) dist � 0, ∂yF(xk+1, yk+1) − ∇yd(xk+1, yk+1)λk+1 y � ≤ ǫk, (53) ∥[d(xk+1, yk+1)]+∥ ≤ 2ρ−1 k δ−1 d (ǫ0 + LF )Dy, (54) |⟨λk+1 y , d(xk+1, yk+1)⟩| ≤ 2ρ−1 k δ−1 d (ǫ0 + LF )Dy max{∥λ0 y∥, 2δ−1 d (ǫ0 + LF)Dy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (55) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that (xk+1, yk+1, λk+1 x , λk+1 y ) is generated by Algorithm 1 for some 0 ≤ k ∈ K−1 with ρk satisfying (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Since (xk+1, yk+1) is an ǫk-stationary point of (9), it then follows from Definition 1 that dist � 0, ∂xL(xk+1, yk+1, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) � ≤ ǫk, dist � 0, ∂yL(xk+1, yk+1, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) � ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (56) Observe from Algorithm 1 that λk+1 y = [λk y + ρkd(xk+1, yk+1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In view of this, (3) and (51), one has ∂xL(xk+1, yk+1, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) = ∂xF(xk+1, yk+1) + ∇c(xk+1)[λk x + ρkc(xk+1)]+ − ∇xd(xk+1, yk+1)[λk y + ρkd(xk+1, yk+1)]+ = ∂xF(xk+1, yk+1) + ∇c(xk+1)˜λk+1 x − ∇xd(xk+1, yk+1)λk+1 y , ∂yL(xk+1, yk+1, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) = ∂yF(xk+1, yk+1) − ∇yd(xk+1, yk+1)λk+1 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 11 These relations together with (56) imply that (52) and (53) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice from Algorithm 1 that 0 < τ < 1, which together with (50) implies that (44) holds for ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows that (45) holds, which immediately yields (54) and ∥λk+1 y ∥ ≤ 2δ−1 d (ǫ0 + LF )Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (57) Claim that ∥λk y∥ ≤ max{∥λ0 y∥, 2δ−1 d (ǫ0 + LF)Dy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (58) Indeed, (58) clearly holds if k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We now assume that k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice from Algorithm 1 that ρk−1 = τρk, which together with (50) implies that (44) holds with k replaced by k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and Lemma 3 with k replaced by k − 1, one can conclude that ∥λk y∥ ≤ 2δ−1 d (ǫ0 + LF)Dy and hence (58) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We next show that (55) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Indeed, by λk+1 y ≥ 0, (47), (54), (57) and (58), one has ⟨λk+1 y , d(xk+1, yk+1)⟩ ≤ ⟨λk+1 y , [d(xk+1, yk+1)]+⟩ ≤ ∥λk+1 y ∥∥[d(xk+1, yk+1)]+∥ (54)(57) ≤ 4ρ−1 k δ−2 d (ǫ0 + LF)2D2 y, ⟨λk+1 y , d(xk+1, yk+1)⟩ (47) ≥ ⟨λk+1 y , −ρ−1 k λk y⟩ ≥ −ρ−1 k ∥λk+1 y ∥∥λk y∥ ≥ −2ρ−1 k δ−1 d (ǫ0 + LF)Dy max{∥λ0 y∥, 2δ−1 d (ǫ0 + LF)Dy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' These relations imply that (55) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1, 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let {(λk x, λk y)}k∈K be generated by Algo- rithm 1, L, f ∗ low, Dy and Fhi be defined in (3), (18), (19) and (20), LF be given in Assumption 1, and ǫ0, τ, ρk, Λ and xk init be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Then for all 0 ≤ k ∈ K − 1, we have max y L(xk init, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) ≤ LFDy + Fhi + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (59) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In view of (6), (8), (20) and ∥λk x∥ ≤ Λ (see Algorithm 1), one has Lx(xk init, yk, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (8) ≤ Lx(xnf, yk, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (6) = F(xnf, yk) + 1 2ρk � ∥[λk x + ρkc(xnf)]+∥2 − ∥λk x∥2� ≤ F(xnf, yk) + 1 2ρk � (∥λk x∥ + ρk∥[c(xnf )]+∥)2 − ∥λk x∥2� = F(xnf, yk) + ∥λk x∥∥[c(xnf )]+∥ + 1 2ρk∥[c(xnf)]+∥2 (20) ≤ Fhi + Λ∥[c(xnf)]+∥ + 1 2ρk∥[c(xnf)]+∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (60) In addition, one can observe from Algorithm 1 that ǫk > τε for all 0 ≤ k ∈ K − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and the choice of ρk in Algorithm 1, we obtain that ρk = ǫ−1 k < τ −1ε−1 for all 0 ≤ k ∈ K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows from this, (3), (6), (19), (41), (60), ∥[c(xnf)]+∥ ≤ √ε ≤ 1, and the Lipschitz continuity 12 of F that max y L(xk init, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (3)(6) = max y � Lx(xk init, y, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) − 1 2ρk � ∥[λk y + ρkd(xk init, y)]+∥2 − ∥λk y∥2�� ≤ max y � Lx(xk init, y, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) + 1 2ρk ∥λk y∥2 � (6) = max y � F(xk init, y) − F(xk init, yk) + Lx(xk init, yk, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) + 1 2ρk ∥λk y∥2 � ≤ max y∈Y LF∥y − yk∥ + Lx(xk init, yk, λk x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) + 1 2ρk ∥λk y∥2 ≤ LF Dy + Fhi + Λ∥[c(xnf)]+∥ + 1 2ρk∥[c(xnf)]+∥2 + 1 2∥λ0 y∥2 + Fhi − f ∗ low + Dyǫ0 1 − τ ≤ LF Dy + Fhi + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ , where the second inequality follows from LF-Lipschitz continuity of F (see Assumption 1(i)), the third inequality follows from (19), (41) and (60), and the last inequality follows from ρk < τ −1ε−1 and ∥[c(xnf )]+∥ ≤ √ε ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1, 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let Lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' f ∗ low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Fhi and Flow be defined in (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (19) and (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' LF be given in Assumption 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Λ and λ0 y be given in Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and αk = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � 4ǫk/(DyLk) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (61) δk = (2 + α−1 k )LkD2 x + max {ǫk/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' αkLk/4} D2 y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (62) Mk = 16 max {1/(2Lk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' min {Dy/ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4/(αkLk)}} ρk [(3Lk + ǫk/(2Dy))2/ min{Lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫk/(2Dy)} + 3Lk + ǫk/(2Dy)]−2 ǫ2 k × � δk + 2α−1 k � Fhi − Flow + Λ2 2ρk + 3 2∥λ0 y∥2 + 3(Fhi − f ∗ low + Dyǫ0) 1 − τ + ρkd2 hi + ǫkDy 4 + LkD2 x �� (63) Tk = � 16 � LF Dy + Fhi − f ∗ low + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ + Λ2 2ρk + ǫkDy 4 � Lkǫ−2 k + 8(1 + 4D2 yL2 kǫ−2 k )ρ−1 k − 1 � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (64) Nk = �� 96 √ 2 � 1 + (24Lk + 4ǫk/Dy) L−1 k �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � DyLkǫ−1 k � × ((Tk + 1)(log Mk)+ + Tk + 1 + 2Tk log(Tk + 1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (65) Then for all 0 ≤ k ∈ K − 1, Algorithm 1 finds an ǫk-stationary point (xk+1, yk+1) of problem (9) that satisfies max y L(xk+1, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) ≤ LFDy + Fhi + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ + ǫkDy 4 + 1 2ρk � L−1 k ǫ2 k + 4D2 yLk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (66) Moreover, the total number of evaluations of ∇f, ∇c, ∇d and proximal operator of p and q performed in iteration k of Algorithm 1 is no more than Nk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Observe from (1) and (3) that problem (9) can be viewed as min x max y {h(x, y) + p(x) − q(y)}, where h(x, y) = f(x, y) + 1 2ρk � ∥[λk x + ρkc(x)]+∥2 − ∥λk x∥2� − 1 2ρk � ∥[λk y + ρkd(x, y)]+∥2 − ∥λk y∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice that ∇xh(x, y) = ∇xf(x, y) + ∇c(x)[λk x + ρkc(x)]+ + ∇xd(x, y)[λk y + ρkd(x, y)]+, ∇yh(x, y) = ∇yf(x, y) + ∇yd(x, y)[λk y + ρkd(x, y)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It follows from Assumption 1(iii) that ∥∇c(x)∥ ≤ Lc, ∥∇d(x, y)∥ ≤ Ld ∀(x, y) ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In view of the above relations, (7) and Assumption 1, one can observe that ∇c(x)[λk x +ρkc(x)]+ is (ρkL2 c + ρkchiL∇c + ∥λk x∥L∇c)-Lipschitz continuous on X, and ∇d(x, y)[λk y + ρkd(x, y)]+ is (ρkL2 d + ρkdhiL∇d + ∥λk y∥L∇d)-Lipschitz continuous on X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using these and the fact that ∇f(x, y) is L∇f-Lipschitz continuous on X × Y, we can see that h(x, y) is Lk-smooth on X × Y for all 0 ≤ k ∈ K − 1, where Lk is given in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Consequently, it follows from Theorem 2 that Algorithm 3 can be suitably applied to problem (9) for finding an ǫk-stationary point (xk+1, yk+1) of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, by (3), (18), (36), (37) and ∥λk x∥ ≤ Λ (see Algorithm 1), one has min x max y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (3)(36) = min x max y � Ly(x, y, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) + 1 2ρk � ∥[λk x + ρkc(x)]+∥2 − ∥λk x∥2�� (37) ≥ min x � f ∗(x) + 1 2ρk � ∥[λk x + ρkc(x)]+∥2 − ∥λk x∥2�� (18) ≥ f ∗ low − 1 2ρk ∥λk x∥2 ≥ f ∗ low − Λ2 2ρk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (67) Let (x∗, y∗) be an optimal solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows that c(x∗) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using this, (3), (20) and (41), we obtain that min x max y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) ≤ max y L(x∗, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (3) = max y � F(x∗, y) + 1 2ρk � ∥[λk x + ρkc(x∗)]+∥2 − ∥λk x∥2� − 1 2ρk � ∥[λk y + ρkd(x∗, y)]+∥2 − ∥λk y∥2�� ≤ max y � F(x∗, y) − 1 2ρk � ∥[λk y + ρkd(x∗, y)]+∥2 − ∥λk y∥2�� (20) ≤ Fhi + 1 2ρk ∥λk y∥2 (41) ≤ Fhi + 1 2∥λ0 y∥2 + Fhi − f ∗ low + Dyǫ0 1 − τ , (68) where the second inequality is due to c(x∗) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Moreover, it follows from this, (3), (7), (20), (41), λk y ∈ R ˜m + and ∥λk x∥ ≤ Λ that min (x,y)∈X×Y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) (3) ≥ min (x,y)∈X×Y � F(x, y) − 1 2ρk ∥λk x∥2 − 1 2ρk ∥[λk y + ρkd(x, y)]+∥2 � ≥ min (x,y)∈X×Y � F(x, y) − 1 2ρk ∥λk x∥2 − 1 2ρk � ∥λk y∥ + ρk∥[d(x, y)]+∥ �2� ≥ min (x,y)∈X×Y � F(x, y) − 1 2ρk ∥λk x∥2 − ρ−1 k ∥λk y∥2 − ρk∥[d(x, y)]+∥2 � ≥ Flow − Λ2 2ρk − ∥λ0 y∥2 − 2(Fhi − f ∗ low + Dyǫ0) 1 − τ − ρkd2 hi, (69) 14 where the second inequality is due to λk y ∈ R ˜m + and the last inequality is due to (7), (20), (41) and ∥λk x∥ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' To complete the rest of the proof, let H(x, y) = L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk), H∗ = min x max y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk), (70) Hlow = min (x,y)∈X×Y L(x, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (71) In view of these, (59), (67), (68), (69), we obtain that max y H(xk init, y) (59) ≤ LFDy + Fhi + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ , f ∗ low − Λ2 2ρk (67) ≤ H∗ (68) ≤ Fhi + 1 2∥λ0 y∥2 + Fhi − f ∗ low + Dyǫ0 1 − τ , Hlow (69) ≥ Flow − Λ2 2ρk − ∥λ0 y∥2 − 2(Fhi − f ∗ low + Dyǫ0) 1 − τ − ρkd2 hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using these and Theorem 2 (see Appendix A) with x0 = xk init, Dp = Dx, Dq = Dy, ǫ = ǫk, ǫ0 = ǫk/(2√ρk), L∇h = Lk, α = αk, δ = δk, and H, H∗, Hlow given in (70) and (71), we can conclude that Algorithm 3 performs at most Nk evaluations of ∇f, ∇c, ∇d and proximal operator of p and q for finding an ǫk-stationary point of problem (9) satisfying (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumptions 1, 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let f ∗ low, Dy, Fhi and �L be defined in (18), (19), (20) and (23), LF , Lc, δc, θf and θa be given in Assumptions 1 and 3, and ǫ0, τ, ρk, Λ and λ0 y be given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that (xk+1, λk+1 x ) is generated by Algorithm 1 for some 0 ≤ k ∈ K − 1 with ρk ≥ max � θ−1 a Λ, θ−2 f � 2LF Dy + 2Fhi − 2f ∗ low + 2Λ + τ −1 + ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ + ǫ0Dy 2 + L−2 c + 4D2 y �L + Λ2� , 4∥λ0 y∥2 δ2 dτ + 8(Fhi − f ∗ low + Dyǫ0) δ2 dτ(1 − τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (72) Let ˜λk+1 x = [λk x + ρkc(xk+1)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (73) Then we have ∥[c(xk+1)]+∥ ≤ ρ−1 k δ−1 c � LF + 2Ldδ−1 d (ǫ0 + LF )Dy + ǫ0 � , (74) |⟨˜λk+1 x , c(xk+1)⟩| ≤ ρ−1 k δ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF)Dy + ǫ0) max{δ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF)Dy + ǫ0), Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (75) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' One can observe from (3), (18), (36) and (37) that max y L(xk+1, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) = max y Ly(xk+1, y, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) + 1 2ρk � ∥[λk x + ρkc(xk+1)]+∥2 − ∥λk x∥2� (37) ≥ f ∗(xk+1) + 1 2ρk � ∥[λk x + ρkc(xk+1)]+∥2 − ∥λk x∥2� (18) ≥ f ∗ low + 1 2ρk � ∥[λk x + ρkc(xk+1)]+∥2 − ∥λk x∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 15 By this inequality, (66) and ∥λk x∥ ≤ Λ, one has ∥[λk x + ρkc(xk+1)]+∥2 ≤ 2ρk max y L(xk+1, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) − 2ρkf ∗ low + ∥λk x∥2 ≤ 2ρk max y L(xk+1, y, λk x, λk y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk) − 2ρkf ∗ low + Λ2 (66) ≤ 2ρkLF Dy + 2ρkFhi + 2ρkΛ + ρk(τ −1 + ∥λ0 y∥2) + 2ρk(Fhi − f ∗ low + Dyǫ0) 1 − τ + ρkǫkDy 2 + L−1 k ǫ2 k + 4D2 yLk − 2ρkf ∗ low + Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' This together with ρ2 k∥[c(xk+1)]+∥2 ≤ ∥[λk x + ρkc(xk+1)]+∥2 implies that ∥[c(xk+1)]+∥2 ≤ ρ−1 k � 2LF Dy + 2Fhi − 2f ∗ low + 2Λ + τ −1 + ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ + ǫkDy 2 � + ρ−2 k � L−1 k ǫ2 k + 4D2 yLk + Λ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (76) In addition, we observe from (10), (23), (41), ρk ≥ 1 and ∥λk x∥ ≤ Λ that for all 0 ≤ k ≤ K, ρkL2 c ≤ Lk = L∇f + ρkL2 c + ρkchiL∇c + ∥λk x∥L∇c + ρkL2 d + ρkdhiL∇d + ∥λk y∥L∇d ≤ L∇f + ρkL2 c + ρkchiL∇c + ΛL∇c + ρkL2 d + ρkdhiL∇d + L∇d � ρk � ∥λ0y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ � ≤ ρk �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (77) Using this relation, (72), (76), ρk ≥ 1 and ǫk ≤ ǫ0, we have ∥[c(xk+1)]+∥2 ≤ ρ−1 k � 2LF Dy + 2Fhi − f ∗ low + 2Λ + τ −1 + ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ + ǫkDy 2 � + ρ−2 k � (ρkL2 c)−1ǫ2 k + 4ρkD2 y�L + Λ2� ≤ ρ−1 k � 2LF Dy + 2Fhi − f ∗ low + 2Λ + τ −1 + ∥λ0 y∥2 + 2(Fhi − f ∗ low + Dyǫ0) 1 − τ + ǫ0Dy 2 � + ρ−1 k � L−2 c + 4D2 y �L + Λ2� (72) ≤ θ2 f, which together with (11) implies that xk+1 ∈ F(θf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It follows from xk+1 ∈ F(θf) and Assumption 3(i) that there exists some vx such that ∥vx∥ = 1 and vT x ∇ci(xk+1) ≤ −δc for all i ∈ A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa), where A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa) is defined in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let ¯ A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa) = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' , ˜n}\\A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice from (11) that ci(xk+1) < −θa for all i ∈ ¯ A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In addition, observe from (72) that ρk ≥ θ−1 a Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using these and ∥λk x∥ ≤ Λ, we obtain that (λk x + ρkc(xk+1))i ≤ Λ − ρkθa ≤ 0 for all i ∈ ¯ A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and the fact that vT x ∇ci(xk+1) ≤ −δc for all i ∈ A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' θa), one has vT x ∇c(xk+1)˜λk+1 x (73) = vT x ∇c(xk+1)[λk x + ρkc(xk+1)]+ = ˜n � i=1 vT x ∇ci(xk+1)([λk x + ρkc(xk+1)]+)i = � i∈A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='θa) vT x ∇ci(xk+1)([λk x + ρkc(xk+1)]+)i + � i∈ ¯ A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='θa) vT x ∇ci(xk+1)([λk x + ρkc(xk+1)]+)i ≤ −δc � i∈A(xk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='θa) ([λk x + ρkc(xk+1)]+)i = −δc ˜n � i=1 ([λk x + ρkc(xk+1)]+)i (73) = −δc∥˜λk+1 x ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (78) Since (xk+1, yk+1) is an ǫk-stationary point of (9), it follows from (3) and (56) that there exists some s ∈ ∂xF(xk+1, yk+1) such that ∥s + ∇c(xk+1)[λk x + ρkc(xk+1)]+ − ∇xd(xk+1, yk+1)[λk y + ρkd(xk+1, yk+1)]+∥ ≤ ǫk, 16 which along with (73) and λk+1 y = [λk y + ρxd(xk+1, yk+1)]+ implies that ∥s + ∇c(xk+1)˜λk+1 x − ∇xd(xk+1, yk+1)λk+1 y ∥ ≤ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this, (78) and ∥vx∥ = 1, one has ǫk ≥ ∥s + ∇c(xk+1)˜λk+1 x − ∇xd(xk+1, yk+1)λk+1 y ∥ · ∥vx∥ ≥ ⟨s + ∇c(xk+1)˜λk+1 x − ∇xd(xk+1, yk+1)λk+1 y , −vx⟩ = −⟨s − ∇xd(xk+1, yk+1)λk+1 y , vx⟩ − vT x ∇c(xk+1)˜λk+1 x (78) ≥ − � ∥s∥ + ∥∇xd(xk+1, yk+1)∥∥λk+1 y ∥ � ∥vx∥ + δc∥˜λk+1 x ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ≥ −LF − Ld∥λk+1 y ∥ + δc∥˜λk+1 x ∥1, where the last inequality is due to ∥vx∥ = 1 and Assumptions 1(i) and 1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Notice from (72) that (44) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows from (45) that ∥λk+1 y ∥ ≤ 2δ−1 d (ǫ0 + LF )Dy, which together with the above inequality and ǫk ≤ ǫ0 yields ∥˜λk+1 x ∥ ≤ ∥˜λk+1 x ∥1 ≤ δ−1 c (LF + Ld∥λk+1 y ∥ + ǫk) ≤ δ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF )Dy + ǫ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (79) By this and (73), one can observe that ∥[c(xk+1)]+∥ ≤ ρ−1 k ∥[λk x + ρkc(xk+1)]+∥ = ρ−1 k ∥˜λk+1 x ∥ ≤ ρ−1 k δ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF )Dy + ǫ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hence, (74) holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We next show that (75) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Indeed, by ˜λk+1 x ≥ 0, (74) and (79), one has ⟨˜λk+1 x , c(xk+1)⟩ ≤ ⟨˜λk+1 x , [c(xk+1)]+⟩ ≤ ∥˜λk+1 x ∥∥[c(xk+1)]+∥ (74)(79) ≤ ρ−1 k δ−2 c (LF + 2Ldδ−1 d (ǫ0 + LF )Dy + ǫ0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (80) Using a similar argument as for the proof of (47), we have −⟨˜λk+1 x , ρ−1 k λk x⟩ ≤ ⟨˜λk+1 x , c(xk+1)⟩, which along with ∥λk x∥ ≤ Λ and (79) yields ⟨˜λk+1 x , c(xk+1)⟩ ≥ −ρ−1 k ∥˜λk+1 x ∥∥λk x∥ ≥ −ρ−1 k δ−1 c (LF + 2Ldδ−1 d (ǫ0 + LF )Dy + ǫ0)Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' The relation (75) then follows from this and (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) Observe from the definition of K in (22) and ǫk = ǫ0τ k that K is the smallest nonnegative integer such that ǫK ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hence, Algorithm 1 terminates and outputs (xK+1, yK+1) after K + 1 outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It follows from these and ρk = ǫ−1 k that ǫK ≤ ε and ρK ≥ ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and (28), one can see that (50) and (72) holds for k = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows from Lemmas 4 and 7 that (29)-(34) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (ii) Let K and N be given in (22) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Recall from Lemma 6 that the number of evaluations of ∇f, ∇c, ∇d, proximal operator of p and q performed by Algorithm 3 at iteration k of Algorithm 1 is at most Nk, where Nk is given in (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By this and statement (i) of this theorem, one can observe that the total number of evaluations of ∇f, ∇c, ∇d, proximal operator of p and q performed in Algorithm 1 is no more than �K k=0 Nk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' As a result, to prove statement (ii) of this theorem, it suffices to show that �K k=0 Nk ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Recall from (77) 17 and Algorithm 1 that ρkL2 c ≤ Lk ≤ ρk �L and ρk ≥ 1 ≥ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Using these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (61),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (62),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (63) and (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' we obtain that 1 ≥ αk ≥ min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � 4ǫk/(ρkDy �L) � ≥ ǫ1/2 k ρ−1/2 k ˆα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (81) δk ≤ (2 + ǫ−1/2 k ρ1/2 k ˆα−1)ρk �LD2 x + max{1/Dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρk �L/4}D2 y ≤ ǫ−1/2 k ρ3/2 k ˆδ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (82) Mk ≤ 16 max � 1/(2ρkL2 c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4/(ǫ1/2 k ρ−1/2 k ˆαρkL2 c) � ρk � (3ρk �L + 1/(2Dy))2/ min{ρkL2c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫk/(2Dy)} + 3ρk �L + 1/(2Dy) �−2 ǫ2 k × � ǫ−1/2 k ρ3/2 k ˆδ + 2ǫ−1/2 k ρ1/2 k ˆα−1� Fhi − Flow + Λ2 2 + 3 2∥λ0 y∥2 + 3(Fhi − f ∗ low + Dyǫ0) 1 − τ + ρkd2 hi + Dy 4 + ρk �LD2 x �� (83) ≤ 16ǫ−1/2 k ρ−1/2 k max � 1/(2L2 c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4/(ˆαL2 c) � ρk ǫ2 kρ−4 k � (3�L + 1/(2Dy))2/ min{L2c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1/(2Dy)} + 3�L + 1/(2Dy) �−2 ǫ2 k × (ǫ−1/2 k ρ3/2 k ) � ˆδ + 2ˆα−1 × � Fhi − Flow + Λ2 2 + 3 2∥λ0 y∥2 + 3(Fhi − f ∗ low + Dyǫ0) 1 − τ + d2 hi + Dy 4 + �LD2 x �� ≤ ǫ−5 k ρ6 k � M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Tk ≤ � 16 � LF Dy + Fhi − f ∗ low + Λ + 1 2(τ −1 + ∥λ0 y∥2) + Fhi − f ∗ low + Dyǫ0 1 − τ + Λ2 2 + Dy 4 � ǫ−2 k ρk �L + 8(1 + 4D2 yρ2 k �L2ǫ−2 k )ρ−1 k − 1 � + ≤ ǫ−2 k ρk �T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' where (83) follows from (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (81),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (82),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ρkL2 c ≤ Lk ≤ ρk �L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and ρk ≥ 1 ≥ ǫk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' By the above inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (65),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (77),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' �T ≥ 1 and ρk ≥ 1 ≥ ǫk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' one has K � k=0 Nk ≤ K � k=0 �� 96 √ 2 � 1 + � 24ρk �L + 4/Dy � /(ρkL2 c) �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dyρk �Lǫ−1 k � × � (ǫ−2 k ρk �T + 1)(log(ǫ−5 k ρ6 k � M))+ + ǫ−2 k ρk �T + 1 + 2ǫ−2 k ρk �T log(ǫ−2 k ρk �T + 1) � ≤ K � k=0 �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy�L � ǫ−1/2 k ρ1/2 k × ǫ−2 k ρk � ( �T + 1)(log(ǫ−5 k ρ6 k � M))+ + �T + 1 + 2 �T log(ǫ−2 k ρk �T + 1) � ≤ K � k=0 �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy�L � × ǫ−5/2 k ρ3/2 k �T � 2(log(ǫ−5 k ρ6 k � M))+ + 2 + 2 log(2ǫ−2 k ρk �T) � ≤ K � k=0 �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy�L � �T × ǫ−5/2 k ρ3/2 k � 14 log ρk − 14 log ǫk + 2(log � M)+ + 2 + 2 log(2 �T) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (84) By the definition of K in (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' one has τ K ≥ τε/ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Also, notice from Algorithm 1 that 18 ρk = ǫ−1 k = (ǫ0τ k)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' It then follows from these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (35) and (84) that K � k=0 Nk ≤ K � k=0 �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy �L � �T × ǫ−4 k � 28 log(1/ǫk) + 2(log � M)+ + 2 + 2 log(2 �T ) � = �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy �L � �T × K � k=0 ǫ−4 0 τ −4k � 28k log(1/τ) + 28 log(1/ǫ0) + 2(log � M)+ + 2 + 2 log(2 �T) � ≤ �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy �L � �T × K � k=0 ǫ−4 0 τ −4k � 28K log(1/τ) + 28 log(1/ǫ0) + 2(log � M)+ + 2 + 2 log(2 �T ) � ≤ �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy �L � �Tǫ−4 0 × τ −4K(1 − τ 4)−1 � 28K log(1/τ) + 28 log(1/ǫ0) + 2(log � M)+ + 2 + 2 log(2 �T ) � ≤ �� 96 √ 2 � 1 + � 24�L + 4/Dy � /L2 c �� + 2 � max � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � Dy �L � �Tǫ−4 0 (1 − τ 4)−1 × (τε/ǫ0)−4 � 28K log(1/τ) + 28 log(1/ǫ0) + 2(log � M)+ + 2 + 2 log(2 �T ) � (35) = N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' where the second last inequality is due to �K k=0 τ −4k ≤ τ −4K/(1 − τ 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and the last inequality is due to τ K ≥ τε/ǫ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hence, statement (ii) of this theorem holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Antonakopoulos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Belmega, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mertikopoulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Adaptive extra-gradient meth- ods for min-max optimization and games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In The International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Birgin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mart´ınez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Practical Augmented Lagrangian Methods for Constrained Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' SIAM, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Birgin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mart´ınez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Complexity and performance of an augmented Lagrangian algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Methods and Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', 35(5):885–920, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Cesa-Bianchi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lugosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Prediction, learning, and games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Cambridge University Press, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' An augmented Lagrangian method for non-Lipschitz nonconvex programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', 55(1):168–193, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [6] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Proximal gradient descent-ascent: variable con- vergence under K�L geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='04653, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Clarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Optimization and nonsmooth analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' SIAM, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Shaw, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xiao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chen, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' SBEED: Convergent reinforcement learning with nonlinear function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1125–1134, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 19 [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Optimality conditions and numerical algorithms for a class of linearly constrained minimax optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='09185, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [10] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dai and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Optimality conditions for constrained minimax optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='09730, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Du, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xiao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Stochastic variance reduction methods for policy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 1049–1058, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Duchi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Namkoong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Variance-based regularization with convex objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Journal of Machine Learning Research, 20(1):2450–2504, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Gidel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Berard, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Vignoud, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Vincent, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lacoste-Julien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A variational inequality perspective on generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Goktas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Greenwald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Convex-concave min-max stackelberg games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:2991–3003, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [15] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Pouget-Abadie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mirza, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Warde-Farley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ozair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Courville, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Generative adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 2672–2680, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Shlens, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Explaining and harnessing adversarial exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Learning Representations, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Grapiglia and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Yuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' On the complexity of an augmented Lagrangian method for nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', 41(2):1508–1530, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [18] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Yuan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Yan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Fast objective & duality gap convergence for nonconvex-strongly-concave min-max problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='06889, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Pei, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Accelerated zeroth-order momentum methods from mini to minimax optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='08170, 3, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Kanzow and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Steck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' An example comparing the standard and safeguarded augmented Lagrangian methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', 45(6):598–603, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Kong and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Monteiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' An accelerated inexact proximal point method for solving nonconvex-concave min-max problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' SIAM Journal on Optimization, 31(4):2558–2585, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Jin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' On gradient descent ascent for nonconvex-concave minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 6083–6093, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [23] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Jin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Near-optimal algorithms for minimax optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Conference on Learning Theory, pages 2738–2779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A single-loop gradient descent and perturbed ascent algorithm for nonconvex func- tional constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 14315–14357, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Tsaknakis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hong, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hybrid block successive approximation for one-sided non-convex min-max problems: algorithms and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' IEEE Transactions on Signal Processing, 68:3676–3691, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 20 [26] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' First-order penalty methods for bilevel optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='01716, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [27] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' An augmented Lagrangian approach for sparse principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', 135(1-2):149–193, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ye, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Huang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Stochastic recursive gradient descent ascent for stochastic nonconvex-strongly-concave minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:20566–20577, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Madry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Makelov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Schmidt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Tsipras, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Vladu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Towards deep learning models resistant to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Learning Repre- sentations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [30] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mateos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Bazerque, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Giannakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Distributed sparse linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' IEEE Transactions on Signal Processing, 58:5262–5276, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [31] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Nachum, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chow, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dai, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' DualDICE: Behavior-agnostic estimation of dis- counted stationary distribution corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 2315–2325, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Nocedal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Springer, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Nouiehed, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sanjabi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lee, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Razaviyayn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Solving a class of non- convex min-max games using iterative first order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Qiu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ye, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Single-timescale stochastic nonconvex-concave optimization for smooth nonlinear td learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='10103, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Rakhlin and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sridharan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Optimization, learning, and games with predictable se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pages 3066–3074, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sahin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Eftekhari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Alacaoglu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Latorre, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Cevher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' An inexact augmented Lagrangian framework for nonconvex optimization with nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sanjabi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Razaviyayn, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' On the convergence and robustness of training gans with regularized optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Process- ing Systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Shafieezadeh-Abadeh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Esfahani, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Kuhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Distributionally robust logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, page 1576–1584, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Shamma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Cooperative Control of Distributed Multi-Agent Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wiley-Interscience, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sinha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Namkoong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Duchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Certifying some distributional robustness with principled adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ren, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sadigh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Multi-agent generative adversarial imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in neural information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [42] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Syrgkanis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Agarwal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Luo, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Schapire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Fast convergence of regularized learning in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, page 2989–2997, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 21 [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Taskar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lacoste-Julien, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Structured prediction via the extragradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, page 1345–1352, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [44] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Tsaknakis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Hong, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Minimax problems with coupled linear constraints: computational complexity, duality and solution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='11210, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Fardad, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Adversarial at- tack generation empowered by min-max optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [46] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Ward and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Borwein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Nonsmooth calculus in finite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' SIAM Journal on control and optimization, 25(5):1312–1340, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [47] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xian, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A faster decentralized algorithm for non- convex minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [48] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xie and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Complexity of proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=', 86(3):1–30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [49] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Caramanis, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Robustness and regularization of support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Journal of Machine Learning Research, 10:1485–1510, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [50] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Neufeld, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Larson, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Schuurmans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Maximum margin clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, page 1537–1544, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [51] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Liang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Gradient free minimax optimization: Variance reduction and faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='09361, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [52] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Lan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A unified single-loop alternating gradient projec- tion algorithm for nonconvex-concave and convex-nonconcave minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='02032, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [53] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Primal dual alternating proximal gradient algorithms for nonsmooth nonconvex minimax problems with coupled linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content='04672, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Xiao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Sun, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A single-loop smoothed gradient descent-ascent algorithm for nonconvex-concave min-max problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Advances in Neural Information Pro- cessing Systems, 33:7377–7389, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' A A first-order method for nonconvex-concave minimax prob- lem In this part we present a first-order method proposed in [26, Algorithm 2] for finding an ǫ- stationary point of the nonconvex-concave minimax problem H∗ = min x max y {H(x, y) := h(x, y) + p(x) − q(y)} , (85) which has at least one optimal solution and satisfies the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (i) p : Rn → R ∪ {∞} and q : Rm → R ∪ {∞} are proper convex functions and continuous on dom p and dom q, respectively, and moreover, dom p and dom q are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (ii) The proximal operator associated with p and q can be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 22 (iii) h is L∇h-smooth on dom p × dom q, and moreover, h(x, ·) is concave for any x ∈ dom p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' For ease of presentation, we define Dp = max{∥u − v∥ ��u, v ∈ dom p}, Dq = max{∥u − v∥ ��u, v ∈ dom q}, (86) Hlow = min{H(x, y)|(x, y) ∈ dom p × dom q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (87) Given an iterate (xk, yk), the first-order method [26, Algorithm 2] finds the next iterate (xk+1, yk+1) by applying a modified optimal first-order method [26, Algorithm 1] to the strongly- convex-strongly-concave minimax problem min x max y � hk(x, y) = h(x, y) − ǫ∥y − y0∥2/(4Dq) + L∇h∥x − xk∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (88) For ease reference, we next present the modified optimal first-order method [26, Algorithm 1] in Algorithm 2 below for solving the strongly-convex-strongly-concave minimax problem min x max y �¯h(x, y) + p(x) − q(y) � , (89) where ¯h(x, y) is σx-strongly-convex-σy-strongly-concave and L∇¯h-smooth on dom p × dom q for some σx, σy > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' In Algorithm 2, the functions ˆh, ak x and ak y are defined as follows: ˆh(x, y) = ¯h(x, y) − σx∥x∥2/2 + σy∥y∥2/2, ak x(x, y) = ∇xˆh(x, y) + σx(x − σ−1 x zk g)/2, ak y(x, y) = −∇yˆh(x, y) + σyy + σx(y − yk g)/8, where yk g and zk g are generated at iteration k of Algorithm 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 23 Algorithm 2 A modified optimal first-order method for problem (89) Input: τ > 0, ¯z0 = z0 f ∈ −σxdom p,4 ¯y0 = y0 f ∈ dom q, (z0, y0) = (¯z0, ¯y0), ¯α = min � 1, � 8σy/σx � , ηz = σx/2, ηy = min {1/(2σy), 4/(¯ασx)}, βt = 2/(t + 3), ζ = � 2 √ 5(1 + 8L∇¯h/σx) �−1, γx = γy = 8σ−1 x , and ˆζ = min{σx, σy}/L2 ∇¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' do 2: (zk g , yk g) = ¯α(zk, yk) + (1 − ¯α)(zk f, yk f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 3: (xk,−1, yk,−1) = (−σ−1 x zk g, yk g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4: xk,0 = proxζγxp(xk,−1 − ζγxak x(xk,−1, yk,−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 5: yk,0 = proxζγyq(yk,−1 − ζγyak y(xk,−1, yk,−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 6: bk,0 x = 1 ζγx (xk,−1 − ζγxak x(xk,−1, yk,−1) − xk,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 7: bk,0 y = 1 ζγy (yk,−1 − ζγyak y(xk,−1, yk,−1) − yk,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 8: t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 9: while γx∥ak x(xk,t, yk,t)+bk,t x ∥2+γy∥ak y(xk,t, yk,t)+bk,t y ∥2 > γ−1 x ∥xk,t−xk,−1∥2+γ−1 y ∥yk,t−yk,−1∥2 do 10: xk,t+1/2 = xk,t + βt(xk,0 − xk,t) − ζγx(ak x(xk,t, yk,t) + bk,t x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 11: yk,t+1/2 = yk,t + βt(yk,0 − yk,t) − ζγy(ak y(xk,t, yk,t) + bk,t y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 12: xk,t+1 = proxζγxp(xk,t + βt(xk,0 − xk,t) − ζγxak x(xk,t+1/2, yk,t+1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 13: yk,t+1 = proxζγyq(yk,t + βt(yk,0 − yk,t) − ζγyak y(xk,t+1/2, yk,t+1/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 14: bk,t+1 x = 1 ζγx (xk,t + βt(xk,0 − xk,t) − ζγxak x(xk,t+1/2, yk,t+1/2) − xk,t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 15: bk,t+1 y = 1 ζγy (yk,t + βt(yk,0 − yk,t) − ζγyak y(xk,t+1/2, yk,t+1/2) − yk,t+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 16: t ← t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 17: end while 18: (xk+1 f , yk+1 f ) = (xk,t, yk,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 19: (zk+1 f , wk+1 f ) = (∇xˆh(xk+1 f , yk+1 f ) + bk,t x , −∇yˆh(xk+1 f , yk+1 f ) + bk,t y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 20: zk+1 = zk + ηzσ−1 x (zk+1 f − zk) − ηz(xk+1 f + σ−1 x zk+1 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 21: yk+1 = yk + ηyσy(yk+1 f − yk) − ηy(wk+1 f + σyyk+1 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 22: xk+1 = −σ−1 x zk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 23: ˜xk+1 = proxˆζp(xk+1 − ˆζ∇x¯h(xk+1, yk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 24: ˜yk+1 = proxˆζq(yk+1 + ˆζ∇y¯h(xk+1, yk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 25: Terminate the algorithm and output (˜xk+1, ˜yk+1) if ∥ˆζ−1(xk+1 − ˜xk+1, ˜yk+1 − yk+1) − (∇¯h(xk+1, yk+1) − ∇¯h(˜xk+1, ˜yk+1))∥ ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 26: end for We are now ready to present the first-order method [26, Algorithm 2] for finding an ǫ- stationary point of (85) in Algorithm 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4For convenience, −σxdom p stands for the set {−σxu|u ∈ dom p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 24 Algorithm 3 A first-order method for problem (85) Input: ǫ > 0, ǫ0 ∈ (0, ǫ/2], (ˆx0, ˆy0) ∈ dom p × dom q, (x0, y0) = (ˆx0, ˆy0), and ǫk = ǫ0/(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 1: for k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' do 2: Call Algorithm 2 with ¯h ← hk, τ ← ǫk, σx ← L∇h, σy ← ǫ/(2Dq), L∇¯h ← 3L∇h+ǫ/(2Dq), ¯z0 = z0 f ← −σxxk, ¯y0 = y0 f ← yk, and denote its output by (xk+1, yk+1), where hk is given in (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 3: Terminate the algorithm and output (xǫ, yǫ) = (xk+1, yk+1) if ∥xk+1 − xk∥ ≤ ǫ/(4L∇h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4: end for The following theorem presents the iteration complexity of Algorithm 3, whose proof is given in [26, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Theorem 2 (Complexity of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Suppose that Assumption 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Let H∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' H Dp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and Hlow be defined in (85),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' (86) and (87),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' L∇h be given in Assumption 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫ0 and x0 be given in Algorithm 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' and α = min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � 4ǫ/(DqL∇h) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' δ = (2 + α−1)L∇hD2 p + max {ǫ/Dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' αL∇h/4} D2 q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' K = � 16(max y H(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' y) − H∗ + ǫDq/4)L∇hǫ−2 + 32ǫ2 0(1 + 4D2 qL2 ∇hǫ−2)ǫ−2 − 1 � + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' N = �� 96 √ 2 � 1 + (24L∇h + 4ǫ/Dq) L−1 ∇h �� + 2 � � 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' � DqL∇hǫ−1 � × � (K + 1) � log 4 max � 1 2L∇h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' min � Dq ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 4 αL∇h �� � δ + 2α−1(H∗ − Hlow + ǫDq/4 + L∇hD2 p) � [(3L∇h + ǫ/(2Dq))2/ min{L∇h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' ǫ/(2Dq)} + 3L∇h + ǫ/(2Dq)]−2 ǫ2 0 � + + K + 1 + 2K log(K + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Then Algorithm 3 terminates and outputs an ǫ-stationary point (xǫ, yǫ) of (85) in at most K +1 outer iterations that satisfies max y H(xǫ, y) ≤ max y H(ˆx0, y) + ǫDq/4 + 2ǫ2 0 � L−1 ∇h + 4D2 qL∇hǫ−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' Moreover, the total number of evaluations of ∇h and proximal operator of p and q performed in Algorithm 3 is no more than N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etA0T4oBgHgl3EQfHP_m/content/2301.02060v1.pdf'} diff --git a/f9E5T4oBgHgl3EQfhg82/content/tmp_files/2301.05641v1.pdf.txt b/f9E5T4oBgHgl3EQfhg82/content/tmp_files/2301.05641v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8122049406b9d1dacccf9cfcb915277272e6f609 --- /dev/null +++ b/f9E5T4oBgHgl3EQfhg82/content/tmp_files/2301.05641v1.pdf.txt @@ -0,0 +1,1775 @@ +1 + +Emergence of Urban Heat Traps from the Intersection of Human Mobility and Heat +Hazard Exposure in Cities +Xinke Huang1*, Ali Mostafavi2 +1 Ph.D. Student, Zachry Department of Civil and Environmental Engineering, Texas A&M +University, College Station, United States; e-mail: adahuang@tamu.edu +2 Associate Professor, Urban Resilience.AI Lab Zachry Department of Civil and Environmental +Engineering, Texas A&M University, College Station, United States; e-mail: +amostafavi@civil.tamu.edu + +Abstract +Understanding the relationship between spatial structures of cities and environmental hazard +exposures (such as urban heat) is essential for urban health and sustainability planning. However, +a critical knowledge gap exists in terms of the extent to which socio-spatial networks shaped by +human mobility exacerbate or alleviate urban heat exposures of populations in cities. In this +study, we utilize location-based data to construct human mobility networks in twenty +metropolitan areas in the U.S. The human mobility networks are analyzed in conjunction with +the urban heat characteristics of spatial areas. We identify areas with high and low urban heat +exposure and evaluate visitation patterns of populations residing in high and low urban heat areas +to other spatial areas with similar and dissimilar urban heat exposure. The results reveal the +presence of urban heat traps in the majority of the studied metropolitan areas in which +populations residing in high heat exposure areas primarily visit areas with high heat exposure. +The results also show a small percentage of human mobility to produce urban heat escalate + + + + +2 + +(visitations from low heat areas to high heat areas) and heat escapes (movements from high heat +areas to low heat areas). The findings from this study provide a better understanding of urban +heat exposure in cities based on patterns of human mobility. These finding contribute to a +broader understanding of the intersection of human network dynamics and environmental hazard +exposures in cities to inform more integrated urban design and planning to promote health and +sustainability. + +Keywords: urban heat exposure, demographic segregation, income segregation, urban +centrality, spatial structures. + +Introduction +The characterization of the spatial environmental hazards in cities is essential for urban +sustainability and health plans and policies (Shen et al., 2011, Seo et al., 2019, Hunter et al., +2019). Among all the environmental hazards, heat is one of the major hazards. Damages of heat +include increased mortality and morbidity due to extremely high air temperatures (Kim & +Brown, 2021), stronger heat-related health threats in urban areas (Li et al., 2016), and increased +energy consumption (Xie et al., 2019). However, comparing to other environmental hazards, +such as air pollution, urban heat did not draw enough attention in the existing literature (Bao et +al., 2022, Glencross et al., 2020, Venter et al., 2020). Within the studies of urban heat, limited +attentions were paid to human network dynamics that could expand the reach of environmental +hazard exposures (Coccia, 2020). Current heat-related studies mostly focused on index-based, +which is an isolated measurement of individual locations (Andrade & Szlafsztein, 2018, Jha & +Gundimeda, 2019, Orioli et al., 2019). Research gap exists in terms of how to understand the + +3 + +spatial distribution of urban heat and people’s respond to the heat from a network-based +perspective. In particular, human mobility shapes the spatial structures of cities and could extend +the reach of environmental hazards beyond hazard hotspots. In a recent study, Fan et al. (2022) +examined the intersection of human mobility and air pollution exposure and found that human +mobility expands the reach of air pollution exposure. This study highlights the significance of +characterizing environmental hazard exposures based on considering human mobility networks +in cities (Fan et al., 2022). In the context of urban heat exposure, Yin et al. (2021) proposed a +dynamic urban thermal exposure index to account for human mobility in specifying urban heat +exposure. While the index-based approach proposed by Yin et al. (2021) captures mobility-based +heat exposure, it does not capture fundamental properties arising at the intersection of human +mobility and spatial heat exposure that extend or alleviate heat exposure. Recognizing this gap, +in this paper, we define and examine three properties at the intersection of urban heat and human +mobility (Figure 1): (1) heat traps: in which populations residing in high heat areas visit other +high heat areas; (2) heat escapes: in which populations residing in high heat areas visit low heat +areas; and (3) heat escalates: in which populations residing in low heat areas visit high heat +areas. In fact, these properties are emergent properties arising from the intersection of human +mobility networks and the spatial distribution of heat hazards in cities. Accordingly, the study +aims to address the following research questions: to what extent human mobility would +exacerbate urban heat exposure (prominence of heat traps), alleviate heat exposure (heat +escapes), or expand the reach of heat exposure (heat escalates)? To address these questions, we +utilize aggregated and anonymized location-based data to construct the human mobility network +(origin-destination network in which origin is the home census tracts of trips and destination is +the visitation census tract of trips) for twenty metropolitan areas in the U.S. to examine the + +4 + +proportion of trips from high heat areas to other high heat areas and low heat areas. Accordingly, +we analyze the prominence of heat traps, escapes, and escalates across different cities to evaluate +cross-city similarities and differences. + + +Figure 1. Conceptual representation of urban heat traps, escalates, and escapes arising from the +intersection of human mobility and heat exposure. + +Background +Urban heat (UH), or the urban heat island effect, refers to the phenomenon where urban areas +have higher temperatures than surrounding rural areas due to the heat generated by human +activity and the lack of vegetation to absorb that heat. To understand and mitigate UH effect, +researchers have identified multiple factors. For example, some studies found that tree density is +correlated with UH (Ziter et al., 2019, Rahman et al., 2020, Morabito et al., 2021), that high tree +density potentially decreases urban heat phenomenon. Transportation is another factor, less + +HighUrbanHeat +ModerateUrbanHeat +LowUrbanHeat +NoUrbanHeat +Home +CensusTract +Escalates +Visitation +Census Tract +Escapes +. +Traps5 + +movement of transportation can reduce the extent of changes in temperatures in urban areas (Hu +et al., 2019, Ali et al., 2021, Angelevska et al., 2021). Moreover, population density also +contributes to the urban heat effect, population loss can have a mitigating effect on the UH effect +(Zhou et al., 2018, Manoli et al., 2019, Peng et al., 2022). However, those studies focused on +examining a single factor with UH, that they ignored the ability for human to adjust living +environment by moving to different locations. + +Human mobility datasets have been widely used in multiple hazards, including hurricane (Li et +al., 2020, Rajput et al., 2020, Dargin et al., 2021, Li & Mostafavi, 2022, Paradkar et al., 2022), +flooding (Esparza et al., 2022, Farahmand et al., 2022a, Farahmand et al., 2022b, Mostafavi & +Yuan, 2022, Ridha et al., 2022, Yuan et al., 2022a, Yuan et al., 2022b), and infectious diseases +(Fan et al., 2021, Ma et al., 2022, Li et al., 2021, Rajput et al., 2022). These studies have found +human mobility data was useful to understand people’s reaction to hazards (Lai et al., 2019). For +example, when hurricane comes, people in the similar social media networks were likely to make +the same evacuation decisions (Jiang et al., 2019). During the COVID 19 pandemic, the +confirmed cases were found highly correlated with human mobility that places with higher +activities had more covid cases (Coleman et al., 2022, Huang et al., 2020). These studies have +recognized that people can successfully change the level of hazard exposure by moving to a +different location. + +The majority of human mobility and hazard studies have focused on the relationship between +human mobility patterns and the likelihood of exposure to natural hazards, infectious diseases, +and environmental pollutants. However, the current literature does not adequately investigate the + +6 + +relationship between human mobility and urban heat (Smith et al., 2019). In this context, +mobility can play a significant role in determining the likelihood of exposure to urban heat. +Therefore, understanding the relationship between human mobility and UH can be useful in +developing strategies to reduce the impact of urban heat on individuals and communities, which +is the focus of this study. +Data Description +Study Context +We collected mobility data in February 2020 in twenty metropolitan areas (Table 1) in the U.S. +to construct human mobility networks. The rationale for selecting February 2020 is that it was +just before the start of the COVID-19 pandemic, and the patterns of human mobility would +represent the standard patterns of mobility. + +Table 1. Metropolitan Areas + +Metropolitan Areas +State +1 +Phoenix +Arizona +2 +Los Angeles +California +3 +Denver +Colorado +4 +Washington DC +District of Columbia +5 +Orlando +Florida +6 +Miami +Florida +7 +Atlanta +Georgia +8 +Chicago +Illinois +9 +Boston +Massachusetts + +7 + +10 +Detroit +Michigan +11 +Minneapolis +Minnesota +12 +Rochester +New York +13 +Columbus +Ohio +14 +Portland +Oregon +15 +Pittsburgh +Pennsylvania +16 +Philadelphia +Pennsylvania +17 +Memphis +Tennessee +18 +Houston +Texas +19 +Dallas +Texas +20 +Seattle +Washington + +Data sources +The heat exposure data were obtained from the United States Surface Urban Heat Island database +(Chakraborty et al., 2020). For all census tracts in the U.S. urbanized regions, this dataset +includes yearly, summer, and winter daytime and nighttime Land Surface Temperature (LST), +Digital Elevation Model (DEM), and Normalized Difference Vegetation Index (NDVI) data, as +well as the mean values for the whole urbanized area (Chakraborty et al., 2020). The UHI dataset +in the urbanized areas was determined by remote sensing data, such as Moderate Resolution +Imaging Spectroradiometer (MODIS) and Global Multi-Resolution Terrain Elevation Data +(GMTED), including 55,871 census tracts organized into 497 urbanized areas, covering roughly +78 percent of the population of the United States (Chakraborty et al., 2020). Our study used the +mean values for Urban Heat Islands (UHIs) as the measurement of UH for the chosen + +8 + +metropolitan areas. We used quantile breaks to split the UHI data into three clusters and defined +them as low UHI area, median UHI area, and high UHI area, respectively. + +The location-based data is provided by Spectus (formerly known as Cuebiq), a platform for +mobility data. Spectus provides privacy-protected and anonymized location datasets by +collecting data from smart devices whose owners have authorized location data collection. +Spectus constructs its geo-location dataset by collaborating with application developers to collect +high-resolution datasets using Bluetooth, GPS, WiFi, and IoT signals. Each day, more than one +hundred data points are gathered for each anonymous user, allowing a more accurate +understanding of human movement and visitation patterns. Spectus collects data on around 15 +million daily active users in the U.S. High privacy policy standards are set to enable data +collection and use of data responsibly and ethically. Users are allowed to opt out of location +sharing at any stage, and all information is obtained transparently with consent. All data provided +by Spectus is de-identified to ensure anonymity and endures further privacy improvements, such +as removing sensitive points of interest and obscuring dwelling locations at the census block +group level. In addition to delivering location-based data at the device level, Spectus aggregates +data using artificial intelligence and machine learning algorithms. By offering access to an +auditable and on-premise sandbox environment, Spectus' platform for responsible data sharing +allows us to query anonymized, aggregated, and privacy-enhanced data (Wang et al., 2019). In +this study, we used one of the aggregated datasets from Spectus, the Device Location database, +to determine the Census tracts of devices' home locations. The Device Location table includes a +timestamp, a privacy-compliant device ID, and geoinformation at the device level. To evaluate + +9 + +UH exposure, we used population activity in February 2020, which reflects a steady-state period +with no events that could affect population activity and movement. + +Methods +Mobility network from the home Census tract to the visitation Census tract +Data processing consisted of utilizing Spectus data to construct the human mobility network +models. Specifically, it involves two steps. First step is to identify each device’s home tract. The +second step is to construct the mobility networks. A device’s home tract was determined based +on its dwell times, as Spectus provides dwell time at each location. + +By using unique identifiers for each device, Spectus can collect each visitor’s destination tract +and aggregate the number of visits from one tract to another tract. Accordingly, we construct the +monthly mobility network model of each city, which captures the number of visits from home +tracts to visitation tracts. In this network, each node is a tract and the links are number of trips +observed between each pair of tracts. + +The Ratio of Urban Heat Traps, Escalates, and Escapes +In each metropolitan area, we used quantile breaks dividing Census tracts into low UH areas, +median UH areas, and high UH areas. In this study, we only considered low and high UH areas. +We aggregated human mobility dataset to summarize the number of trips between low and high +UH areas. As noted earlier, we define heat traps as high UH areas whose populations visit places +in other high UH areas. Similarly, heat escalates are low UH exposure areas whose populations +visit places in high UH areas. And heat escapes are high UH exposure areas whose populations + +10 + +visit places in low UH areas. The ratio of UH traps, escalates, and escapes of each tract is +calculated by summing the trips in each category (high to high, low to high, and high to low, +respectively) and dividing by the total trips associated with each home tract. The ratio of heat +escalates is computed using Equation 1: + +𝑅𝐿𝑜𝑤𝑖,𝑗 = +𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷ℎ𝑖𝑔ℎ𝑖,𝑗 +𝑇𝑂𝑇𝑖 + +(1) +where, 𝑅𝐿𝑜𝑤𝑖,𝑗 refers to the ratio of trips visiting from low UH tract i to high UH j, +𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷ℎ𝑖𝑔ℎ𝑖,𝑗refers to the total number of trips from low UH tract i to high UH tract j, +and 𝑇𝑂𝑇𝑖 refers to the total number of trips starting from origin tract i. Similarly, the ratio of +trips visiting from high UH tract to low UH tract and the ratio of trips visiting from high UH +tract to high UH tracts are computed using Equations 2 and 3, respectively: + +𝑅𝐻𝑖𝑔ℎ𝑖,𝑗 = +𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷𝑙𝑜𝑤𝑖,𝑗 +𝑇𝑂𝑇𝑖 + +(2) +𝑅𝐻𝑖𝑔ℎ𝑖,𝑗 = +𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷ℎ𝑖𝑔ℎ𝑖,𝑗 +𝑇𝑂𝑇𝑖 + +(3) + +Classifying Cities +For each metropolitan area, we first calculated the total number of tracts in high and low UH +exposures based on the UH dataset. Then, we classified cities as heat traps, heat escalates, and +heat escapes based on the percentage of trips in each category. If more than half of trips in the +city were heat trap type, we classified this cities as urban heat traps. Similarly, if the city has + +11 + +more than half heat escalate trips or heat escape trips, the city was classified as a heat escalate +city or heat escape city, respectively. + +Results +Patterns across Cities +Table 2 presents the list of metropolitan areas, and their percentage of trips in each category (i.e., +high to high, low to high, and high to low). The high UH and low UH percentages divide the +total number of census tracts by the number of census tracts in high UH areas and low UH areas. +Note that the total number of census tracts with trips from high to low and with trips from high to +high is the same, but the ratio of trips visiting from high UHI census tract i to low UHI census +tract j are significantly different (Equation (2) and (3)). The metropolitan classifications are +based on the percentage of low-to-high trips, high-to-low trips, and high-to-high trips, as stated +in the previous section. + +Table 2. Metropolitan areas with the total number of census tracts (CT), different UH visiting +patterns count and percentage, and classification of the metropolitan areas. +MSA +Total # +of CT +Total # +CT in +high +UHI +areas +High +UHI +% +Total # +CT in +low +UHI +areas +Low +UHI +% +Total # +CT with +trips from +low to +high +Low +to +high +trips +% +Total # +CT with +trips from +high to +low +Total # of +CT with +trips from +high to +high +High +to +low +% +High +to +high +% +Classifications +Atlanta, GA +885 +186 +0.21 +280 +0.32 +37 +0.13 +75 +75 +0.4 +0.4 +trap +Boston, MA +947 +343 +0.36 +264 +0.28 +10 +0.04 +128 +133 +0.37 +0.37 +trap + +12 + +Chicago, IL +1,923 +945 +0.49 +301 +0.16 +168 +0.56 +739 +739 +0.78 +0.78 +trap +Columbus, +OH +340 +155 +0.46 +67 +0.2 +21 +0.31 +155 +155 +1 +1 +escalate & trap +Dallas. TX +1122 +575 +0.51 +110 +0.1 +42 +0.38 +282 +282 +0.49 +0.49 +trap & escape +DC +179 +56 +0.31 +49 +0.27 +49 +1 +56 +56 +1 +1 +escalate & trap +Denver, CO +581 +218 +0.38 +98 +0.17 +5 +0.05 +96 +96 +0.44 +0.44 +escape +Detroit, MI +1,158 +658 +0.57 +183 +0.16 +31 +0.17 +408 +408 +0.62 +0.62 +trap +Houston, TX 908 +434 +0.48 +130 +0.14 +86 +0.66 +402 +402 +0.93 +0.93 +trap +Los +Angeles, CA +2788 +1462 +0.52 +351 +0.13 +284 +0.81 +1179 +1179 +0.81 +0.81 +escalate & trap +Memphis, +TN +221 +93 +0.42 +51 +0.23 +50 +0.98 +93 +93 +1 +1 +escalate & trap +Miami, FL +1206 +514 +0.43 +232 +0.19 +97 +0.42 +291 +291 +0.57 +0.57 +escalate & trap +Minneapolis +, MN +683 +306 +0.45 +120 +0.18 +31 +0.26 +170 +170 +0.56 +0.56 +trap +Orlando, FL +299 +99 +0.33 +58 +0.19 +26 +0.45 +88 +88 +0.89 +0.89 +escalate & trap +Philadelphia +, PA +968 +279 +0.29 +274 +0.28 +20 +0.07 +238 +238 +0.85 +0.85 +trap +Phoenix, AZ 893 +327 +0.37 +165 +0.18 +157 +0.95 +326 +326 +1 +1 +escalate & trap + +13 + +Pittsburgh, +PA +599 +190 +0.32 +203 +0.34 +99 +0.49 +168 +169 +0.88 +0.88 +escalate & trap +Portland, +OR +334 +178 +0.53 +58 +0.17 +25 +0.43 +106 +106 +0.6 +0.6 +escalate & trap +Rochester, +NY +206 +102 +.0.50 +17 +0.08 +12 +0.71 +102 +102 +1 +1 +escalate & trap +Seattle, WA +660 +200 +0.3 +155 +0.23 +97 +0.63 +120 +120 +0.6 +0.6 +escalate & trap + +Cities with High Urban Heat Traps +The Los Angeles metropolitan area shows significant urban heat traps. Figure 2A maps the UH +in Los Angeles. Three orange shades represent three levels of UH. The darker the shade is, the +more severe UH was observed. The metropolitan area has 13 percent of the tracts in low UH +areas, mainly located on the north and east, while 52 percent of the metropolitan area is in high +UH areas (dark orange). Figure 2B to 2D shows the ratio of trips between low UH tracts and +high UH tracts, which break into four categories for better visualization. Light blue shows a low +ratio of trips, and dark blue shows a high ratio of trips. All the following figures are presented in +the same plot format as Figure 2A and 2B to 2D. + +Figure 2B shows the ratio of trips visiting from low UH tracts to high UH tracts. A high ratio of +low-to-high trips from 0.22 to 0.35 occurs in the north, which means that a significant number of +people living in low UH areas are visiting high UH areas in the north. Figure 2D shows the ratio +of trips from high UH tracts to low UH tracts with a higher ratio of trips, 0.05 to 0.11, occurring +in the northwest and southwest. This means that a relatively high number of people living in high + +14 + +UH areas are visiting low UH areas in the northwest and southwest. Figure 2C shows the ratio +of trips visiting from high UH tracts to high UH tracts. 81 percent of all the tracts in high UH +areas have trips trapped inside high UH areas with the ratio of trips from 0.30 to 0.92, meaning +lots of people suffering UH did not move to relief their UH exposure. These urban heat traps are +in the northwest and central of Los Angeles, with an especially high ratio from 0.76 to 0.92 in +the central. Figure 2D shows the ratio of trips visiting from high UH areas to low UH areas with +ratio of trips from 0 to 0.11. + + +(A) Distribution of urban heat + (B) The ratio of trips from low UH to high UH + +(C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH + + +6.29,0.22 +0.22,2.96 +2.96,5.790.01, 0.06 +0.06, 0.12 +0.12,0.22 +0.22,0.350.30,0.54 +0.54, 0.67 +0.67,0.76 +0.76,0.920.00,0.02 +0.02,0.03 +0.03,0.05 +0.05,0.1115 + +Figure 2. Urban Heat Traps and Trips in Los Angeles Metropolitan Area. (A) shows that 52 +percent of tracts are in high UH areas across Los Angeles. (C) 81 percent of tracts in high UH +areas have trips to other high UH tracts, representing that Los Angeles is a metropolitan area +with urban heat traps. + +Similarly, the Chicago metropolitan area shows strong urban heat traps as well. Figure 3A maps +the UH in Chicago. Chicago has 16 percent of its tract in low UH areas, while 49 percent of its +tracts are in high UH areas. Figure 3B shows the ratio of trips visiting from low UH tracts to high +UH tracts. A higher ratio of trips 0.17 to 0.24 occurs on the coast of Lake Michigan, meaning +that a significant number of people living in low UH areas are visiting high UH areas on the +coast of Lake Michigan. Figure 3D shows trips from high UH tracts to low UH tracts with a ratio +as high as 0.08 to 0.13 occurring in the east. Figure 3C shows the ratio of trips visiting from high +UH tracts to high UH tracts, with the ratio of trips from 0.44 to 0.91, which means that a large +number of people living in high UH areas are visiting other high UH areas within the Chicago +metropolitan area. About 78 percent of Chicago tracts in high UH areas have trips trapped inside +high UH areas. Most of the UH traps are in the west of Chicago. At the same time, central +Chicago presents an exceptionally high heat trap ratio, ranging from 0.79 to 0.91. Figure 3D. +shows the ratio of trips visiting from high UH tracts to low UH tracts, with the ratio of trips from +0 to 0.13. This means that a relatively low number of people living in high UH areas are visiting +low UH areas within the Chicago metropolitan area. + +Comparing the UH traps between Chicago and Los Angeles, we can see that the traps in Chicago +are clustered in one place, while in Los Angeles are distributed into multiple clusters. + +16 + + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +-6.07,-0.02 +-0.02,2.65 +2.65,5.670.03, 0.09 +0.09, 0.13 +0.13, 0.17 +0.17,0.240.44,0.61 +0.61,0.71 +0.71,0.79 +0.79,0.910.00,0.02 +0.02,0.04 +0.04,0.08 +0.08,0.1317 + + +Figure 3. UH Traps and Trips in the Chicago Metropolitan Area. (A) 16 percent of tracts are in +low UH areas and 49 percent of tracts are in high UH areas across Chicago. (C) 78 percent of +tracts in high UH areas have trips to high UH tracts, representing that Chicago is a metropolitan +area with urban heat traps. + +Figures 3 and 4 show that the Los Angeles and Chicago metropolitan areas both have significant +urban heat traps. In Los Angeles, 52 percent of all the tracts are in high UH areas, while in +Chicago, 49 percent are in high UH areas. The figures also show that trips from low UH areas to +high UH areas are more frequent in the north of both cities, while trips from high UH areas to +low UH areas are more common in the northwest and southwest of Los Angeles, and the east of +Chicago. Additionally, the figures show that both cities present high heat trap trips, with around +80 percent of tracts with heat trap trips. This indicates that people in the high UH areas are likely +not visiting the low UH areas to escape the heat, but instead are staying in other high UH areas. + +Cities with Low Urban Heat Traps +Boston Metropolitan shows low urban heat traps. Figure 4A maps the UH in Boston. About 28 +percent of tracts in Boston are in low UH areas, while 36 percent of tracts have high UH. Most of +these high UH tracts are clustered in central Boston. Figure 4B shows the ratio of trips visiting +from low UH tracts to high UH tracts. The ratio of such trips is from 0.04 to 0.19 and only occur +in 4 percent of all the tracts with low UH. Figure 4C shows the ratio of trips visiting from high +UH tracts to high UH tracts. This ratio ranges from 0.30 to 0.90. About 37 percent of tracts with +high UH have trips trapped inside high UH areas. This percentage is relatively small when +comparing to Los Angeles (81 percent) and Chicago (78 percent). Figure 4D shows the trips + +18 + +from high UH areas to low UH areas with ratio from 0 to 0.02. These results indicate that people +living in low UH areas in the Boston metropolitan area are not frequently visiting high UH areas, +which could be an indication of a fewer heat traps. + +(A) Distribution of UH + (B) The ratio of trips from low UH to high UH + + +(C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH + +5.30,0.73 +0.73,3.17 +3.17,6.320.04,0.05 +0.05,0.08 +0.08,0.12 +0.12, 0.190.30,0.48 +0.48,0.66 +0.66, 0.77 +0.77,0.900.00,0.00 +0.00,0.01 +0.01,0.01 +0.01,0.0219 + + +Figure 4. UH Traps and Trips in Boston Metropolitan Area. (A) 28 percent of tracts are low UH +areas and 38 percent are in high UH areas across Boston. (C) 37 percent of tracts in high UH +areas have trips to high UH tracts, representing that Boston is a metropolitan area with low urban +heat traps. + +Similarly, the Atlanta Metropolitan also shows low UH traps. Figure 5A maps the UH in Atlanta. +Atlanta has 32 percent of the tracts in low UH areas, while 21 percent are in high UH areas. +Figure 5B shows the ratio of trips visiting from low UH tracts to high UH tracts. The ratio of +such trips is from 0.01 to 0.16 and only occurred in 13 percent of all the low UH tracts. Figure +5C shows the ratio of trips visiting from high UH tracts to high UH tracts with ratios from 0.37 +to 0.87. About 40 percent of high UH tracts have heat trap trips. This number is similar with +Boston and is relatively small comparing to Los Angeles and Chicago. Figure 5D shows the trips +from high UH areas to low UH areas, ranging from 0.01 to 0.12. This ratio is small but more +significant than that of Boston, which means that comparing to Boston, more heat escape trips +exist in Atlanta. + +20 + + +(A) Distribution of UH + (B) The ratio of trips from low UH to high UH + + +(C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH + +Figure 5. UH Traps and Trips in Atlanta Metropolitan Area. (A) 32 percent of tracts are low UH +and 21 percent are high UH across Atlanta. (C)40 percent of tracts in high UH areas have trips to +high UH tracts, representing that Atlanta is a metropolitan area with low UH traps. + + +3.35,0.37 +0.37,2.31 +2.31,5.600.02,0.04 +0.04,0.06 +0.06,0.10 +0.10,0.160.37,0.47 +0.47,.0.62 +0.62,0.71 +0.71,0.870.01,0.03 +0.03,0.05 +0.05,0.08 +0.08,0.1221 + +Figures 4 and 5 show that both Boston and Atlanta have relatively low UH comparing to Los +Angeles and Chicago. In Boston, only 36 percent of tracts are in high UH, while in Atlanta, only +21 percent of the tracts are in high UH. The figures also show that trips from low UH areas to +high UH areas are relatively rare in both cities, only 4 percent and 13 percent in low UH tracts in +Boston and Atlanta, respectively. In both cities, the percentages of tracts with trips trapped inside +high UH areas are lower than in Los Angeles and Chicago. + +Cities with High Urban Heat Escapes +The Minneapolis Metropolitan Area shows high UH escapes. Figure 6A maps the UH in +Minneapolis. The metropolitan area has 18 percent of its tracts in low UH areas, while 45 percent +are in high UH areas. Figure 6B shows the ratio of trips from low UH tracts to high UH tracts. +This ratio ranges from 0.03 to 0.34, occurring in 26 percent of low UH tracts. Figure 6C shows +the ratio of trips from high UH tracts to high UH tracts with the ratios from 0.41 to 0.86. Figure +6D shows the ratio of trips from high UH tracts to low UH tracts. This ratio is between 0.01 and +0.13, occurring in 56 percent of high UH tracts. Comparing this high UH to low UH ratio with +other cities, Minneapolis shows strong UH escape, indicating that a significant number of people + +22 + +living in high UH areas are visiting low UH areas in the Minneapolis metropolitan area. + +(A) Distribution of UH + (B) The ratio of trips from low UH to high UH + +(C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH + +Figure 6. UH Traps and Trips in Minneapolis Metropolitan Area. (A) 18 percent of tracts are +low UH areas and 45 percent are in high UH areas across Minneapolis. (D) 56 percent of tracts + +5.27,-0.12 +-0.12,2.35 +2.35,5.030.03, 0.08 +0.08,0.15 +0.15, 0.26 +0.26, 0.340.41,0.53 +0.53,0.66 +0.66,0.74 +0.74,0.860.01,0.02 +0.02,0.04 +0.04,0.08 +0.08,0.1323 + +in high UH areas have trips to low UH tracts, representing that Minneapolis has high heat +escapes trips. + +Similarly, the Dallas Metropolitan Area also shows high heat escapes. Figure 7A maps the UH in +Dallas. Dallas has 10 percent of its tracts in low UH areas, while 50 percent of its tracts are in +high UH areas. Figure 7A shows that the high UH tracts form multiple clusters across the city. +Figure 7B shows the ratio of trips from low UH tracts to high UH tracts. This ratio is between +0.07 and 0.28, occurring in 38 percent of the low UH tracts. Figure 7C shows the ratio of trips +from high UH tracts to high UH tracts with ratios from 0.39 to 0.82. Figure 7D shows the ratio of +trips from high UH tracts to low UH tracts. The ratio of trips from high UH tracts to low UH +tracts is notable, ranging from 0.00 to 0.16, in 49 percent of high UH tracts. This indicate that +Dallas has strong urban heat escapes trips. + + +(A) Distribution of UH + (B) The ratio of trips from low UH to high UH + +-5.15, -0.50 +-0.50,1.50 +1.50,3.500.07,0.10 +0.10, 0.15 +0.15, 0.21 +0.21, 0.2824 + + + +(C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH + +Figure 7. UH Traps and Trips in the Dallas Metropolitan Area. (A) 10 percent of tracts are low +UH areas and 51 percent are in high UH areas across Dallas. (D) 49 percent of high UH tracts +have trips to low UH tracts, representing that Dallas is a metropolitan area with high urban heat +escapes. + +Figures 6 and 7 show that both Minneapolis and Dallas have significant urban heat escapes, with +a higher ratio of trips from high UH tracts to low UH tracts when comparing to other +metropolitan areas, such as Boston (37 percent) and Atlanta (24 percent). This indicates that +people in the high UH areas travel to the low UH areas to escape the heat. + +Additionally, this study offered important insights by examining the factors of distinctive +characteristics underline spatial structures (Angel & Blei, 2016), facility distribution (Pereira et +al., 2013), income, and racial segregation, as in Appendix C. However, no statistical significance +was found between heat traps and attributes of demographic segregation. This interpolates that + +0.39,0.52 +0.52, 0.63 +0.63,0.71 +0.71,0.820.00,0.02 +0.02, 0.04 +0.04,0.08 +0.08,0.1625 + +an urban heat trap is an emergent property (Georgiou, 2003) that cannot be attributed to the +centrality of city facilities and demographics. Therefore, we observe that human mobility leads +to the creation of traps, not escapes or escalates. Maybe people are more likely to go to places +where they are more familiar. + +Discussion and Concluding Remarks +This study utilized large-scale, high-resolution location-intelligence data to identify and quantify +the urban heat (UH) exposure and people’s response based on human mobility networks in urban +areas. This study analyzed the intersection of UH and human mobility by examining the UH +dataset and trips between tracts in February 2020 in twenty metropolitan areas. The study +identified and analyzed three properties: heat traps, heat escapes, and heat escalate by +quantifying the trips between tracts in high UH areas and low UH areas. This study found that +not many cities have heat escapes or heat escalates trips. Heat escapes were found in +Minneapolis and heat escalates were found in Los Angeles. A potential reason might be that +people are more likely to stay in their resident areas. + +Researchers and professionals are well aware of the diverse effects that UH can have heat-related +diseases, such as respiratory difficulties among urban populations (Huang et al., 2020). However, +there is little knowledge about the extent to which human mobility exacerbates UH. This study +offers an innovative, data-driven method and metrics for using large-scale location intelligence +data to assess UH exposure. This study evaluates the intersection of human mobility and the +spatial distribution of urban heat. In addition, this study defines three important characteristics of +people’s potential response to UH based on trip destinations. Specifically, heat traps refer to + +26 + +population residing in high UH areas visit other high UH areas; heat escalates refer to population +residing in low UH areas visit high UH areas and thus escalate their heat exposure; and heat +escapes refer to population residing in high UH areas visit low UH areas and thus escape from +their local heat. Defining these three different responses to UH can help researchers understand +different characteristics of the urban areas. + +There are several limitations of this study. First, this study is based on smartphone data. +Smartphone users who allowed such location data collection is a biased sample. Visitors who do +not own smartphones, such as children, teenagers, the elderly, and those with lower income, +were less likely to be included in the data, which may create biases (Esmalian et al., 2021, Song +et al., 2022). Additionally, efforts could be made to ensure that the sample of smartphone users is +representative of the population as a whole, such as by using stratified sampling or weighting the +data to account for any biases. We partially address this limitation by utilizing Spectus data, +which has been demonstrated to contain a representative sample of users (Li & Mostafavi, 2022). +Second, the mobility data does not include the visiting time for the destinations, which may +cause mis-labeling of trip purposes. Future researchers could leverage other sources of data, such +as surveys or observational studies, to further validate traveling information. + +This study offers important insights to city designers and city planners. The three important +characteristics of traps, escalates, and escapes are likely related to how heat exposure can affect +people in different parts of a city. Better understandings of people’s movements and associated +heat exposure can provide city planner information for future city development. These +characteristics may include factors such as the availability of shade and other forms of shelter, + +27 + +the accessibility of air conditioning and other cooling mechanisms, and the presence of social +networks and support systems that can help people cope with heat waves and other extreme +weather events. By understanding these characteristics, it may be possible to develop strategies +and interventions that can help reduce the risks associated with heat exposure in urban +environments. + +Data Availability +All data were collected through a CCPA- and GDPR-compliant framework and utilized for +research purposes. The data that support the findings of this study are available from Spectus, but +restrictions apply to the availability of these data, which were used under license for the current +study. The data can be accessed upon request submitted on spectus.ai. Other data we use in this +study are all publicly available. + +Code Availability +The code that supports the findings of this study is available from the corresponding author upon +request. + +Declaration of Interests: none + +Acknowledgement +This material is based in part upon work supported by the National Science Foundation under +Grant CMMI-1846069 (CAREER), Texas A&M University X-Grant 699, and the Microsoft +Azure AI for Public Health grant. The authors also would like to acknowledge the data support + +28 + +from Spectus. Any opinions, findings, conclusions or recommendations expressed in this +material are those of the authors and do not necessarily reflect the views of the National Science +Foundation, Texas A&M University, Microsoft Azure, or Spectus. + +Reference +Ali, G., Abbas, S., Qamer, F. M., Wong, M. S., Rasul, G., Irteza, S. M., & Shahzad, N. (2021). +Environmental impacts of shifts in energy, emissions, and urban heat island during the +COVID-19 lockdown across Pakistan. Journal of Cleaner Production, 291, 125806. +https://doi.org/10.1016/j.jclepro.2021.125806 +Andrade, M. M. N. de, & Szlafsztein, C. F. (2018). Vulnerability assessment including tangible +and intangible components in the index composition: An Amazon case study of flooding +and flash flooding. Science of The Total Environment, 630, 903–912. +https://doi.org/10.1016/j.scitotenv.2018.02.271 +Angel, S., & Blei, A. M. (2016). The spatial structure of American cities: The great majority of +workplaces are no longer in CBDs, employment sub-centers, or live-work communities. +Cities, 51, 21–35. https://doi.org/10.1016/j.cities.2015.11.031 +Angelevska, B., Atanasova, V., & Andreevski, I. (2021). Urban air quality guidance based on +measures categorization in road transport. Civil Engineering Journal, 7(2), 253–267. +https://doi.org/10.28991/cej-2021-03091651 +Bao, R., & Zhang, A. (2020). Does lockdown reduce air pollution? Evidence from 44 cities in +northern China. Science of The Total Environment, 731, 139052. +https://doi.org/10.1016/j.scitotenv.2020.139052 + +29 + +Chakraborty, T. (2020). United states surface urban heat island database [Data set]. Mendeley. +https://doi.org/10.17632/X9MV4KRNM2.2 +Chakraborty, T., Hsu, A., Manya, D., & Sheriff, G. (2020). A spatially explicit surface urban +heat island database for the United States: Characterization, uncertainties, and possible +applications. ISPRS Journal of Photogrammetry and Remote Sensing, 168, 74–88. +https://doi.org/10.1016/j.isprsjprs.2020.07.021 +Coccia, M. (2020). An index to quantify environmental risk of exposure to future epidemics of +the COVID-19 and similar viral agents: Theory and practice. Environmental Research, 191, +110155. https://doi.org/10.1016/j.envres.2020.110155 +Coleman, N., Gao, X., DeLeon, J., & Mostafavi, A. (2022). Human activity and mobility data +reveal disparities in exposure risk reduction indicators among socially vulnerable +populations during COVID-19 for five U.S. metropolitan cities. Scientific Reports, 12(1), +15814. https://doi.org/10.1038/s41598-022-18857-7 +Dargin, J. S., Fan, C., & Mostafavi, A. (2021). Vulnerable populations and social media use in +disasters: Uncovering the digital divide in three major U.S. hurricanes. International +Journal of Disaster Risk Reduction, 54, 102043. https://doi.org/10.1016/j.ijdrr.2021.102043 +Esmalian, A., Dong, S., Coleman, N., & Mostafavi, A. (2021). Determinants of risk disparity due +to infrastructure service losses in disasters: A household service gap model. Risk Analysis, +41(12), 2336–2355. https://doi.org/10.1111/risa.13738 +Esparza, M., Farahmand, H., Brody, S., & Mostafavi, A. (2022). Examining data imbalance in +crowdsourced reports for improving flash flood situational awareness. arXiv. +https://doi.org/10.48550/arXiv.2207.05797 + +30 + +Fan, C., Chien, Y.-H., & Mostafavi, A. (2022). Human mobility disproportionately extends pm2. +5 emission exposure for low income populations. arXiv. +https://doi.org/10.48550/arXiv.2205.15381 +Fan, C., Lee, R., Yang, Y., & Mostafavi, A. (2021). Fine-grained data reveal segregated mobility +networks and opportunities for local containment of COVID-19. Scientific Reports, 11(1), +16895. https://doi.org/10.1038/s41598-021-95894-8 +Farahmand, H., Liu, X., Dong, S., Mostafavi, A., & Gao, J. (2022). A network observability +framework for sensor placement in flood control networks to improve flood situational +awareness and risk management. Reliability Engineering & System Safety, 221, 108366. +https://doi.org/10.1016/j.ress.2022.108366 +Farahmand, H., Wang, W., Mostafavi, A., & Maron, M. (2022). Anomalous human activity +fluctuations from digital trace data signal flood inundation status. Environment and +Planning B: Urban Analytics and City Science, 49(7), 1893–1911. +https://doi.org/10.1177/23998083211069990 +Georgiou, I. (2003). The idea of emergent property. Journal of the Operational Research +Society, 54(3), 239–247. https://doi.org/10.1057/palgrave.jors.2601520 +Glencross, D. A., Ho, T.-R., Camiña, N., Hawrylowicz, C. M., & Pfeffer, P. E. (2020). Air +pollution and its effects on the immune system. Free Radical Biology and Medicine, 151, +56–68. https://doi.org/10.1016/j.freeradbiomed.2020.01.179 +Hu, Y., Hou, M., Jia, G., Zhao, C., Zhen, X., & Xu, Y. (2019). Comparison of surface and +canopy urban heat islands within megacities of eastern China. ISPRS Journal of +Photogrammetry and Remote Sensing, 156, 160–168. +https://doi.org/10.1016/j.isprsjprs.2019.08.012 + +31 + +Huang, H., Yang, H., Deng, X., Zeng, P., Li, Y., Zhang, L., & Zhu, L. (2020). Influencing +mechanisms of urban heat island on respiratory diseases. Iranian Journal of Public Health. +https://doi.org/10.18502/ijph.v48i9.3023 +Huang, X., Li, Z., Jiang, Y., Li, X., & Porter, D. (2020). Twitter reveals human mobility +dynamics during the COVID-19 pandemic. PLOS ONE, 15(11), e0241957. +https://doi.org/10.1371/journal.pone.0241957 +Hunter, R. F., Cleland, C., Cleary, A., Droomers, M., Wheeler, B. W., Sinnett, D., +Nieuwenhuijsen, M. J., & Braubach, M. (2019). Environmental, health, wellbeing, social +and equity effects of urban green space interventions: A meta-narrative evidence synthesis. +Environment International, 130, 104923. https://doi.org/10.1016/j.envint.2019.104923 +Jha, R. K., & Gundimeda, H. (2019). An integrated assessment of vulnerability to floods using +composite index – A district level analysis for Bihar, India. International Journal of +Disaster Risk Reduction, 35, 101074. https://doi.org/10.1016/j.ijdrr.2019.101074 +Jiang, Y., Li, Z., & Cutter, S. L. (2019). Social network, activity space, sentiment, and +evacuation: What can social media tell us? Annals of the American Association of +Geographers, 109(6), 1795–1810. https://doi.org/10.1080/24694452.2019.1592660 +Kim, S. W., & Brown, R. D. (2021). Urban heat island (Uhi) intensity and magnitude +estimations: A systematic literature review. Science of The Total Environment, 779, 146389. +https://doi.org/10.1016/j.scitotenv.2021.146389 +Kim, Y., Yeo, H., & Kim, Y. (2022). Estimating urban spatial temperatures considering +anthropogenic heat release factors focusing on the mobility characteristics. Sustainable +Cities and Society, 85, 104073. https://doi.org/10.1016/j.scs.2022.104073 + +32 + +Lai, S., Farnham, A., Ruktanonchai, N. W., & Tatem, A. J. (2019). Measuring mobility, disease +connectivity and individual risk: a review of using mobile phone data and mHealth for +travel medicine. Journal of travel medicine, 26(3), taz019. +Li, B., & Mostafavi, A. (2022). Location intelligence reveals the extent, timing, and spatial +variation of hurricane preparedness. Scientific Reports, 12(1), 16121. +https://doi.org/10.1038/s41598-022-20571-3 +Li, D., Sun, T., Liu, M., Wang, L., & Gao, Z. (2016). Changes in wind speed under heat waves +enhance urban heat islands in the beijing metropolitan area. Journal of Applied Meteorology +and Climatology, 55(11), 2369–2375. https://doi.org/10.1175/JAMC-D-16-0102.1 +Li, Q., Hannibal, B., Mostafavi, A., Berke, P., Woodruff, S., & Vedlitz, A. (2020). Examining of +the actor collaboration networks around hazard mitigation: A hurricane harvey study. +Natural Hazards, 103(3), 3541–3562. https://doi.org/10.1007/s11069-020-04142-1 +Li, Q., Yang, Y., Wang, W., Lee, S., Xiao, X., Gao, X., Oztekin, B., Fan, C., & Mostafavi, A. +(2021). Unraveling the dynamic importance of county-level features in trajectory of +COVID-19. Scientific Reports, 11(1), 13058. https://doi.org/10.1038/s41598-021-92634-w +Ma, J., Li, B., Li, Q., Fan, C., & Mostafavi, A. (2022). Attributed network embedding model for +exposing covid-19 spread trajectory archetypes. arXiv. +https://doi.org/10.48550/arXiv.2209.09448 +Manoli, G., Fatichi, S., Schläpfer, M., Yu, K., Crowther, T. W., Meili, N., Burlando, P., Katul, +G. G., & Bou-Zeid, E. (2019). Magnitude of urban heat islands largely explained by climate +and population. Nature, 573(7772), 55–60. https://doi.org/10.1038/s41586-019-1512-9 +Morabito, M., Crisci, A., Guerri, G., Messeri, A., Congedo, L., & Munafò, M. (2021). Surface +urban heat islands in Italian metropolitan cities: Tree cover and impervious surface + +33 + +influences. Science of The Total Environment, 751, 142334. +https://doi.org/10.1016/j.scitotenv.2020.142334 +Mostafavi, A., & Yuan, F. (2022). Smart flood resilience: Harnessing community-scale big data +for predictive flood risk monitoring, rapid impact assessment, and situational awareness +(No. EGU22-781). Copernicus Meetings. https://doi.org/10.5194/egusphere-egu22-781 +Orioli, R., Antonucci, C., Scortichini, M., Cerza, F., Marando, F., Ancona, C., Manes, F., Davoli, +M., Michelozzi, P., Forastiere, F., & Cesaroni, G. (2019). Exposure to residential greenness +as a predictor of cause-specific mortality and stroke incidence in the rome longitudinal +study. Environmental Health Perspectives, 127(2), 027002. +https://doi.org/10.1289/EHP2854 +Peng, X., Zhou, Y., Fu, X., & Xu, J. (2022). Study on the spatial-temporal pattern and evolution +of surface urban heat island in 180 shrinking cities in China. Sustainable Cities and Society, +84, 104018. https://doi.org/10.1016/j.scs.2022.104018 +Pereira, R. H. M., Nadalin, V., Monasterio, L., & Albuquerque, P. H. M. (2013). Urban +centrality: A simple index: urban centrality: a simple index. Geographical Analysis, 45(1), +77–89. https://doi.org/10.1111/gean.12002 +Rahman, M. A., Stratopoulos, L. M. F., Moser-Reischl, A., Zölch, T., Häberle, K.-H., Rötzer, T., +Pretzsch, H., & Pauleit, S. (2020). Traits of trees for cooling urban heat islands: A meta- +analysis. Building and Environment, 170, 106606. +https://doi.org/10.1016/j.buildenv.2019.106606 +Rajput, A. A., Li, Q., Gao, X., & Mostafavi, A. (2022). Revealing critical characteristics of +mobility patterns in New York City during the onset of COVID-19 pandemic. Frontiers in +Built Environment, 7, 180. + +34 + +Rajput, A. A., Li, Q., Zhang, C., & Mostafavi, A. (2020). Temporal network analysis of inter- +organizational communications on social media during disasters: A study of Hurricane +Harvey in Houston. International Journal of Disaster Risk Reduction, 46, 101622. +https://doi.org/10.1016/j.ijdrr.2020.101622 +Ridha, T., Ross, A. D., & Mostafavi, A. (2022). Climate change impacts on infrastructure: Flood +risk perceptions and evaluations of water systems in coastal urban areas. International +Journal of Disaster Risk Reduction, 73, 102883. https://doi.org/10.1016/j.ijdrr.2022.102883 +Seo, S., Choi, S., Kim, K., Kim, S. M., & Park, S. M. (2019). Association between urban green +space and the risk of cardiovascular disease: A longitudinal study in seven Korean +metropolitan areas. Environment International, 125, 51–57. +https://doi.org/10.1016/j.envint.2019.01.038 +Shen, L.-Y., Jorge Ochoa, J., Shah, M. N., & Zhang, X. (2011). The application of urban +sustainability indicators – A comparison between various practices. Habitat International, +35(1), 17–29. https://doi.org/10.1016/j.habitatint.2010.03.006 +Smith, A., Bates, P. D., Wing, O., Sampson, C., Quinn, N., & Neal, J. (2019). New estimates of +flood exposure in developing countries using high-resolution population data. Nature +Communications, 10(1), 1814. https://doi.org/10.1038/s41467-019-09282-y +Song, Y., Newman, G., Huang, X., & Ye, X. (2022). Factors influencing long-term city park +visitations for mid-sized US cities: A big data study using smartphone user mobility. +Sustainable Cities and Society, 80, 103815. https://doi.org/10.1016/j.scs.2022.103815 +Venter, Z. S., Aunan, K., Chowdhury, S., & Lelieveld, J. (2020). COVID-19 lockdowns cause +global air pollution declines. Proceedings of the National Academy of Sciences, 117(32), +18984–18990. https://doi.org/10.1073/pnas.2006853117 + +35 + +Wang, F., Wang, J., Cao, J., Chen, C., & Ban, X. (Jeff). (2019). Extracting trips from multi- +sourced data for mobility pattern analysis: An app-based data example. Transportation +Research Part C: Emerging Technologies, 105, 183–202. +https://doi.org/10.1016/j.trc.2019.05.028 +Xie, N., Li, H., Abdelhady, A., & Harvey, J. (2019). Laboratorial investigation on optical and +thermal properties of cool pavement nano-coatings for urban heat island mitigation. +Building and Environment, 147, 231–240. https://doi.org/10.1016/j.buildenv.2018.10.017 +Yin, Y., Grundstein, A., Mishra, D. R., Ramaswamy, L., Hashemi Tonekaboni, N., & Dowd, J. +(2021). DTEx: A dynamic urban thermal exposure index based on human mobility patterns. +Environment International, 155, 106573. https://doi.org/10.1016/j.envint.2021.106573 +Yuan, F., Fan, C., Farahmand, H., Coleman, N., Esmalian, A., Lee, C.-C., Patrascu, F. I., Zhang, +C., Dong, S., & Mostafavi, A. (2022). Smart flood resilience: Harnessing community-scale +big data for predictive flood risk monitoring, rapid impact assessment, and situational +awareness. Environmental Research: Infrastructure and Sustainability, 2(2), 025006. +https://doi.org/10.1088/2634-4505/ac7251 +Yuan, F., Xu, Y., Li, Q., & Mostafavi, A. (2022). Spatio-temporal graph convolutional networks +for road network inundation status prediction during urban flooding. Computers, +Environment and Urban Systems, 97, 101870. +https://doi.org/10.1016/j.compenvurbsys.2022.101870 +Zhou, D., Bonafoni, S., Zhang, L., & Wang, R. (2018). Remote sensing of the urban heat island +effect in a highly populated urban agglomeration area in East China. Science of The Total +Environment, 628–629, 415–429. https://doi.org/10.1016/j.scitotenv.2018.02.074 + +36 + +Ziter, C. D., Pedersen, E. J., Kucharik, C. J., & Turner, M. G. (2019). Scale-dependent +interactions between tree canopy cover and impervious surfaces reduce daytime urban heat +during summer. Proceedings of the National Academy of Sciences, 116(15), 7575–7580. +https://doi.org/10.1073/pnas.1817561116 + +Appendices +Appendix A. Urban Heat and Human Mobility Ratios in Cities + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + + +4.43,0.58 +0.58,3.11 +3.11,6.620.01,0.04 +0.04,0.08 +0.08,0.15 +0.15,0.350.34, 0.54 +0.54, 0.65 +0.65, 0.73 +0.73,0.880.00,0.02 +0.02,0.04 +0.04,0.08 +0.08,0.2137 + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 1. Urban Heat Traps and Trips in Houston Metropolitan Area. (A) shows that 14 +percent of census tracts are low UHI areas, and 48 percent are in high UHI areas across Houston. +(C) 93 percent of census tracts in high urban heat areas have trips to high urban heat census tract, +representing that Houston is a metropolitan area with high urban heat escapes. + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +-5.23, 0.04 +0.04,2.72 +2.72,7.020.02,0.07 +0.07,0.13 +0.13,0.18 +0.18,0.2738 + + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 2. Urban Heat Traps and Trips in Detroit Metropolitan Area. (A) shows that 16 +percent of census tracts are low UHI areas, and 57 percent are in high UHI areas across Detroit. +(C) 62 percent of census tracts in high urban heat areas have trips to high urban heat census tract, +representing that Detroit is a metropolitan area with high urban heat escapes. + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +0.41,0.61 +0.61,0.73 +0.73,0.81 +0.81,0.920.00,0.01 +0.01,0.03 +0.03,0.06 +0.06,0.133.50,-0.21 +-0.21,1.09 +1.09,2.710.01,0.05 +0.05,0.09 +0.09,0.14 +0.14,0.2139 + + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 3. Urban Heat Traps and Trips in Phoenix Metropolitan Area. (A) shows that 18 +percent of census tracts are low UHI areas, and 37 percent are in high UHI areas across Phoenix. +(B) 95 percent of census tracts in low urban heat areas have trips to high urban heat census tract +with ratio of trips as high as 0.21. (C) 100 percent of census tracts in low urban heat areas have +trips to high urban heat census tract, representing that Phoenix is a metropolitan area with high +urban heat escapes and high urban heat escalates. + +0.28,0.52 +0.52,0.61 +0.61,0.68 +0.68,0.840.01,0.03 +0.03,0.06 +0.06,0.10 +0.10,0.1740 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + + +-0.45, 1.89 +1.89, 3.34 +3.34, 4.930.01,0.04 +0.04, 0.07 +0.07,0.10 +0.10,0.140.34, 0.48 +0.48, 0.56 +0.56, 0.63 +0.63, 0.770.02, 0.06 +0.06, 0.09 +0.09, 0.13 +0.13,0.2541 + +Figure A - 4. Urban Heat Traps and Trips in Washington DC Metropolitan Area. (A) shows that +27 percent of census tracts are low UHI areas, and 31 percent are in high UHI areas across +Washington DC. (B) 100 percent of census tracts in low urban heat areas have trips to high urban +heat census tract with ratio of trips as high as 0.14. (C) 100 percent of census tracts in low urban +heat areas have trips to high urban heat census tract, representing that Washington DC is a +metropolitan area with high urban heat escapes and high urban heat escalates. + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +2.98,0.32 +0.32,2.24 +2.24,4.540.05,0.08 +0.08,0.13 +0.13,0.16 +0.16,0.1942 + + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 5. Urban Heat Traps and Trips in Columbus Metropolitan Area. (A) shows that 20 +percent of census tracts are low UHI areas, and 46 percent are in high UHI areas across +Columbus. (B) 31 percent of census tracts in low urban heat areas have trips to high urban heat +census tract with ratio of trips as high as 0.19. (C) 100 percent of census tracts in low urban heat +areas have trips to high urban heat census tract, representing that Columbus is a metropolitan +area with high urban heat escapes and high urban heat escalates. + +0.44, 0.51 +0.51, 0.61 +0.61,0.68 +0.68,0.820.01,0.02 +0.02,0.04 +0.04,0.06 +0.06,0.1143 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 6. Urban Heat Traps and Trips in Pittsburgh Metropolitan Area. (A) shows that 34 +percent of census tracts are low UHI areas, and 32 percent are in high UHI areas across +Pittsburgh. (B) 49 percent of census tracts in low urban heat areas have trips to high urban heat +census tract with ratio of trips as high as 0.22. (C) 88 percent of census tracts in low urban heat +areas have trips to high urban heat census tract, representing that Pittsburgh is a metropolitan +area with high urban heat escalates and high urban heat traps. + +-2.21, 0.24 +0.24,1.84 +1.84,5.010.01,0.04 +0.04, 0.07 +0.07,0.12 +0.12,0.220.29, 0.48 +0.48,0.60 +0.60, 0.67 +0.67,0.850.02,0.06 +0.06, 0.11 +0.11,0.18 +0.18,0.3144 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 7. Urban Heat Traps and Trips in Philadelphia Metropolitan Area. (A) shows that 28 +percent of census tracts are low UHI areas, and 29 percent are in high UHI areas across +Philadelphia. (C) 85 percent of census tracts in low urban heat areas have trips to high urban heat +census tract, representing that Philadelphia is a metropolitan area with high urban heat traps. + +-2.46, 0.94 +0.94,3.37 +3.37,6.960.05, 0.10 +0.10,0.14 +0.14,0.20 +0.20, 0.260.47, 0.61 +0.61,0.70 +0.70,0.77 +0.77,0.870.00, 0.00 +0.00,0.01 +0.01,0.01 +0.01,0.0345 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 8. Urban Heat Traps and Trips in Memphis Metropolitan Area. (A) shows that 23 +percent of census tracts are low UHI areas, and 42 percent are in high UHI areas across +Memphis. (B) 98 percent of census tracts in low urban heat areas have trips to high urban heat +census tract with ratio of trips as high as 0.21 (C)100 percent of census tracts in low urban heat +areas have trips to high urban heat census tract, representing that Memphis is a metropolitan area +with high urban heat escalates and high urban heat traps. + +-3.50,-0.23 +-0.23,1.63 +1.63,3.820.02,0.07 +0.07,0.11 +0.11,0.15 +0.15,0.210.30,0.44 +0.44,0.59 +0.59,0.68 +0.68,0.800.02,0.05 +0.05,0.09 +0.09,0.14 +0.14,0.2046 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 9. Urban Heat Traps and Trips in Orlando Area. (A) shows that 19 percent of census +tracts are low UHI areas, and 33 percent are in high UHI areas across Orlando. (B) 45 percent of +census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips +as high as 0.24 (C)89 percent of census tracts in low urban heat areas have trips to high urban +heat census tract, representing that Orlando is a metropolitan area with high urban heat escalates +and high urban heat traps. + + + + +4.27,-0.43 +-0.43,1.60 +1.60,4.290.02,0.06 +0.06,0.11 +0.11,0.16 +0.16,0.240.38, 0.52 +0.52,0.62 +0.62,0.69 +0.69,0.830.02,0.04 +0.04,0.06 +0.06,0.12 +0.12,0.1947 + + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +4.59,0.24 +0.24,2.74 +2.74,5.790.05, 0.10 +0.10, 0.15 +0.15, 0.22 +0.22, 0.300.38, 0.55 +0.55, 0.64 +0.64, 0.73 +0.73, 0.850.01, 0.03 +0.03, 0.05 +0.05, 0.08 +0.08, 0.1548 + + +Figure A - 10. Urban Heat Traps and Trips in Miami Area(A) shows that 23 percent of census +tracts are low UHI areas, and 43 percent are in high UHI areas across Miami. (B) 42 percent of +census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips +as high as 0.30 (C)57 percent of census tracts in low urban heat areas have trips to high urban +heat census tract, representing that Miami is a metropolitan area with high urban heat escalates +and high urban heat traps. + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +-5.61, -0.04 +0.04,2.19 +2.19,5.330.01,0.05 +0.05, 0.10 +0.10,0.16 +0.16, 0.2349 + + + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 11. Urban Heat Traps and Trips in Seattle Area. (A) shows that 23 percent of census +tracts are low UHI areas, and 30 percent are in high UHI areas across Seattle. (C)60 percent of +census tracts in low urban heat areas have trips to high urban heat census tract, representing that +Seattle is a metropolitan area with low urban heat traps. + +0.25, 0.42 +0.42, 0.55 +0.55, 0.64 +0.64, 0.750.02,0.04 +0.04, 0.07 +0.07,0.11 +0.11, 0.1750 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 12. Urban Heat Traps and Trips in Rochester Area. (A) shows that 8 percent of +census tracts are low UHI areas, and 50 percent are in high UHI areas across Rochester. (B) 71 +percent of census tracts in low urban heat areas have trips to high urban heat census tract with +ratio of trips as high as 0.19 (C) 100 percent of census tracts in low urban heat areas have trips to + +-7.48, -1.82 +-1.82, 2.13 +2.13, +6.150.06, 0.06 +0.06, 0.08 +0.08, 0.11 +0.11, 0.190.46, 0.55 +0.55, 0.65 +0.65, 0.72 +0.72, 0.820.00, 0.01 +0.01, 0.02 +0.02,0.03 +0.03, 0.0451 + +high urban heat census tract, representing that Rochester is a metropolitan area with high urban +heat escalates and high urban heat traps. + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +Figure A - 13. Urban Heat Traps and Trips in Portland Area. (A) shows that 17 percent of +census tracts are low UHI areas, and 53 percent are in high UHI areas across Portland. (B) 43 +percent of census tracts in low urban heat areas have trips to high urban heat census tract with +ratio of trips as high as 0.28. (C)60 percent of census tracts in high urban heat areas have trips to +high urban heat census tract, representing that Portland is a metropolitan area with high urban +heat escalates and high urban heat traps. + + +4.22,-0.05 +-0.05,2.17 +2.17,4.360.05,0.08 +0.08,0.12 +0.12,0.18 +0.18,0.280.50,0.60 +0.60,0.69 +0.69, 0.76 +0.76,0.850.01,0.03 +0.03,0.05 +0.05,0.07 +0.07,0.1252 + + +(A) Distribution of urban heat + (B) The ratio of trips from low UHI to high UHI + +(C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI + +-3.84,-0.19 +-0.19,1.52 +1.52,3.480.07, 0.07 +0.07,0.10 +0.10, 0.12 +0.12,0.130.56, 0.65 +0.65,0.73 +0.73, 0.80 +0.80,0.890.00, 0.00 +0.00,0.01 +0.01,0.01 +0.01,0.0253 + +Figure A - 14. Urban Heat Traps and Trips in Denver Area. (A) shows that 17 percent of census +tracts are low UHI areas, and 38 percent are in high UHI areas across Denver. (C) 44 percent of +census tracts in high urban heat areas have trips to high urban heat census tract, representing that +Denver is a metropolitan area with low urban heat traps. + +Appendix B. Urban Centrality Index, Income, White, and Non-white Gini indices + +Table B. Urban Centrality index (UCI), Spatial distribution of urban heat index, Income, White, +and None-white Gini indices in each metropolitan area. +MSA +UHI Spatial +Gini +UCI +Income +Gini +White Gini +Non-white +Gini +Atlanta, GA +0.76 +0.49 +0.47 +0.71 +0.72 +Boston, MA +0.59 +0.32 +0.48 +0.59 +0.53 +Chicago, IL +0.56 +0.42 +0.48 +0.66 +0.65 +Columbus, OH +0.50 +0.56 +0.46 +0.54 +0.58 +Dallas. TX +0.56 +0.51 +0.47 +0.50 +0.45 +Denver, CO +0.71 +0.75 +0.45 +0.50 +0.49 +Detroit, MI +0.48 +0.40 +0.47 +0.76 +0.80 + +54 + +Houston, TX +0.39 +0.50 +0.48 +0.50 +0.46 +Los Angeles, CA +0.41 +0.42 +0.49 +0.55 +0.48 +Memphis, TN +0.92 +0.64 +0.50 +0.10 +0.22 +Miami, FL +0.55 +0.41 +0.51 +0.48 +0.58 +Minneapolis, MN +0.62 +0.57 +0.44 +0.50 +0.54 +Orlando, FL +0.99 +0.53 +0.47 +0.43 +0.44 +Philadelphia, PA +0.56 +0.35 +0.48 +0.71 +0.68 +Phoenix, AZ +0.85 +0.72 +0.39 +0.48 +0.47 +Pittsburgh, PA +0.80 +0.47 +0.48 +0.53 +0.61 +Portland, OR +0.54 +0.71 +0.45 +0.29 +0.37 +Rochester, NY +0.77 +0.47 +0.46 +0.60 +0.65 +Seattle, WA +0.82 +0.55 +0.47 +0.39 +0.43 +Washington DC +0.52 +0.43 +0.45 +0.66 +0.68 + +UHI Gini index is calculated based on the average UHI indices in each metropolitan area. The +higher the UHI Gini index, the more clustered UHI areas. UCI, also known as the urban + +55 + +centrality index, assesses the centrality of a certain area (city, metropolitan area, region, country, +etc.) on a continuum ranging from extreme monocentric to extreme polycentric (Pereira et al., +2013). UCI values vary from 0 to 1, with 0 expressing the highest level of polycentricity and 1 +the highest level of monocentricity. Income, White, and non-White Gini indices are retrieved +from the American Community Survey database administrated by US Census Bureau ("United +States Census Bureau,") 5-year data. These Gini indices vary from 0 to 1, with 0 representing +perfectly not segregated neighborhoods and 1 representing perfectly segregated neighborhoods. + +Appendix C: Statistical Significance +Table C. Statistical Significance of traps escalates and escapes vs. Urban Centrality Index, +Spatial Distribution of Urban Heat Index, and Income, White, and non-white Gini indices. There +is no statistical significance between traps and demographic segregation. + +UHI Spatial Gini +UCI +Income Gini +White Gini +Non-White Gini +trap +0.01 +-0.24 +0.37 +-0.09 +0.01 +escalade +0.15 +-0.11 +0.05 +-0.08 +0.03 +escape +0.08 +0.1 +-0.14 +0.17 +0.12 + + diff --git a/f9E5T4oBgHgl3EQfhg82/content/tmp_files/load_file.txt b/f9E5T4oBgHgl3EQfhg82/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a578ea742b47321857e819a4249a4466b39a5fd --- /dev/null +++ b/f9E5T4oBgHgl3EQfhg82/content/tmp_files/load_file.txt @@ -0,0 +1,1983 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf,len=1982 +page_content='1 Emergence of Urban Heat Traps from the Intersection of Human Mobility and Heat Hazard Exposure in Cities Xinke Huang1*, Ali Mostafavi2 1 Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Student, Zachry Department of Civil and Environmental Engineering, Texas A&M University, College Station, United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' e-mail: adahuang@tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='edu 2 Associate Professor, Urban Resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='AI Lab Zachry Department of Civil and Environmental Engineering, Texas A&M University, College Station, United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' e-mail: amostafavi@civil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='tamu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='edu Abstract Understanding the relationship between spatial structures of cities and environmental hazard exposures (such as urban heat) is essential for urban health and sustainability planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' However, a critical knowledge gap exists in terms of the extent to which socio-spatial networks shaped by human mobility exacerbate or alleviate urban heat exposures of populations in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In this study, we utilize location-based data to construct human mobility networks in twenty metropolitan areas in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The human mobility networks are analyzed in conjunction with the urban heat characteristics of spatial areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' We identify areas with high and low urban heat exposure and evaluate visitation patterns of populations residing in high and low urban heat areas to other spatial areas with similar and dissimilar urban heat exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The results reveal the presence of urban heat traps in the majority of the studied metropolitan areas in which populations residing in high heat exposure areas primarily visit areas with high heat exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The results also show a small percentage of human mobility to produce urban heat escalate 2 (visitations from low heat areas to high heat areas) and heat escapes (movements from high heat areas to low heat areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The findings from this study provide a better understanding of urban heat exposure in cities based on patterns of human mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These finding contribute to a broader understanding of the intersection of human network dynamics and environmental hazard exposures in cities to inform more integrated urban design and planning to promote health and sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Keywords: urban heat exposure, demographic segregation, income segregation, urban centrality, spatial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Introduction The characterization of the spatial environmental hazards in cities is essential for urban sustainability and health plans and policies (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2011, Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019, Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Among all the environmental hazards, heat is one of the major hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Damages of heat include increased mortality and morbidity due to extremely high air temperatures (Kim & Brown, 2021), stronger heat-related health threats in urban areas (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2016), and increased energy consumption (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' However, comparing to other environmental hazards, such as air pollution, urban heat did not draw enough attention in the existing literature (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022, Glencross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020, Venter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Within the studies of urban heat, limited attentions were paid to human network dynamics that could expand the reach of environmental hazard exposures (Coccia, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Current heat-related studies mostly focused on index-based, which is an isolated measurement of individual locations (Andrade & Szlafsztein, 2018, Jha & Gundimeda, 2019, Orioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Research gap exists in terms of how to understand the 3 spatial distribution of urban heat and people’s respond to the heat from a network-based perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In particular, human mobility shapes the spatial structures of cities and could extend the reach of environmental hazards beyond hazard hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In a recent study, Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022) examined the intersection of human mobility and air pollution exposure and found that human mobility expands the reach of air pollution exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This study highlights the significance of characterizing environmental hazard exposures based on considering human mobility networks in cities (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In the context of urban heat exposure, Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021) proposed a dynamic urban thermal exposure index to account for human mobility in specifying urban heat exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' While the index-based approach proposed by Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021) captures mobility-based heat exposure, it does not capture fundamental properties arising at the intersection of human mobility and spatial heat exposure that extend or alleviate heat exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Recognizing this gap, in this paper, we define and examine three properties at the intersection of urban heat and human mobility (Figure 1): (1) heat traps: in which populations residing in high heat areas visit other high heat areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2) heat escapes: in which populations residing in high heat areas visit low heat areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' and (3) heat escalates: in which populations residing in low heat areas visit high heat areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In fact, these properties are emergent properties arising from the intersection of human mobility networks and the spatial distribution of heat hazards in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Accordingly, the study aims to address the following research questions: to what extent human mobility would exacerbate urban heat exposure (prominence of heat traps), alleviate heat exposure (heat escapes), or expand the reach of heat exposure (heat escalates)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' To address these questions, we utilize aggregated and anonymized location-based data to construct the human mobility network (origin-destination network in which origin is the home census tracts of trips and destination is the visitation census tract of trips) for twenty metropolitan areas in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' to examine the 4 proportion of trips from high heat areas to other high heat areas and low heat areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Accordingly, we analyze the prominence of heat traps, escapes, and escalates across different cities to evaluate cross-city similarities and differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Conceptual representation of urban heat traps, escalates, and escapes arising from the intersection of human mobility and heat exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Background Urban heat (UH), or the urban heat island effect, refers to the phenomenon where urban areas have higher temperatures than surrounding rural areas due to the heat generated by human activity and the lack of vegetation to absorb that heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' To understand and mitigate UH effect, researchers have identified multiple factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' For example, some studies found that tree density is correlated with UH (Ziter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019, Rahman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020, Morabito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021), that high tree density potentially decreases urban heat phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Transportation is another factor, less HighUrbanHeat ModerateUrbanHeat LowUrbanHeat NoUrbanHeat Home CensusTract Escalates Visitation Census Tract Escapes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Traps5 movement of transportation can reduce the extent of changes in temperatures in urban areas (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019, Ali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021, Angelevska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Moreover, population density also contributes to the urban heat effect, population loss can have a mitigating effect on the UH effect (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2018, Manoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' However, those studies focused on examining a single factor with UH, that they ignored the ability for human to adjust living environment by moving to different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Human mobility datasets have been widely used in multiple hazards, including hurricane (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020, Rajput et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020, Dargin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021, Li & Mostafavi, 2022, Paradkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022), flooding (Esparza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022, Farahmand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022a, Farahmand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022b, Mostafavi & Yuan, 2022, Ridha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022, Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022a, Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022b), and infectious diseases (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021, Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021, Rajput et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These studies have found human mobility data was useful to understand people’s reaction to hazards (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' For example, when hurricane comes, people in the similar social media networks were likely to make the same evacuation decisions (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' During the COVID 19 pandemic, the confirmed cases were found highly correlated with human mobility that places with higher activities had more covid cases (Coleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These studies have recognized that people can successfully change the level of hazard exposure by moving to a different location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The majority of human mobility and hazard studies have focused on the relationship between human mobility patterns and the likelihood of exposure to natural hazards, infectious diseases, and environmental pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' However, the current literature does not adequately investigate the 6 relationship between human mobility and urban heat (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In this context, mobility can play a significant role in determining the likelihood of exposure to urban heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Therefore, understanding the relationship between human mobility and UH can be useful in developing strategies to reduce the impact of urban heat on individuals and communities, which is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Data Description Study Context We collected mobility data in February 2020 in twenty metropolitan areas (Table 1) in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' to construct human mobility networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The rationale for selecting February 2020 is that it was just before the start of the COVID-19 pandemic, and the patterns of human mobility would represent the standard patterns of mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Metropolitan Areas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Metropolitan Areas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Phoenix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Arizona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Los Angeles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='California ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Denver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Colorado ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Washington DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='District of Columbia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Orlando ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Florida ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Miami ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Florida ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Atlanta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Georgia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Chicago ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Illinois ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Boston ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Massachusetts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Detroit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Michigan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Minneapolis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Minnesota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Rochester ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='New York ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Columbus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Ohio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Portland ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Oregon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Pittsburgh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Pennsylvania ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Philadelphia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Pennsylvania ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Memphis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Tennessee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Houston ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Texas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Dallas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Texas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Seattle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Washington ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='Data sources The heat exposure data were obtained from the United States Surface Urban Heat Island database (Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' For all census tracts in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' urbanized regions, this dataset includes yearly, summer, and winter daytime and nighttime Land Surface Temperature (LST), Digital Elevation Model (DEM), and Normalized Difference Vegetation Index (NDVI) data, as well as the mean values for the whole urbanized area (Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The UHI dataset in the urbanized areas was determined by remote sensing data, such as Moderate Resolution Imaging Spectroradiometer (MODIS) and Global Multi-Resolution Terrain Elevation Data (GMTED), including 55,871 census tracts organized into 497 urbanized areas, covering roughly 78 percent of the population of the United States (Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Our study used the mean values for Urban Heat Islands (UHIs) as the measurement of UH for the chosen 8 metropolitan areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' We used quantile breaks to split the UHI data into three clusters and defined them as low UHI area, median UHI area, and high UHI area, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The location-based data is provided by Spectus (formerly known as Cuebiq), a platform for mobility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Spectus provides privacy-protected and anonymized location datasets by collecting data from smart devices whose owners have authorized location data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Spectus constructs its geo-location dataset by collaborating with application developers to collect high-resolution datasets using Bluetooth, GPS, WiFi, and IoT signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Each day, more than one hundred data points are gathered for each anonymous user, allowing a more accurate understanding of human movement and visitation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Spectus collects data on around 15 million daily active users in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' High privacy policy standards are set to enable data collection and use of data responsibly and ethically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Users are allowed to opt out of location sharing at any stage, and all information is obtained transparently with consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' All data provided by Spectus is de-identified to ensure anonymity and endures further privacy improvements, such as removing sensitive points of interest and obscuring dwelling locations at the census block group level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In addition to delivering location-based data at the device level, Spectus aggregates data using artificial intelligence and machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=" By offering access to an auditable and on-premise sandbox environment, Spectus' platform for responsible data sharing allows us to query anonymized, aggregated, and privacy-enhanced data (Wang et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=" In this study, we used one of the aggregated datasets from Spectus, the Device Location database, to determine the Census tracts of devices' home locations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The Device Location table includes a timestamp, a privacy-compliant device ID, and geoinformation at the device level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' To evaluate 9 UH exposure, we used population activity in February 2020, which reflects a steady-state period with no events that could affect population activity and movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Methods Mobility network from the home Census tract to the visitation Census tract Data processing consisted of utilizing Spectus data to construct the human mobility network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Specifically, it involves two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' First step is to identify each device’s home tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The second step is to construct the mobility networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A device’s home tract was determined based on its dwell times, as Spectus provides dwell time at each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' By using unique identifiers for each device, Spectus can collect each visitor’s destination tract and aggregate the number of visits from one tract to another tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Accordingly, we construct the monthly mobility network model of each city, which captures the number of visits from home tracts to visitation tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In this network, each node is a tract and the links are number of trips observed between each pair of tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The Ratio of Urban Heat Traps, Escalates, and Escapes In each metropolitan area, we used quantile breaks dividing Census tracts into low UH areas, median UH areas, and high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In this study, we only considered low and high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' We aggregated human mobility dataset to summarize the number of trips between low and high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' As noted earlier, we define heat traps as high UH areas whose populations visit places in other high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Similarly, heat escalates are low UH exposure areas whose populations visit places in high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' And heat escapes are high UH exposure areas whose populations 10 visit places in low UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The ratio of UH traps, escalates, and escapes of each tract is calculated by summing the trips in each category (high to high, low to high, and high to low, respectively) and dividing by the total trips associated with each home tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The ratio of heat escalates is computed using Equation 1: 𝑅𝐿𝑜𝑤𝑖,𝑗 = 𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷ℎ𝑖𝑔ℎ𝑖,𝑗 𝑇𝑂𝑇𝑖 (1) where, 𝑅𝐿𝑜𝑤𝑖,𝑗 refers to the ratio of trips visiting from low UH tract i to high UH j, 𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷ℎ𝑖𝑔ℎ𝑖,𝑗refers to the total number of trips from low UH tract i to high UH tract j, and 𝑇𝑂𝑇𝑖 refers to the total number of trips starting from origin tract i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Similarly, the ratio of trips visiting from high UH tract to low UH tract and the ratio of trips visiting from high UH tract to high UH tracts are computed using Equations 2 and 3, respectively: 𝑅𝐻𝑖𝑔ℎ𝑖,𝑗 = 𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷𝑙𝑜𝑤𝑖,𝑗 𝑇𝑂𝑇𝑖 (2) 𝑅𝐻𝑖𝑔ℎ𝑖,𝑗 = 𝐶𝑒𝑛𝑠𝑢𝑠 𝑇𝑟𝑎𝑐𝑡𝐷ℎ𝑖𝑔ℎ𝑖,𝑗 𝑇𝑂𝑇𝑖 (3) Classifying Cities For each metropolitan area, we first calculated the total number of tracts in high and low UH exposures based on the UH dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Then, we classified cities as heat traps, heat escalates, and heat escapes based on the percentage of trips in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' If more than half of trips in the city were heat trap type, we classified this cities as urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Similarly, if the city has 11 more than half heat escalate trips or heat escape trips, the city was classified as a heat escalate city or heat escape city, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Results Patterns across Cities Table 2 presents the list of metropolitan areas, and their percentage of trips in each category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', high to high, low to high, and high to low).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The high UH and low UH percentages divide the total number of census tracts by the number of census tracts in high UH areas and low UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Note that the total number of census tracts with trips from high to low and with trips from high to high is the same, but the ratio of trips visiting from high UHI census tract i to low UHI census tract j are significantly different (Equation (2) and (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The metropolitan classifications are based on the percentage of low-to-high trips, high-to-low trips, and high-to-high trips, as stated in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Metropolitan areas with the total number of census tracts (CT), different UH visiting patterns count and percentage, and classification of the metropolitan areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' MSA Total # of CT Total # CT in high UHI areas High UHI % Total # CT in low UHI areas Low UHI % Total # CT with trips from low to high Low to high trips % Total # CT with trips from high to low Total # of CT with trips from high to high High to low % High to high % Classifications Atlanta, GA 885 186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21 280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='32 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 75 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='4 trap Boston, MA 947 343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='36 264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='28 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 128 133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37 trap 12 Chicago, IL 1,923 945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='49 301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16 168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='56 739 739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='78 trap Columbus, OH 340 155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='46 67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='31 155 155 1 1 escalate & trap Dallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' TX 1122 575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='51 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='38 282 282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='49 trap & escape DC 179 56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='31 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='27 49 1 56 56 1 1 escalate & trap Denver, CO 581 218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='38 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='17 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 96 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='44 escape Detroit, MI 1,158 658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='57 183 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='95 326 326 1 1 escalate & trap 13 Pittsburgh, PA 599 190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='32 203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='34 99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='49 168 169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='88 escalate & trap Portland, OR 334 178 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='6 escalate & trap Cities with High Urban Heat Traps The Los Angeles metropolitan area shows significant urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 2A maps the UH in Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Three orange shades represent three levels of UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The darker the shade is, the more severe UH was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The metropolitan area has 13 percent of the tracts in low UH areas, mainly located on the north and east, while 52 percent of the metropolitan area is in high UH areas (dark orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 2B to 2D shows the ratio of trips between low UH tracts and high UH tracts, which break into four categories for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Light blue shows a low ratio of trips, and dark blue shows a high ratio of trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' All the following figures are presented in the same plot format as Figure 2A and 2B to 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 2B shows the ratio of trips visiting from low UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A high ratio of low-to-high trips from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='22 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='35 occurs in the north, which means that a significant number of people living in low UH areas are visiting high UH areas in the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 2D shows the ratio of trips from high UH tracts to low UH tracts with a higher ratio of trips, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11, occurring in the northwest and southwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This means that a relatively high number of people living in high 14 UH areas are visiting low UH areas in the northwest and southwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 2C shows the ratio of trips visiting from high UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 81 percent of all the tracts in high UH areas have trips trapped inside high UH areas with the ratio of trips from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='30 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='92, meaning lots of people suffering UH did not move to relief their UH exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These urban heat traps are in the northwest and central of Los Angeles, with an especially high ratio from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='76 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='92 in the central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1115 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Los Angeles Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 52 percent of tracts are in high UH areas across Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 81 percent of tracts in high UH areas have trips to other high UH tracts, representing that Los Angeles is a metropolitan area with urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Similarly, the Chicago metropolitan area shows strong urban heat traps as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 3A maps the UH in Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Chicago has 16 percent of its tract in low UH areas, while 49 percent of its tracts are in high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 3B shows the ratio of trips visiting from low UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A higher ratio of trips 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='17 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24 occurs on the coast of Lake Michigan, meaning that a significant number of people living in low UH areas are visiting high UH areas on the coast of Lake Michigan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 3D shows trips from high UH tracts to low UH tracts with a ratio as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 occurring in the east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 3C shows the ratio of trips visiting from high UH tracts to high UH tracts, with the ratio of trips from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='44 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='91, which means that a large number of people living in high UH areas are visiting other high UH areas within the Chicago metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' About 78 percent of Chicago tracts in high UH areas have trips trapped inside high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Most of the UH traps are in the west of Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' At the same time, central Chicago presents an exceptionally high heat trap ratio, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='79 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' shows the ratio of trips visiting from high UH tracts to low UH tracts, with the ratio of trips from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This means that a relatively low number of people living in high UH areas are visiting low UH areas within the Chicago metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Comparing the UH traps between Chicago and Los Angeles, we can see that the traps in Chicago are clustered in one place, while in Los Angeles are distributed into multiple clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 16 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1317 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UH Traps and Trips in the Chicago Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) 16 percent of tracts are in low UH areas and 49 percent of tracts are in high UH areas across Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 78 percent of tracts in high UH areas have trips to high UH tracts, representing that Chicago is a metropolitan area with urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figures 3 and 4 show that the Los Angeles and Chicago metropolitan areas both have significant urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In Los Angeles, 52 percent of all the tracts are in high UH areas, while in Chicago, 49 percent are in high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The figures also show that trips from low UH areas to high UH areas are more frequent in the north of both cities, while trips from high UH areas to low UH areas are more common in the northwest and southwest of Los Angeles, and the east of Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Additionally, the figures show that both cities present high heat trap trips, with around 80 percent of tracts with heat trap trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This indicates that people in the high UH areas are likely not visiting the low UH areas to escape the heat, but instead are staying in other high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Cities with Low Urban Heat Traps Boston Metropolitan shows low urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 4A maps the UH in Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' About 28 percent of tracts in Boston are in low UH areas, while 36 percent of tracts have high UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Most of these high UH tracts are clustered in central Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 4B shows the ratio of trips visiting from low UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The ratio of such trips is from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19 and only occur in 4 percent of all the tracts with low UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 4C shows the ratio of trips visiting from high UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This ratio ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='30 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' About 37 percent of tracts with high UH have trips trapped inside high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This percentage is relatively small when comparing to Los Angeles (81 percent) and Chicago (78 percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 4D shows the trips 18 from high UH areas to low UH areas with ratio from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These results indicate that people living in low UH areas in the Boston metropolitan area are not frequently visiting high UH areas, which could be an indication of a fewer heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of UH (B) The ratio of trips from low UH to high UH (C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='73,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='17,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='48,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='66, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='77,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='0219 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UH Traps and Trips in Boston Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) 28 percent of tracts are low UH areas and 38 percent are in high UH areas across Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 37 percent of tracts in high UH areas have trips to high UH tracts, representing that Boston is a metropolitan area with low urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Similarly, the Atlanta Metropolitan also shows low UH traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 5A maps the UH in Atlanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Atlanta has 32 percent of the tracts in low UH areas, while 21 percent are in high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 5B shows the ratio of trips visiting from low UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The ratio of such trips is from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16 and only occurred in 13 percent of all the low UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 5C shows the ratio of trips visiting from high UH tracts to high UH tracts with ratios from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' About 40 percent of high UH tracts have heat trap trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This number is similar with Boston and is relatively small comparing to Los Angeles and Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 5D shows the trips from high UH areas to low UH areas, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This ratio is small but more significant than that of Boston, which means that comparing to Boston, more heat escape trips exist in Atlanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 20 (A) Distribution of UH (B) The ratio of trips from low UH to high UH (C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UH Traps and Trips in Atlanta Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) 32 percent of tracts are low UH and 21 percent are high UH across Atlanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C)40 percent of tracts in high UH areas have trips to high UH tracts, representing that Atlanta is a metropolitan area with low UH traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='35,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='31,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='47,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='62,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='71,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1221 Figures 4 and 5 show that both Boston and Atlanta have relatively low UH comparing to Los Angeles and Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In Boston, only 36 percent of tracts are in high UH, while in Atlanta, only 21 percent of the tracts are in high UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The figures also show that trips from low UH areas to high UH areas are relatively rare in both cities, only 4 percent and 13 percent in low UH tracts in Boston and Atlanta, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In both cities, the percentages of tracts with trips trapped inside high UH areas are lower than in Los Angeles and Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Cities with High Urban Heat Escapes The Minneapolis Metropolitan Area shows high UH escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 6A maps the UH in Minneapolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The metropolitan area has 18 percent of its tracts in low UH areas, while 45 percent are in high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 6B shows the ratio of trips from low UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This ratio ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='34, occurring in 26 percent of low UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 6C shows the ratio of trips from high UH tracts to high UH tracts with the ratios from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='41 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 6D shows the ratio of trips from high UH tracts to low UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This ratio is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13, occurring in 56 percent of high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Comparing this high UH to low UH ratio with other cities, Minneapolis shows strong UH escape, indicating that a significant number of people 22 living in high UH areas are visiting low UH areas in the Minneapolis metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of UH (B) The ratio of trips from low UH to high UH (C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UH Traps and Trips in Minneapolis Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) 18 percent of tracts are low UH areas and 45 percent are in high UH areas across Minneapolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (D) 56 percent of tracts 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='27, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='12,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='35,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='26, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='41,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='53,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='66,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='74,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1323 in high UH areas have trips to low UH tracts, representing that Minneapolis has high heat escapes trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Similarly, the Dallas Metropolitan Area also shows high heat escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 7A maps the UH in Dallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Dallas has 10 percent of its tracts in low UH areas, while 50 percent of its tracts are in high UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 7A shows that the high UH tracts form multiple clusters across the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 7B shows the ratio of trips from low UH tracts to high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This ratio is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='28, occurring in 38 percent of the low UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 7C shows the ratio of trips from high UH tracts to high UH tracts with ratios from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='39 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figure 7D shows the ratio of trips from high UH tracts to low UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The ratio of trips from high UH tracts to low UH tracts is notable, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16, in 49 percent of high UH tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This indicate that Dallas has strong urban heat escapes trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of UH (B) The ratio of trips from low UH to high UH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='50,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='50,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2824 (C) The ratio of trips from high UH to high UH (D) The ratio of trips from high UH to low UH Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UH Traps and Trips in the Dallas Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) 10 percent of tracts are low UH areas and 51 percent are in high UH areas across Dallas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (D) 49 percent of high UH tracts have trips to low UH tracts, representing that Dallas is a metropolitan area with high urban heat escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Figures 6 and 7 show that both Minneapolis and Dallas have significant urban heat escapes, with a higher ratio of trips from high UH tracts to low UH tracts when comparing to other metropolitan areas, such as Boston (37 percent) and Atlanta (24 percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This indicates that people in the high UH areas travel to the low UH areas to escape the heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Additionally, this study offered important insights by examining the factors of distinctive characteristics underline spatial structures (Angel & Blei, 2016), facility distribution (Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2013), income, and racial segregation, as in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' However, no statistical significance was found between heat traps and attributes of demographic segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This interpolates that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='39,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='63,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='71,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1625 an urban heat trap is an emergent property (Georgiou, 2003) that cannot be attributed to the centrality of city facilities and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Therefore, we observe that human mobility leads to the creation of traps, not escapes or escalates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Maybe people are more likely to go to places where they are more familiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Discussion and Concluding Remarks This study utilized large-scale, high-resolution location-intelligence data to identify and quantify the urban heat (UH) exposure and people’s response based on human mobility networks in urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This study analyzed the intersection of UH and human mobility by examining the UH dataset and trips between tracts in February 2020 in twenty metropolitan areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The study identified and analyzed three properties: heat traps, heat escapes, and heat escalate by quantifying the trips between tracts in high UH areas and low UH areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This study found that not many cities have heat escapes or heat escalates trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Heat escapes were found in Minneapolis and heat escalates were found in Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A potential reason might be that people are more likely to stay in their resident areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Researchers and professionals are well aware of the diverse effects that UH can have heat-related diseases, such as respiratory difficulties among urban populations (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' However, there is little knowledge about the extent to which human mobility exacerbates UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This study offers an innovative, data-driven method and metrics for using large-scale location intelligence data to assess UH exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This study evaluates the intersection of human mobility and the spatial distribution of urban heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' In addition, this study defines three important characteristics of people’s potential response to UH based on trip destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Specifically, heat traps refer to 26 population residing in high UH areas visit other high UH areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' heat escalates refer to population residing in low UH areas visit high UH areas and thus escalate their heat exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' and heat escapes refer to population residing in high UH areas visit low UH areas and thus escape from their local heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Defining these three different responses to UH can help researchers understand different characteristics of the urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' There are several limitations of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' First, this study is based on smartphone data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Smartphone users who allowed such location data collection is a biased sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Visitors who do not own smartphones, such as children, teenagers, the elderly, and those with lower income, were less likely to be included in the data, which may create biases (Esmalian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2021, Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Additionally, efforts could be made to ensure that the sample of smartphone users is representative of the population as a whole, such as by using stratified sampling or weighting the data to account for any biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' We partially address this limitation by utilizing Spectus data, which has been demonstrated to contain a representative sample of users (Li & Mostafavi, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Second, the mobility data does not include the visiting time for the destinations, which may cause mis-labeling of trip purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Future researchers could leverage other sources of data, such as surveys or observational studies, to further validate traveling information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' This study offers important insights to city designers and city planners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The three important characteristics of traps, escalates, and escapes are likely related to how heat exposure can affect people in different parts of a city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Better understandings of people’s movements and associated heat exposure can provide city planner information for future city development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These characteristics may include factors such as the availability of shade and other forms of shelter, 27 the accessibility of air conditioning and other cooling mechanisms, and the presence of social networks and support systems that can help people cope with heat waves and other extreme weather events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' By understanding these characteristics, it may be possible to develop strategies and interventions that can help reduce the risks associated with heat exposure in urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Data Availability All data were collected through a CCPA- and GDPR-compliant framework and utilized for research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The data that support the findings of this study are available from Spectus, but restrictions apply to the availability of these data, which were used under license for the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The data can be accessed upon request submitted on spectus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Other data we use in this study are all publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Code Availability The code that supports the findings of this study is available from the corresponding author upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Declaration of Interests: none Acknowledgement This material is based in part upon work supported by the National Science Foundation under Grant CMMI-1846069 (CAREER), Texas A&M University X-Grant 699, and the Microsoft Azure AI for Public Health grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The authors also would like to acknowledge the data support 28 from Spectus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation, Texas A&M University, Microsoft Azure, or Spectus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Reference Ali, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Abbas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Qamer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Wong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Rasul, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Irteza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Shahzad, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environmental impacts of shifts in energy, emissions, and urban heat island during the COVID-19 lockdown across Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Journal of Cleaner Production, 291, 125806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='jclepro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='125806 Andrade, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' de, & Szlafsztein, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Vulnerability assessment including tangible and intangible components in the index composition: An Amazon case study of flooding and flash flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Science of The Total Environment, 630, 903–912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='271 Angel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Blei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The spatial structure of American cities: The great majority of workplaces are no longer in CBDs, employment sub-centers, or live-work communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Cities, 51, 21–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='031 Angelevska, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Atanasova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Andreevski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban air quality guidance based on measures categorization in road transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Civil Engineering Journal, 7(2), 253–267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='28991/cej-2021-03091651 Bao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Does lockdown reduce air pollution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Evidence from 44 cities in northern China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Science of The Total Environment, 731, 139052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='139052 29 Chakraborty, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' United states surface urban heat island database [Data set].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Mendeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='17632/X9MV4KRNM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2 Chakraborty, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Hsu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Manya, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Sheriff, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A spatially explicit surface urban heat island database for the United States: Characterization, uncertainties, and possible applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing, 168, 74–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='isprsjprs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='021 Coccia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' An index to quantify environmental risk of exposure to future epidemics of the COVID-19 and similar viral agents: Theory and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environmental Research, 191, 110155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05797 30 Fan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Chien, Y.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Fine-grained data reveal segregated mobility networks and opportunities for local containment of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Scientific Reports, 11(1), 16895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1038/s41598-021-95894-8 Farahmand, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Dong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A network observability framework for sensor placement in flood control networks to improve flood situational awareness and risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Reliability Engineering & System Safety, 221, 108366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='ress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='108366 Farahmand, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Maron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Anomalous human activity fluctuations from digital trace data signal flood inundation status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environment and Planning B: Urban Analytics and City Science, 49(7), 1893–1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1177/23998083211069990 Georgiou, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The idea of emergent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Journal of the Operational Research Society, 54(3), 239–247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1057/palgrave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='jors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2601520 Glencross, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Ho, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Camiña, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Hawrylowicz, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Pfeffer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Air pollution and its effects on the immune system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Free Radical Biology and Medicine, 151, 56–68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='freeradbiomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='179 Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Hou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Jia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Comparison of surface and canopy urban heat islands within megacities of eastern China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing, 156, 160–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='isprsjprs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='012 31 Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zeng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhang, L.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='18502/ijph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='v48i9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='3023 Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Porter, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Twitter reveals human mobility dynamics during the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' PLOS ONE, 15(11), e0241957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='0241957 Hunter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Cleland, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Cleary, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Droomers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Wheeler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Sinnett, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Nieuwenhuijsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Braubach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environmental, health, wellbeing, social and equity effects of urban green space interventions: A meta-narrative evidence synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environment International, 130, 104923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='envint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='104923 Jha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Gundimeda, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' An integrated assessment of vulnerability to floods using composite index – A district level analysis for Bihar, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' International Journal of Disaster Risk Reduction, 35, 101074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='ijdrr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='101074 Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Cutter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Social network, activity space, sentiment, and evacuation: What can social media tell us?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Annals of the American Association of Geographers, 109(6), 1795–1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1080/24694452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1592660 Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban heat island (Uhi) intensity and magnitude estimations: A systematic literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Science of The Total Environment, 779, 146389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='146389 Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Yeo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Estimating urban spatial temperatures considering anthropogenic heat release factors focusing on the mobility characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Sustainable Cities and Society, 85, 104073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='104073 32 Lai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Farnham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Ruktanonchai, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Tatem, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Measuring mobility, disease connectivity and individual risk: a review of using mobile phone data and mHealth for travel medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Journal of travel medicine, 26(3), taz019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Gao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Changes in wind speed under heat waves enhance urban heat islands in the beijing metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Journal of Applied Meteorology and Climatology, 55(11), 2369–2375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1175/JAMC-D-16-0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1 Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Hannibal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Berke, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Woodruff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Vedlitz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Examining of the actor collaboration networks around hazard mitigation: A hurricane harvey study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Natural Hazards, 103(3), 3541–3562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1007/s11069-020-04142-1 Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Xiao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Oztekin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Fan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Fan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Attributed network embedding model for exposing covid-19 spread trajectory archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09448 Manoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Fatichi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Schläpfer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Crowther, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Meili, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Burlando, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Katul, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Bou-Zeid, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Magnitude of urban heat islands largely explained by climate and population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Nature, 573(7772), 55–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1038/s41586-019-1512-9 Morabito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Crisci, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Guerri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Messeri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Congedo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Munafò, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Surface urban heat islands in Italian metropolitan cities: Tree cover and impervious surface 33 influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Science of The Total Environment, 751, 142334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='142334 Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Yuan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Smart flood resilience: Harnessing community-scale big data for predictive flood risk monitoring, rapid impact assessment, and situational awareness (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' EGU22-781).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Copernicus Meetings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='5194/egusphere-egu22-781 Orioli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Antonucci, C.' metadata={'source': 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Michelozzi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Forastiere, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Cesaroni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Exposure to residential greenness as a predictor of cause-specific mortality and stroke incidence in the rome longitudinal study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environmental Health Perspectives, 127(2), 027002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1289/EHP2854 Peng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Fu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' 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Albuquerque, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban centrality: A simple index: urban centrality: a simple index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Geographical Analysis, 45(1), 77–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1111/gean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='12002 Rahman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Stratopoulos, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Moser-Reischl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zölch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Häberle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Rötzer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Pretzsch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Pauleit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Traits of trees for cooling urban heat islands: A meta- analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Building and Environment, 170, 106606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='buildenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='106606 Rajput, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Gao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Temporal network analysis of inter- organizational communications on social media during disasters: A study of Hurricane Harvey in Houston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' International Journal of Disaster Risk Reduction, 46, 101622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='ijdrr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='101622 Ridha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Ross, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Climate change impacts on infrastructure: Flood risk perceptions and evaluations of water systems in coastal urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' International Journal of Disaster Risk Reduction, 73, 102883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='ijdrr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='102883 Seo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Association between urban green space and the risk of cardiovascular disease: A longitudinal study in seven Korean metropolitan areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environment International, 125, 51–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='envint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='038 Shen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Jorge Ochoa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Shah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The application of urban sustainability indicators – A comparison between various practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Habitat International, 35(1), 17–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='habitatint.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Quinn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Neal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' New estimates of flood exposure in developing countries using high-resolution population data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Nature Communications, 10(1), 1814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1038/s41467-019-09282-y Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Newman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Ye, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Factors influencing long-term city park visitations for mid-sized US cities: A big data study using smartphone user mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Sustainable Cities and Society, 80, 103815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='103815 Venter, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Aunan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Chowdhury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Lelieveld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' COVID-19 lockdowns cause global air pollution declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 117(32), 18984–18990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2006853117 35 Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Ban, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (Jeff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Extracting trips from multi- sourced data for mobility pattern analysis: An app-based data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies, 105, 183–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='trc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='028 Xie, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Abdelhady, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Harvey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Laboratorial investigation on optical and thermal properties of cool pavement nano-coatings for urban heat island mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Building and Environment, 147, 231–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='buildenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='017 Yin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Grundstein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Mishra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Ramaswamy, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Hashemi Tonekaboni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Dowd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' DTEx: A dynamic urban thermal exposure index based on human mobility patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environment International, 155, 106573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='envint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='106573 Yuan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Fan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Farahmand, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Coleman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Esmalian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Patrascu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Dong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Smart flood resilience: Harnessing community-scale big data for predictive flood risk monitoring, rapid impact assessment, and situational awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Environmental Research: Infrastructure and Sustainability, 2(2), 025006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1088/2634-4505/ac7251 Yuan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Mostafavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Spatio-temporal graph convolutional networks for road network inundation status prediction during urban flooding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Computers, Environment and Urban Systems, 97, 101870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='compenvurbsys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='101870 Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Bonafoni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Remote sensing of the urban heat island effect in a highly populated urban agglomeration area in East China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Science of The Total Environment, 628–629, 415–429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='scitotenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='074 36 Ziter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Pedersen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', Kucharik, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', & Turner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Scale-dependent interactions between tree canopy cover and impervious surfaces reduce daytime urban heat during summer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 116(15), 7575–7580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1817561116 Appendices Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat and Human Mobility Ratios in Cities (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='43,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='58,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='34, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='73,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2137 (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Houston Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 14 percent of census tracts are low UHI areas, and 48 percent are in high UHI areas across Houston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 93 percent of census tracts in high urban heat areas have trips to high urban heat census tract, representing that Houston is a metropolitan area with high urban heat escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='72,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='18,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2738 (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Detroit Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 16 percent of census tracts are low UHI areas, and 57 percent are in high UHI areas across Detroit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 62 percent of census tracts in high urban heat areas have trips to high urban heat census tract, representing that Detroit is a metropolitan area with high urban heat escapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='41,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='73 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2139 (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Phoenix Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 18 percent of census tracts are low UHI areas, and 37 percent are in high UHI areas across Phoenix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 95 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 100 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Phoenix is a metropolitan area with high urban heat escapes and high urban heat escalates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='28,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1740 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='45, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='89, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='34, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2541 Figure A - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Washington DC Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 27 percent of census tracts are low UHI areas, and 31 percent are in high UHI areas across Washington DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 100 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 100 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Washington DC is a metropolitan area with high urban heat escapes and high urban heat escalates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='98,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='32,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1942 (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Columbus Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 20 percent of census tracts are low UHI areas, and 46 percent are in high UHI areas across Columbus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 31 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 100 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Columbus is a metropolitan area with high urban heat escapes and high urban heat escalates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='44, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='51, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1143 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Pittsburgh Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 34 percent of census tracts are low UHI areas, and 32 percent are in high UHI areas across Pittsburgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 49 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 88 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Pittsburgh is a metropolitan area with high urban heat escalates and high urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24,1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='18,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='3144 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Philadelphia Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 28 percent of census tracts are low UHI areas, and 29 percent are in high UHI areas across Philadelphia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 85 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Philadelphia is a metropolitan area with high urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='0345 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Memphis Metropolitan Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 23 percent of census tracts are low UHI areas, and 42 percent are in high UHI areas across Memphis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 98 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='21 (C)100 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Memphis is a metropolitan area with high urban heat escalates and high urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='23,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='63,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='15,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='44,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='59,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2046 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Orlando Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 19 percent of census tracts are low UHI areas, and 33 percent are in high UHI areas across Orlando.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 45 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24 (C)89 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Orlando is a metropolitan area with high urban heat escalates and high urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='27, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='43,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='38, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='62,0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='74,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10, 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1548 Figure A - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Miami Area(A) shows that 23 percent of census tracts are low UHI areas, and 43 percent are in high UHI areas across Miami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 42 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='30 (C)57 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Miami is a metropolitan area with high urban heat escalates and high urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='61, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='10,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='2349 (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Seattle Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 23 percent of census tracts are low UHI areas, and 30 percent are in high UHI areas across Seattle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C)60 percent of census tracts in low urban heat areas have trips to high urban heat census tract, representing that Seattle is a metropolitan area with low urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='42, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='64, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='07,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='1750 (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Rochester Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 8 percent of census tracts are low UHI areas, and 50 percent are in high UHI areas across Rochester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 71 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19 (C) 100 percent of census tracts in low urban heat areas have trips to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='48, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='82, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='13, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='06, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='08 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='0451 high urban heat census tract, representing that Rochester is a metropolitan area with high urban heat escalates and high urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) Distribution of urban heat (B) The ratio of trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI Figure A - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Portland Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 17 percent of census tracts are low UHI areas, and 53 percent are in high UHI areas across Portland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (B) 43 percent of census tracts in low urban heat areas have trips to high urban heat census tract with ratio of trips as high as 0.' 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trips from low UHI to high UHI (C) The ratio of trips from high UHI to high UHI (D) The ratio of trips from high UHI to low UHI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='84, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='19,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='480.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='80,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='0253 Figure A - 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Heat Traps and Trips in Denver Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (A) shows that 17 percent of census tracts are low UHI areas, and 38 percent are in high UHI areas across Denver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' (C) 44 percent of census tracts in high urban heat areas have trips to high urban heat census tract, representing that Denver is a metropolitan area with low urban heat traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Centrality Index, Income, White, and Non-white Gini indices Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Centrality index (UCI), Spatial distribution of urban heat index, Income, White, and None-white Gini indices in each metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' MSA UHI Spatial Gini UCI Income Gini White Gini Non-white Gini Atlanta, GA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='72 Boston, MA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='32 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='43 Washington DC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='68 UHI Gini index is calculated based on the average UHI indices in each metropolitan area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' The higher the UHI Gini index, the more clustered UHI areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UCI, also known as the urban 55 centrality index, assesses the centrality of a certain area (city, metropolitan area, region, country, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=') on a continuum ranging from extreme monocentric to extreme polycentric (Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UCI values vary from 0 to 1, with 0 expressing the highest level of polycentricity and 1 the highest level of monocentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Income, White, and non-White Gini indices are retrieved from the American Community Survey database administrated by US Census Bureau ("United States Census Bureau,") 5-year data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' These Gini indices vary from 0 to 1, with 0 representing perfectly not segregated neighborhoods and 1 representing perfectly segregated neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Appendix C: Statistical Significance Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Statistical Significance of traps escalates and escapes vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' Urban Centrality Index, Spatial Distribution of Urban Heat Index, and Income, White, and non-white Gini indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' There is no statistical significance between traps and demographic segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content=' UHI Spatial Gini UCI Income Gini White Gini Non White Gini trap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E5T4oBgHgl3EQfhg82/content/2301.05641v1.pdf'} +page_content='09 0.' metadata={'source': 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+Published as a conference paper at ICDE 2023 +Towards Long-Term Time-Series Forecasting: +Feature, Pattern, and Distribution +Yan Li§ † ∗, Xinjiang Lu† �, Haoyi Xiong†, Jian Tang¶ ♮, Jiantao Su♮, Bo Jin♯, Dejing Dou† +†Baidu Research, §Zhejiang University, ¶Tsinghua University, +♮China Longyuan Power Group Corp. Ltd., ♯Dalian University of Technology +ly21121@zju.edu.cn, {luxinjiang,xionghaoyi}@baidu.com, {12101779,12091329}@chnenergy.com.cn, +jinbo@dlut.edu.cn, dejingdou@gmail.com +Abstract—Long-term time-series forecasting (LTTF) has be- +come a pressing demand in many applications, such as wind +power supply planning. Transformer models have been adopted +to deliver high prediction capacity because of the high compu- +tational self-attention mechanism. Though one could lower the +complexity of Transformers by inducing the sparsity in point-wise +self-attentions for LTTF, the limited information utilization pro- +hibits the model from exploring the complex dependencies com- +prehensively. To this end, we propose an efficient Transformer- +based model, named Conformer, which differentiates itself from +existing methods for LTTF in three aspects: (i) an encoder- +decoder architecture incorporating a linear complexity without +sacrificing information utilization is proposed on top of sliding- +window attention and Stationary and Instant Recurrent Network +(SIRN); (ii) a module derived from the normalizing flow is devised +to further improve the information utilization by inferring the +outputs with the latent variables in SIRN directly; (iii) the +inter-series correlation and temporal dynamics in time-series +data are modeled explicitly to fuel the downstream self-attention +mechanism. Extensive experiments on seven real-world datasets +demonstrate that Conformer outperforms the state-of-the-art +methods on LTTF and generates reliable prediction results with +uncertainty quantification. +Index Terms—Long-term time-series forecasting, Transformer, +Normalizing Flow +I. INTRODUCTION +Time-series data evolve over time, which can result in +perplexing time evolution patterns over the short- and long- +term. The time evolution nature of time-series data is of great +interest to many downstream tasks including time-series classi- +fication, outlier detection, and time-series forecasting. Among +these tasks, time-series forecasting (TF) has attracted many +researchers and practitioners in a wide range of application +domains, such as transportation and urban planning [1], energy +and smart grid management [2], as well as weather [3] and +disease propagation analysis [4]. +In many real-world application scenarios, given a substantial +amount of time-series data recorded, there is a necessity +to make a decision in advance, such that, with long-term +prediction, the benefits can be maximized while the potential +risks can be avoided. Therefore, in this work, we study the +problem of forecasting time series that looks far into the future, +namely long-term time-series forecasting (LTTF). +∗This work was done when the first author was an intern at Baidu Research +under the supervision of the second author. +While tons of TF methods [5]–[8] have been proposed with +statistical learners, the use of domain knowledge however +seems indispensable to model the temporal dependencies for +TF but also limits the potential in applications. Recently, +deep models [9]–[13] have been proposed for TF, which +can be categorized into two types: the RNN-based and the +Transformer-based models. RNN-based methods capture and +utilize long- and short-term temporal dependencies to make +the prediction, but fail to deliver good performance in long- +term time-series forecasting tasks. Transformer-based models +have achieved promising performance in extracting temporal +patterns for LTTF because of the usage of self-attention +mechanisms. However, such “full” attention mechanisms bring +quadratic computation complexity for TF tasks, which thus +becomes the main bottleneck for Transformer-based models +to solve the long-term time-series forecasting task. +Several works have been devoted to improving the computa- +tion efficiency of self-attention mechanisms and lowering the +complexity of handling a length-L sequence to (O(L log L) +or O(L +√ +L)), such as Logtrans [14], Reformer [12], In- +former [15] and Autoformer [13]. In the NLP field, some +pioneering works have been proposed to reduce the complexity +of self-attention to linear (O(L)), including Longformer [16] +and BigBird [17]. However, these deep models with a linear +complexity might limit the information utilization and strain +the performance of LTTF. Lowering the computational com- +plexity to O(L) without sacrificing information utilization is +a big challenge for LTTF. +In addition to the complexity, as the input length climbs up, +the intricate time-series could exhibit obscure and confusing +temporal patterns, which may lead to unstable prediction +for self-attention-based models. Moreover, multivariate long- +term time-series often embody multiple temporal patterns at +different temporal resolutions, e.g., seconds, minutes, hours, +or days. On the other hand, the intricate and prevailing multi- +dimensional characteristics of the time-series data exhibit +multi-faceted complex correlations among different series. +Therefore, how to make the prediction for LTTF more stable +and disaggregate multiscale dynamics and multivariate depen- +dencies in time-series data are two more challenges. +To this end, our work devotes to the above three challenges +and proposes a novel model based on Transformer for LTTF, +namely Conformer. In particular, Conformer first explicitly ex- +1 +arXiv:2301.02068v1 [cs.LG] 5 Jan 2023 + +Published as a conference paper at ICDE 2023 +plores the inter-series correlations and temporal dependencies +with Fast Fourier Transform (FFT) plus multiscale dynamics +extraction. Then, to address the LTTF problem in a sequence- +to-sequence manner with linear computational complexity, +an encoder-decoder architecture is employed on top of the +sliding-window self-attention mechanism and the proposed +stationary and instant recurrent network (namely, SIRN). More +specifically, the sliding-window attention allows each point to +attend to its local neighbors for reference, such that the self- +attention dedicated to a length-L time-series requires the O(L) +complexity. Besides, to explore global signals in time-series +data without violating the linear complexity, we renovate the +cycle structure of the recurrent neural network (RNN) and +distill stationary and instant patterns in long-term time-series +with the series decomposition model in a recurrent way. +Moreover, to relieve the fluctuation effect caused by the +aleatoric uncertainty [18] of time series data and improve +the prediction reliability for LTTF, we further put efforts to +model the underlying distribution of time-series data. To be +specific, we devise a normalizing flow block to absorb latent +states yielded in the SIRN model and generate the distribution +of future series directly. More specifically, we leverage the +outcome latent state of the encoder, as well as the latent +state of the decoder, as input to initiate the normalizing flow. +Afterward, the latent state of the decoder can be cascaded to +infer the distribution of the target series. Along this line, the +information utilization for LTTF can be further enhanced and +the time-series forecasting can be implemented in a generative +fashion, which is more noise-resistant. +Extensive experiments on seven real-world datasets validate +that Conformer outperforms the state-of-the-art (SOTA) base- +lines with satisfactory margins. To sum up, our contributions +can be highlighted as follows: +• We reduce the complexity of self-attention to O(L) without +sacrificing prediction capacity with the help of windowed +attention and the renovated recurrent network. +• We design a normalizing flow block to infer target series +from hidden states directly, which can further improve the +prediction and equip the output with uncertainty awareness. +• Extensive experiments on five benchmark datasets and two +collected datasets validate the superior long-term time- +series forecasting performance of Conformer. +II. RELATED WORK +A. Methods for Time-Series Forecasting +Many statistical methods have achieved big success in time- +series forecasting (TF). For instance, ARIMA [5] is flexible +to subsume multiple types of time-series but the limited scal- +ability strains its further applications. Vector Autoregression +(VAR) [6], [7] makes significant progress in multivariate TF by +discovering dependencies between high-dimensional variables. +Besides, there exist other traditional methods for the TF +problem, such as SVR [8], SVM [19], etc., which also play +important roles in different fields. +Another line of studies focuses on deep learning methods +for TF, including RNN- and CNN-based models. For example, +LSTM [20] and GRU [21] show their strengths in extracting +the long- and short-term dependencies, LSTNet [1] combines +the CNN and RNN to capture temporal dependencies in the +time-series data, DeepAR [9] utilizes the autoregressive model, +as well as the RNN, to model the distribution of future +time-series. There are also some works focusing on CNN +models [22]–[25], which can capture inner patterns of the +time-series data through convolution. +The Transformer [26] has shown its great superiority in NLP +problems because of its effective self-attention mechanism, +and it has been extended to many different fields successfully. +There are many attempts to apply the Transformer to TF tasks, +and the main idea lies in aiming to break the bottleneck of +efficiency by focusing on the sparsity of the self-attention +mechanism. The LogSparse Transformer [14] allows each +point to attend to itself and its previous points with exponential +step size, Reformer [12] explores the hashing self-attention, +Informer [15] utilizes probability estimation to reduce the +time and memory complexities, Autoformer [13] studies the +auto-correlation mechanism in place of self-attention. All +the above models reduce the complexity of self-attention to +O(L log L). The Sparse Transformer [27] reduces the com- +plexity to O(L +√ +L) with attention matrix factorization. The +very recent Longformer [16] and BigBird [17] adopt a number +of attention patterns and can further reduce the complexity to +O(L). However, the above reduction of complexity is often +at the expense of sacrificing information utilization and the +self-attention mechanism might not be reliable when temporal +patterns are intricate in the LTTF task. +B. Generative Models +There are works attempting to learn the distribution of +future time-series data. Gaussian mixture model (GMM) [28] +can learn the complex probability distribution with the EM +algorithm, but it fails to suit dynamic scenarios. Wu et al. [29] +proposed a generative model for TF by using the dynamic +Gaussian mixture. [30] devises an end-to-end model to make +coherent and probabilistic forecasts by generating the distribu- +tion of parameters. In addition, the authors of [31] proposed +an autoregressive model to learn the distribution of the data +and make the probabilistic prediction. +The variational inference was proposed for generative mod- +eling and introduced latent variables to explain the observed +data [32], which provides more flexibility in the inference. +Both GAN [33] and VAE [34] show their impressive perfor- +mances in distribution inference, but the cumbersome training +process plus the limited generalization to new data hinder them +for wider applications. Normalizing Flows (NFs) are a family +of generative models, an NF is the transformation of a simple +distribution that results in a more complex distribution. NF +models have been applied in many fields successfully to learn +intractable distribution, including image generation, noise +modeling, video generation, audio generation, etc. Conformer +employs the NF as an inner block for LTTF to absorb latent +states in the encoder-decoder architecture, which differentiates +itself from prior works. +2 + +Published as a conference paper at ICDE 2023 +Fig. 1: The framework overview of Conformer. In particular, the encoder extracts local patterns with sliding-window multi-head +attention (MHA) and explores long-term trends and instant patterns with the proposed SIRN module. The decoder then receives +long sequence inputs with the target elements being padded into zeros, measures the weighted composition of multi-faceted +temporal patterns, and generates the prediction for target elements. At last, the normalizing flow block absorbs latent states +yielded in the encoder-decoder architecture and predicts target elements with a chain of invertible transformations directly. +III. PROBLEM STATEMENT +We introduce the problem definition in this section. Given +a length-L time-series X = {x1, x2, · · · xL| xi ∈ Rdx} where +xi is not limited to the univariate case (i.e., dx ≥ 1), the +time series forecasting problem takes a length-Lx time-series +X = {xm+1, · · · , xm+Lx} as input to predict the future +length-Ly time series Y = {xn+1, · · · , xn+Ly} (n = m+Lx +and m = 1, · · · , L − Ly). For the sake of clarity, we denote +Y = {yn+1, · · · , yn+Ly | yj ∈ X}. Long-term time-series +forecasting is to predict the future time-series with larger Ly. +IV. METHODOLOGY +The framework overview of Conformer is shown in Fig. 1. +Conformer mainly consists of three parts: the input representa- +tion block, encoder-decoder architecture, and normalizing flow +block. First, the input representation block preprocesses and +embeds the input time series accordingly. Then, the encoder- +decoder architecture explores the local temporal patterns with +windowed attention from time-series representations and ex- +amines long-term intricate dynamics from both stationary and +instant perspectives with the help of recurrent network and +time-series decomposition. Moreover, to improve information +utilization, the normalizing flow block leverages latent states +in the recurrent network and generates target series from +the latent states directly. The technical details of these three +components will be introduced in the following subsections. +A. Input Representation +The time series data exhibits intricate patterns since multi- +faceted underlying signals are often complex and varying. +Given a length-L time series X, X = {x1, x2, · · · , xL|xi ∈ +Rdx} (dx ≥ 1), we investigate the underlying multi-faceted +relatedness in X from two perspectives, i.e., the “vertical” +feature perspective, and the “horizontal” temporal perspective. +1) Multivariate Correlation: Complex relatedness among +different variables in a multivariate time series hinders the +effectiveness of distinguishing and harnessing important sig- +nals for future series prediction. On the one hand, the impacts +of different variables on forecasting future series differ. For +instance, the heatmaps in Fig. 2 illustrate rhythms of differ- +ent variables in various time-series datasets, it is clear that +(a) Exchange rate. +(b) Wind power. +Fig. 2: Different variables of time-series data evolve at varying +rhythms and dynamics. The details of these datasets can be +found in Section V-A1. +different variables exhibit distinct relatedness to the target +variable, which can also vary over time. On the other hand, +the well-leveraged dependencies among variables can benefit +time-series forecasting. +Fast Fourier Transform (FFT) [35] has been proven to +be effective in discovering the correlations for time series +data [36]–[38]. Inspired by this, we adopt FFT to represent +implicit multivariate correlations of a length-L time series by +exploring the auto-correlation as follows: +MRXX = f −1(f(X)f ∗(X)) , +(1) +where f and f −1 denote FFT and inverse FFT, respectively. +The asterisk represents a conjugate operation. Besides, we em- +ploy Softmax to highlight informative variables accordingly: +WR = Softmax(MRXX ) . +(2) +2) Multiscale Dynamics: Temporal patterns are helpful in +solving the long-term time-series forecasting problem [39]. We +further examine the temporal patterns by means of multiscale +representation. Specifically, a time series can present distinct +temporal patterns at different temporal resolutions. In other +words, more attention should be paid to informative dynamics +extracted at certain temporal resolutions. +To implement the temporal pattern extraction at different +scales, we first devise a temporal resolution set S ⫅ {second, +minute, hour, day, week, month, year} for X. Then the +3 + +Input Representation +Multivariate Correlation +Windowed MHA +Windowed MHA +Initial Distribution +Encoder Input +Decoder Input +Construction + Multiscale Dynamics +0 +0 +0 +0 +SIRN +SIRN +Chain of +Transformations +Multivariate Correlation +Windowed MHA +Windowed MHA +Multiscale Dynamics +SIRN +SIRN +Outputscounty1 + 2.0 + 1.5 + 1.0 +Dimension + 0.5 + 0.0 +0.5 +1.0 +1.5 +0 +500 +1000 +1500 +2000 +2500 +Time Point 2.0 +pred + 1.5 +pred. +1.0 +Dimension +pred + 0.5 + 0.0 +pred +0.5 +pred +1.0 +ture +1.5 +2.0 +0 +500 +1000 +1500 +2000 +2500 +Time PointPublished as a conference paper at ICDE 2023 +sampled time-series set ΓS = {ΓS1, · · · , ΓSK} is obtained, +where K denotes the number of temporal resolutions and ΓSk +is the sequence of sampled timestamps at corresponding tem- +poral resolution Sk. Afterward, each series in ΓS is embedded +into a latent space with d×L dimensionality, such that different +series in ΓS are additive: +˜ΓS = E(ΓS) = {E(ΓS1), · · · , E(ΓSK)} += {˜ΓS1, · · · , ˜ΓSK} , +(3) +where E denotes an embedding operation and ˜ΓSk ∈ Rd×L +represents the embedded series at a certain temporal resolution +Sk. Then the multiscale temporal patterns can be modeled as: +¯ΓS = WS Concat(˜ΓS) + (bS)′ += +K +� +k=1 +WS +k (˜ΓSk)′ + (bS)′ , +(4) +where WS ∈ RL×L×K and bS ∈ Rd×L are trainable weights +and bias, respectively. The prime symbol denotes the matrix +transpose. Besides, WS +k +∈ RL×L denotes the k-th sliced +matrix of WS. +3) Fusing Multivariate and Temporal Dependencies: +Moreover, to make different variables in multivariate time +series more distinguishable w.r.t. their importance for future +series, we further apply the convolution to take temporal +dependencies into account, which is defined as follows: +X v = Wv ⊙ (WR X + X) + bv , +(5) +where ⊙ denotes the convolution operation, and Wv ∈ Rdx×d +and bv ∈ Rd×L denote weights and bias, respectively. +Finally, by combining the above multivariate correlations +and multiscale dynamics with Eqs. (2) and (5), the outcome +time-series representation can be obtained as follows: +X in = X v + ¯ΓS . +(6) +B. Encoder-Decoder Architecture +Our proposed Conformer adopts the encoder-decoder archi- +tecture for long-term time-series forecasting. +1) Attention Mechanism: The standard attention mech- +anism [26] takes a three-tuple (query, key, value) as input +and employs the scaled dot product and Softmax to cal- +culate the weights against the value as: Attn(Q, K, V ) = +Softmax( QKT +√dk )V , where Q ∈ RL×dk, K ∈ RL×dk, and +V ∈ RL×dv represent query, key and value, respectively. +Moreover, the multi-head attention (MHA) [26] employs +projections for the original query, key, and value N times, +and the i-th projected query, key, and value can be obtained +by Qi = QW Q +i , Ki = KW K +i , and Vi = V W V +i , where +W Q +i +∈ Rdk×dk/N, W K +i +∈ Rdk×dK/N, and W V +i +∈ Rdv×dv/N. +Afterward, the attention can be applied to these queries, keys, +and values in parallel, and the outcome is further concatenated +and projected as follows: +hai =Attn(Qi, Ki, Vi), +i = 1, 2, · · · , N +MHA(Q, K, V ) = Concat(ha1, ha2, · · · , haN)W o . +(7) +Sliding-Window Attention. +Duplicated messages exist +across different heads in full self-attention [40]. A time series +often shows a strong locality of reference, thus a great deal of +information about a point can be derived from its neighbors. +Hence, the full attention message might be too redundant for +future series prediction. Given the importance of locality for +TF, the sliding-window attention (with fixed window size w) +allows each point attends to its 1 +2w neighbors on each side. +Thus, the time complexity of this pattern is O(w × L), which +scales linearly with input length. Therefore, we adopt this +windowed attention to realize self-attention. +2) Stationary and Instant Recurrent Network: Although +the windowed attention can reduce the complexity to O(L), +the information utilization could be sacrificed for LTTF due +to point-wise sparse connections. RNNs have achieved big +successes in many sequential data applications [41]–[44] at- +tributed to their capabilities of capturing dynamics in se- +quences via cycles in the network of nodes. To enhance +information utilization without increasing time and memory +complexities, we, therefore, renovate the recurrent network +accordingly. In particular, we not only distill the stationary +(trend) and instant (seasonal) temporal patterns from input +series but also integrate the distilled long-term patterns, as +well as the aforementioned local temporal patterns, into the +time-series representation. The architecture of the proposed +Stationary and Instant Recurrent Network (SIRN) is demon- +strated in Fig. 3a. +Specifically, we feed the input representation to the first +RNN block (followed by a Softmax) to initialize the global +representation and add it to the local representation, as well +as the original input representation, as follows: +X in =SoftMax(RNN(X in)) × X in ++ MHAW (X in) + X in , +(8) +where MHAW (·) denotes the sliding-window attention. Note +that the RNN block (followed by Softmax) in the first term +of Eq. (8) aims to capture the global temporal dependency, +which can supplement the local dependency captured by the +windowed attention. +Though intricate and diverse, the complex temporal patterns +in different time-series data can be roughly divided into +(coarse-grained) stationary trends and (fine-grained) instant +patterns. Along this line, we employ the series decomposition +introduced in [13], [45] to distill stationary and instant patterns +by capturing trend and seasonal parts of the time-series data. +Similar to [13], we adopt the moving average to capture long- +term trends and the residual of the original series subtracting +the moving average as seasonal patterns: +Xt = AvgPool(Padding(X in)), +Xs = X in − Xt, +(9) +where Xt, Xs ∈ RL×dx denote the trend and seasonal parts +of X in, respectively. Then, we use a convolution layer to +embed the seasonal pattern. And, we feed the embedded rep- +resentation, coupled with the local representation, to another +decomposition block for distilling more seasonal patterns. This +distillation process can be implemented in a recurrent way: +4 + +Published as a conference paper at ICDE 2023 +(a) Stationary and instant recurrent network (SIRN). +(b) Normalizing flow framework. +Fig. 3: The architecture of SIRN and the normalizing flow framework. (a) The first RNN block embeds the global information of +input time-series and the second RNN block represents the aggregated trend information extracted by the decomposition block. +The decomposition procedure following the initial decomposition can be repeated multiple times. The latent state yielded by +the first RNN will be utilized in the normalizing flow framework. (b) After initiating the flow of transformations with Eqs. (15) +and (16), the latent state of decoder is adopted to generate the target variable. +X (l) +t , X (l) +s += Decomp(Conv(X (l−1) +s +) ++ MHAW (X in)), l = 1, · · · , η , +(10) +where Decomp denotes Eq. (9), X (0) +s += Xs and X (0) +t += Xt. +On the other hand, the trend parts generated by different +decompositions are merged and fed to the second RNN block. +Finally, the distilled multi-faceted temporal dynamics are fused +to generate the outcome representation: +X out = W(X (η) +s ++ RNN( +η +� +l=0 +X (l) +t )). +(11) +C. Time Series Prediction with Normalizing Flow +The aforementioned SIRN framework adopts RNN to ex- +tract global signals. In addition, the hidden states yielded by +RNN are beneficial for understanding the distribution of time- +series data. Specifically, we design a normalizing-flow block to +learn the distribution of hidden states to increase the reliability +of prediction. +1) Background of Normalizing Flow: +A time series +X = {x1, · · · , xL} can be reconstructed by maximizing the +marginal log-likelihood: log p(X) = �L +i=1 log p(xi). Due to +the intractability of such log-likelihood, a parametric inference +model over the latent variables z, i.e., q(z|x), was introduced. +Then, one can optimize the variational lower bound on the +marginal log-likelihood of each observation x as follows: +log p(x) ⩾Eq(z|x)[log p(x, z) − log q(z, x)] += log p(x) − DKL(q(z|x) || p(z|x)) +=L(x; θ) , +(12) +where DKL(·) denotes the Kullback-Leibler divergence. When +the dimensionality of z climbs up, the diagonal posterior +distribution is often adopted, which is, however, not flexible +enough to match the complex true posterior distributions [46]. +To solve this, the Normalizing Flow [47] was proposed to build +flexible posterior distributions. +Basically, one can start off with an initial random variable +z0 (with a simple distribution, coupled with a known density +function), and then apply a chain of invertible transformations +ft, such that the outcome zT has a more flexible distribution: +z0 ∽ q(z0|x), +zt = ft(zt−1), +t = 1, · · · , T . +(13) +Besides, as long as the Jacobian determinant det +��� +dzt +dzt−1 +��� is +available, the transformation can take the following definition: +ft(zt−1) = zt−1 + u g(wT zt−1 + b) , +(14) +where u, w and b are parameters, and g(·) denotes a nonlinear +function. +2) Normalizing Flow for LTTF: The proposed architec- +ture of normalizing flow in Conformer is shown in Fig. 3b. +Let h denote the hidden state yielded by the first RNN +block in SIRN. Then, draw a random variable from a Gaussian +distribution, i.e., ϵ ∽ N(0, I), and the distribution of the +hidden state in the encoder can be obtained as: +ze = FCN(e) +µ (he) + FCN(e) +σ (he) · ϵ , +(15) +where FCN(e) +µ +and FCN(e) +σ +are two fully connected networks, +he denotes the hidden state in encoder. Afterward, we take +the latent representation ze and the decoder latent state hd as +input to initiate the normalizing flow: +z0 = FCN(d) +µ (hd) + FCN(d) +σ (hd) · ze . +(16) +Now that the normalizing flow can be iterated as follows: +zt = FCN(t) +µ (hd, zt−1) ++ FCN(t) +σ (hd, zt−1) · zt−1, +t = 1, · · · , T . +(17) +Here, we utilize the decoder latent state to cascade the mes- +sage, such that the future series can be generated directly. +D. Loss Function +In order to coordinate with the other parts of Conformer, +the commonly used log-likelihood is substituted for the MSE +(mean squared error) loss function for learning the normalizing +flow framework. In particular, the random variable sampled +from the outcome distribution, i.e., zt, is deemed as the point +estimation of the target series. Then, we adopt MSE loss +5 + +Decomp +Conv +SoftMax ++ +Decomp +RNN ++ +RNNFCN +FCN +FCN +FCN +N(O, I)Published as a conference paper at ICDE 2023 +functions on prediction w.r.t. the target series for both encoder- +decoder architecture and normalizing flow framework. Finally, +the loss function is defined as follows: +L = λ · MSE(Yout, Y) + (1 − λ) · MSE(Zout, Y) +(18) +where Yout and Zout denote the output of decoder and +normalizing flow, respectively, and λ is a trade-off hyper- +parameter balancing the relative contributions of encoder- +decoder and normalizing flow. +V. EXPERIMENTS +A. Experiment Settings +1) Datasets: We conduct experiments on seven datasets +including five benchmark datasets and two collected datasets. +Table I describes some basic statistics of these datasets. +ECL1 was collected in 15-minute intervals from 2011 to +2014. We select the records from 2012 to 2014 since many +zero values exist in 2011 [1]. The processed dataset contains +the hourly electricity consumption of 321 clients. We use +’MT 321’ as the target, and the train/val/test is 12/2/2 months. +Weather2 +was +recorded +in +10-minute +intervals +from +07/2020 to 07/2021. There exist 21 meteorological indicators, +e.g., the amount of rain, humidity, etc. We choose temperature +as the target, and the train/val/test is 10/1/1 months. +Exchange [1] records the daily exchange rates of eight +countries from 1990 to 2016. We use the exchange rates of +Singapore as the target, The train/val/test is 16/2/2 years. +ETT [15] records the electricity transformer temperature. +Every data point consists of six power load features and the +target value is “oil temperture”. This dataset is separated into +{ETTh1, ETTh2} and {ETTm1, ETTm2} for 1-hour-level and +15-minute-level observations, respectively. We use ETTh1 and +ETTm1 as our datasets. The train/val/test are 12/2/2 and 12/1/1 +months for ETTh1 and ETTm1, respectively. +Wind (Wind Power)3 records the generated wind power of +a wind farm in 15-minute intervals from 01/2020 to 07/2021. +The train/val/test is 12/1/1 months. +AirDelay was collected from the “On-Time” database in the +TranStas data library4. We extracted the flights arrived at the +airports in Texas and examined arrival delays in the first month +of the year 2022, and the canceled flights were removed. Note +that the time interval of this dataset is varying. This dataset +was split into train/val/test as 7:1:2. +2) Baselines: We compare Conformer with 9 baselines, +i.e., 5 Transformer methods (Autoformer, Informer, Reformer, +Longformer, and LogTrans), 2 RNN methods (GRU and +LSTNet), and 2 other deep methods (TS2Vec and N-Beats). +• GRU [21]: GRU employs the gating mechanism such that +each recurrent unit adaptively captures temporal signals in +the series. In this work, we adopt a 2-layer GRU. +1https://archive.ics.uci.edu/ml/ datasets/ElectricityLoadDiagrams20112014 +2https://www.bgc-jena.mpg.de/wetter/ +3We collect this dataset and publish it at https://github.com/PaddlePaddle/ +PaddleSpatial/tree/main/paddlespatial/datasets/WindPower. +4https://www.transtats.bts.gov. +The +processed +dataset +is +available +at +https://github.com/PaddlePaddle/PaddleSpatial/tree/main/paddlespatial/ +datasets/AirDelay. +TABLE I: Statistical descriptions of the time-series datasets. +Datasets # Dims. +Time Span +# Points Target Variable Interval +ECL +321 +01/2012 - 12/2014 +26304 +MT 321 +1 hour +Weather +21 +01/2020 - 06/2021 +36761 +Temperature +10 mins +Exchange +8 +01/1990 - 12/2016 +7588 +Country8 +1 day +ETTh1 +7 +07/2016 - 07/2018 +17420 +OT +1 hour +ETTm1 +7 +07/2016 - 07/2018 +69680 +OT +15 mins +Wind +7 +01/2020 - 05/2021 +45550 +Wind Power 15 mins +AirDelay +6 +01/01 - 01/31, 2022 54451 +ArrDelay +– +• LSTNet [1]: LSTNet combines the convolution and recur- +rent networks to extract short-term dependencies among +variables and long-term trends in the time series. Note +that, to simplify the parameter tuning, the highway and +skip connection mechanisms are omitted. +• N-Beats [48]: N-Beats was proposed to address time-series +forecasting via a deep model on top of the backward and +forward residual links and a very deep stack of fully- +connected layers. We implement N-Beats for multivariate +LTTF with suggested settings. +• Reformer [12]: Reformer uses locality-sensitive hashing +(LSH) attention and reversible residual layers to reduce +the computation complexity. We implement Reformer by +setting the bucket length and the number of rounds for +LSH attention as 24 and 4, respectively. +• Longformer [16]: Longformer combines the windowed +attention with a task motivated global attention to scale +up linearly as the sequence length grows. +• LogTrans [14]: LogTrans breaks the memory bottleneck +of Transformer for LTTF via producing queries and keys +with the help of causal convolutional self-attention. The +number of the LogTransformer blocks is set to 2 and the +sub len of the sparse-attention is set to 1. +• Informer [15]: Informer proposes the ProbSparse slef- +attention to reduce time and memory complexities, and +handles the long-term sequence with self-attention distill- +ing operation and generative style decoder. +• Autoformer [13]: Autoformer renovates the series decom- +position with the help of auto-correlation mechanism, and +put the series decomposition as a basic inner block of the +deep model. +• TS2Vec [49]: TS2Vec is a universal framework for learning +representations of time series. It performs contrastive learn- +ing in a hierarchical way over augmented context views, +which leads to the robust contextual representation for each +timestamp. We implement TS2Vec for univariate LTTF +with the suggested settings. +All baselines employ the one-step prediction strategy. For +the RNN-based methods, the number of hidden states is chosen +from {16, 24, 32, 64}. For the Transformer-based methods, +the number of heads of the self-attention is 8 and the di- +mensionality is set as 512 for all attention mechanisms in +the experiments. Moreover, the sampling factor of the self- +attention is set to 1 for both Informer and Autoformer, other +settings are the same as suggested by [13]. All Transformer- +based baselines (except Autoformer) use the same embedding +6 + +Published as a conference paper at ICDE 2023 +TABLE II: Comparisons of multivariate LTTF results (the best and 2nd best scores are boldfaced and underlined, resp.). +Model +Transformer-based +RNN-based +Others +Conformer +Longformer [16] Autoformer [13] Informer [15] Reformer [12] +LSTNet [1] +GRU [21] +N-beats [48] +Metric +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +ECL +96 0.2124 0.3193 0.3156 +0.3939 +0.2018 +0.3100 +0.5423 0.5568 0.9865 0.7795 1.1002 0.8066 0.7292 0.6274 1.3759 0.8753 +192 0.2378 0.3456 0.3371 +0.4169 +0.3579 +0.4277 +0.5304 0.5549 1.0119 0.7831 1.0965 0.8048 1.0093 0.7679 1.3228 0.8660 +384 0.2643 0.3620 0.3976 +0.4183 +0.4670 +0.5019 +0.6429 0.5921 1.0883 0.7867 1.1034 0.8057 1.0548 0.7898 1.3911 0.8709 +768 0.3396 0.4092 0.5651 +0.5182 +0.5525 +0.5598 +0.9534 0.7789 1.0624 0.7913 1.1132 0.8085 1.0651 0.7938 1.3645 0.8585 +Weather +48 0.3216 0.3433 0.3475 +0.3630 +0.4552 +0.4340 +0.3929 0.3231 0.5099 0.4506 0.7852 0.6769 0.6569 0.5438 0.6171 0.5346 +192 0.4129 0.4170 0.4259 +0.4235 +0.4965 +0.4711 +0.4396 0.4332 0.6960 0.5852 0.7858 0.6770 0.7548 0.6025 0.6121 0.5402 +384 0.4997 0.4847 0.5518 +0.5030 +0.5832 +0.5255 +0.5848 0.5197 0.7525 0.6231 0.8063 0.6886 0.7679 0.6070 0.6032 0.5067 +768 0.6146 0.5603 0.6734 +0.5769 +0.6429 +0.5682 +0.7051 0.5881 0.7883 0.6529 0.8303 0.7003 0.7671 0.6100 0.5944 0.5143 +Exchange +48 0.0764 0.2093 0.1736 +0.3314 +0.1431 +0.2892 +0.2310 0.3841 0.3653 0.4952 1.0319 0.8623 1.1399 0.9119 2.1053 1.0750 +96 0.1193 0.2607 0.3519 +0.4829 +0.2021 +0.3586 +0.3079 0.4488 0.9120 0.7731 1.0260 0.8648 1.3953 0.9837 1.8161 0.9896 +192 0.2900 0.4187 0.6145 +0.6393 +0.4249 +0.5486 +0.5902 0.6306 1.1195 0.8713 0.9954 0.8562 1.3754 0.9800 1.8113 0.9899 +384 0.4730 0.5369 0.8105 +0.7513 +1.2798 +0.9983 +0.8630 0.7953 1.2748 0.9435 0.9642 0.8457 1.3801 0.9858 2.4088 1.1708 +ETTm1 +96 0.6854 0.5901 1.0947 +0.7079 +0.8586 +0.6591 +1.0921 0.7023 1.6397 0.9771 1.6250 0.9045 1.7469 0.9714 1.2350 2.3957 +192 0.7856 0.6387 1.2555 +0.7644 +0.9406 +0.6958 +1.2657 0.7898 1.6499 0.9659 1.6012 0.9080 1.7223 0.9592 1.2253 2.3467 +384 0.9298 0.6988 1.2303 +0.7786 +1.1112 +0.7593 +1.3849 0.8459 1.6396 0.9783 1.5063 0.8916 1.5815 0.9124 1.2149 2.2922 +768 0.9835 0.7193 1.2247 +0.7816 +1.2974 +0.7940 +1.3537 0.8492 1.6121 0.9501 1.3637 0.8604 1.3437 0.8425 1.2420 2.3629 +ETTh1 +96 0.6978 0.5623 0.7276 +0.5894 +0.7515 +0.5727 +0.8901 0.6498 0.9794 0.6926 1.1763 0.7884 1.0490 0.7312 1.6426 0.9352 +192 0.8444 0.6249 0.9074 +0.6566 +0.9346 +0.6375 +1.0463 0.7082 1.0157 0.7156 1.1850 0.7870 1.0850 0.7469 1.7524 0.9709 +384 0.9708 0.6767 0.9804 +0.6919 +1.1832 +0.7347 +1.1237 0.7383 1.0673 0.7298 1.2493 0.8082 1.1081 0.7471 1.7703 0.9778 +768 1.0827 0.7275 1.0501 +0.7083 +1.2562 +0.7676 +1.1047 0.7292 1.1105 0.7447 1.5301 0.9207 1.1275 0.7524 1.7656 0.9822 +Wind +48 0.9479 0.6539 0.9605 +0.6767 +1.3522 +0.8099 +1.0056 0.6552 1.1881 0.7949 1.3874 0.9246 1.1599 0.8022 1.5667 0.8727 +96 1.1725 0.7641 1.2467 +0.7698 +1.4859 +0.8702 +1.2371 0.7994 1.3283 0.8529 1.4489 0.9441 1.2797 0.8455 1.6842 0.9074 +192 1.3291 0.8464 1.4829 +0.8487 +1.6118 +0.9172 +1.5022 0.8489 1.4074 0.8980 1.4794 0.9508 1.3779 0.8866 1.6146 0.8839 +384 1.3644 0.8692 1.5479 +0.8830 +1.7363 +0.9585 +1.5002 0.8747 1.4541 0.9190 1.4966 0.9541 1.3818 0.8897 1.5746 0.8551 +768 1.3698 0.8905 1.4995 +0.8954 +1.6629 +0.9426 +1.5152 0.8956 1.5215 0.9540 1.4813 0.9471 1.4580 0.9253 1.6176 0.9009 +AirDelay +96 0.7491 0.5702 0.7746 +0.5984 +0.7959 +0.6041 +0.7663 0.5904 0.7719 0.5961 0.7781 0.6058 0.7675 0.5859 0.7961 0.5940 +192 0.7523 0.5689 0.7770 +0.5980 +0.7865 +0.5934 +0.7659 0.5876 0.7724 0.5960 0.7782 0.6053 0.7705 0.5903 0.8041 0.5961 +384 0.7560 0.5718 0.7876 +0.6051 +0.7896 +0.5889 +0.7729 0.5841 0.7735 0.5963 0.7784 0.6049 0.7739 0.5968 0.8082 0.5988 +768 0.7614 0.5729 0.8081 +0.6209 +0.7952 +0.5921 +0.7868 0.6061 0.7749 0.5956 0.7797 0.6044 0.7750 0.5961 0.8099 0.5981 +method applied to the Informer. As suggested by [13], we +omit the position embedding and keep the value embedding +and timestamp embedding for Autoformer. +3) Implementation Details: Conformer5 includes a 2-layer +encoder and a 1-layer decoder, as well as a 2-layer normalizing +flow block. The window size of the sliding-window attention is +2, and λ in Eq. (18) is set to 0.8. We use an Adam optimizer, +and the initial learning rate is 1 × 10−4. The batch-size is +32 and the training process employs early stopping within 10 +epochs. In addition, we use MAE (mean absolute error) and +MSE (mean squared error) as the evaluation metrics. +An input-Lx-predict-Ly window is applied to roll the train, +validation and test sets with stride one time step, respectively. +This setting is adopted for all datasets. The input length Lx is +96 and the predict length Ly is chosen from {48, 96, 192, +384, 768} on all datasets. The averaged results in 5 runs +are reported. All models are implemented in PyTorch and +trained/tested on a Linux machine with one A100 40GB GPU. +All of the RNN blocks in Conformer are implemented with +GRU. Under the multivariate LTTF setting, we adopt 1-layer +GRU and 2-layer GRU for encoder and decoder, respectively. +Under the univariate LTTF setting, both the encoder and +decoder adopt 1-layer GRU. +5The source code of Conformer is available at https://github.com/ +PaddlePaddle/PaddleSpatial/tree/main/research/Conformer. +B. Prediction Results of Multivariate LTTF +We compare Conformer to other baselines in terms of +MSE and MAE under the multivariate time-series forecasting +setting, and the results are reported in Table II. We can +observe that Conformer outperforms SOTA Transformer-based +models, as well as other competitive methods, under differ- +ent predict-length settings. For example, under the predict- +96 setting, compared to the second best results, Conformer +achieves 41.0% (0.2021→0.1193), 20.2% (0.8586→ 0.6854), +5.2% (1.2371→1.1725) and 4.1% (0.7276→0.6978) MSE +reductions on Exchange, ETTm1, Wind and ETTh1 datasets, +respectively. Besides, when Ly = 384, Conformer achieves +41.6% (0.8105→0.4730), 33.5% (0.3976→0.2643), 16.3% +(1.1112→0.9298) and 9.4% (0.5518→0.4997) MSE reduc- +tions on Exchange, ECL, ETTm1 and Weather datasets, re- +spectively, as well as 28.5% (0.7513→0.5369), 13.5% (0.4183 +→0.3620) and 8.0% (0.7593→0.6988) MAE reductions on +Exchange, ECL and ETTm1 datasets, respectively. Moreover, +when the predict-length Ly is prolonged to 768, Conformer +achieves 39.9% (0.5651→0.3396), 19.7% (1.2247→0.9835) +and 6.0% (1.4580→1.3698) MSE reductions on ECL, ETTm1 +and Wind datasets, respectively, plus 21.0% (0.5182→0.4092) +and 9.4% (0.7940→0.7193) MAE reductions on ECL and +ETTm1 datasets, respectively. +On the other hand, in general, the Transformer-based models +outperform the RNN-based models. This shows the strength +of the self-attention mechanism in extracting intricate temporal +7 + +Published as a conference paper at ICDE 2023 +TABLE III: Multivariate LTTF with time-determined lengths (boldface and underline for the best and 2nd best scores). +Model +Transformer-based +RNN-based +Others +Conformer +Longformer [16] Autoformer [13] Informer [15] Reformer [12] +LSTNet [1] +GRU [21] +N-beats [48] +Metric +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +ETTh1 +1D 0.4158 0.4434 0.3924 +0.4316 0.4248 0.4374 +0.3985 0.4345 0.6987 0.5640 1.1549 0.7713 0.9265 0.6771 0.9027 1.5612 +1W 0.7326 0.5785 0.7583 +0.5997 0.8682 0.6130 +0.8585 0.6362 1.0286 0.7207 1.2254 0.8020 1.1184 0.7571 0.9388 1.6576 +2W +0.8661 0.6400 0.9328 +0.6767 1.1191 0.7126 +1.0686 0.7054 1.0796 0.7378 1.2231 0.7964 1.1095 0.7456 0.9533 1.6852 +1M 0.9845 0.6887 0.9205 +0.6763 1.2151 0.7567 +1.0292 0.6887 1.1092 0.7451 1.6012 0.9525 1.1341 0.7566 1.0084 1.8501 +ETTm1 +1D 0.6854 0.5901 1.0947 +0.7079 0.8586 0.6591 +1.0921 0.7023 1.6397 0.9771 1.6250 0.9045 1.7469 0.9714 1.2350 2.3957 +1W 0.9540 0.7009 1.2416 +0.7912 1.2016 0.7660 +1.4284 0.8551 1.4008 0.8697 1.3692 0.8576 1.3852 0.8551 1.2322 2.3267 +2W 1.0948 0.7583 1.2463 +0.7871 1.5101 0.8469 +1.2857 0.8177 1.2233 0.7952 1.2931 0.8326 1.2245 0.7945 1.2540 2.3790 +TABLE IV: Comparisons of univariate LTTF results (the best and 2nd best scores are boldfaced and underlined, resp.). +Model +Transformer-based +RNN-based +Others +Conformer +Autoformer [13] Informer [15] Reformer [12] LogTrans [14] +LSTNet [1] +GRU [21] +TS2VEC [49] +Metric +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +ECL +96 0.3481 0.4587 0.4457 +0.5295 +0.3182 0.4425 0.7945 0.7286 0.6378 0.6859 0.9617 0.7872 0.8348 0.7358 1.1987 2.1247 +192 0.3565 0.4629 0.5327 +0.5787 +0.3906 0.4968 0.8575 0.7465 0.6373 0.6837 0.9929 0.7904 0.9379 0.7700 1.1557 2.0041 +384 0.3659 0.4656 0.6592 +0.6506 +0.4352 0.5260 0.9269 0.7666 0.6369 0.6807 1.0074 0.7928 0.9685 0.7784 1.1863 2.0864 +768 0.4283 0.4989 0.7821 +0.7190 +0.4987 0.5674 0.9521 0.7711 0.6468 0.6815 1.0429 0.8034 0.9917 0.7847 1.0891 1.7940 +Weather +48 0.0846 0.2025 0.1719 +0.3073 +0.0914 0.2183 0.1809 0.3202 0.3231 0.4460 0.4818 0.5985 0.3660 0.4689 1.6989 3.8099 +192 0.1898 0.3114 0.2365 +0.3562 +0.2008 0.3330 0.3597 0.4651 0.3115 0.4270 0.4947 0.5994 0.4505 0.5232 1.6884 3.7227 +384 0.3006 0.4041 0.3249 +0.4345 +0.3095 0.4267 0.4386 0.5198 0.3500 0.4554 0.5218 0.6171 0.4869 0.5383 1.4820 3.1272 +768 0.4005 0.4811 0.4665 +0.5314 +0.5536 0.5891 0.4493 0.5240 0.4297 0.5065 0.5414 0.6302 0.5397 0.5508 1.7431 3.9300 +Exchange +48 0.0676 0.2059 0.1446 +0.2892 +0.2308 0.3840 0.3655 0.4954 0.2768 0.4338 1.0319 0.8623 1.1399 0.9858 1.2644 2.2181 +96 0.1168 0.2693 0.1369 +0.2935 +0.1827 0.3408 1.5431 1.0742 0.1976 0.3432 1.7163 1.2134 2.2484 1.3663 1.2797 2.2738 +192 0.2107 0.3751 0.4256 +0.5549 +0.3889 0.5021 2.0076 1.2959 0.5285 0.6217 1.6056 1.1823 2.5891 1.5426 1.2846 2.3074 +384 0.4591 0.5770 1.2899 +1.0128 +1.1126 0.7888 2.2899 1.4411 0.5520 0.6415 1.5664 1.1789 2.5353 1.5355 1.2947 2.3176 +ETTm1 +96 0.0655 0.1827 0.0733 +0.1979 +0.0793 0.1945 0.1481 0.2865 0.0752 0.2008 0.1583 0.2952 0.3128 0.4853 1.0667 1.7871 +192 0.0898 0.2237 0.1018 +0.2445 +0.1124 0.2407 0.2135 0.3546 0.0906 0.2277 0.2099 0.3424 0.3007 0.4511 0.9113 1.4667 +384 0.1032 0.2549 0.1175 +0.2746 +0.2643 0.4057 0.2699 0.4092 0.1039 0.2560 0.0980 0.2479 0.2569 0.3891 1.0060 1.7039 +768 0.1194 0.2770 0.2058 +0.3496 +0.4202 0.5488 0.2017 0.3470 0.1219 0.2693 0.1197 0.2778 0.1693 0.3138 0.9859 1.6759 +ETTh1 +96 0.1139 0.2717 0.1484 +0.3163 +0.1517 0.3164 0.3425 0.4650 0.1362 0.3040 0.6936 0.7430 0.4917 0.6045 1.5564 3.2653 +192 0.1452 0.3114 0.1456 +0.3095 +0.1581 0.3264 0.4233 0.5264 0.1435 0.3108 0.8762 0.8584 0.4501 0.5678 1.5088 3.0289 +384 0.1431 0.3071 0.1478 +0.3087 +0.2189 0.3763 0.3917 0.5164 0.1719 0.3378 0.7613 0.7932 0.3959 0.5256 1.2817 2.2929 +768 0.1705 0.3368 0.1733 +0.3404 +0.2999 0.4599 0.3546 0.4922 0.2127 0.3711 0.7940 0.8163 0.3774 0.5125 1.1972 1.8658 +Wind +48 2.6124 1.1886 3.5491 +1.4283 +2.7963 1.1904 3.3011 1.2922 3.3916 1.3999 3.0307 1.2898 2.9602 1.2638 4.0928 1.8116 +96 3.1175 1.3198 4.0628 +1.5638 +3.3353 1.3279 3.5927 1.3374 4.0250 1.5616 3.2913 1.3322 3.3277 1.3292 3.9678 1.7949 +192 3.3957 1.3623 4.2476 +1.5732 +3.6808 1.3659 3.7467 1.3671 4.2043 1.5623 3.5763 1.3773 3.7408 1.3951 4.1021 1.7965 +384 3.5119 1.3748 4.3452 +1.5886 +3.7133 1.3755 3.7970 1.3800 4.3374 1.5979 3.6803 1.3798 3.7605 1.3815 3.9905 1.7902 +768 3.5959 1.3853 4.0653 +1.5428 +3.7967 1.3860 3.8205 1.3857 4.0893 1.5324 3.7261 1.3888 3.8065 1.3851 3.9071 1.7978 +AirDelay +96 0.4687 0.3120 0.4809 +0.3348 +0.5157 0.3954 0.4885 0.3594 0.4722 0.3190 0.5012 0.3870 0.4799 0.3336 0.6866 0.9315 +192 0.4727 0.3167 0.4887 +0.3385 +0.5355 0.4214 0.4848 0.3385 0.4768 0.3266 0.5067 0.3927 0.4874 0.3478 1.1279 1.9009 +384 0.4800 0.3176 0.5028 +0.3402 +0.5563 0.4429 0.4950 0.3530 0.4820 0.3212 0.5142 0.3966 0.4993 0.3662 0.8286 1.2476 +768 0.4894 0.3216 0.5193 +0.3516 +0.6081 0.4895 0.5041 0.3625 0.4953 0.3418 0.5185 0.3932 0.5071 0.3704 1.2700 2.2073 +dependencies in high-dimensional time-series data. Moreover, +MSE and MAE scores of Conformer grows slower as the +predict length prolongs than other baselines indicating better +stability of our proposed model. For the datasets w/ periodicity +(e.g., Weather, ECL) and w/o periodicity (e.g., Exchange), +Conformer consistently delivers good performance, which +suggests the promising generalization ability. In addition, +for the dataset with irregular time intervals (e.g., AirDelay), +Conformer still achieves the best performance consistently, +while the improvements are less significant. This suggests that +the temporal patterns in less-structured time-series data are +more challenging for deep models to capture. +Forecasting with Time-Determined Lengths. We further +evaluate the performance of multivariate LTTF when the input +and output lengths are configured as time-determined intervals, +e.g., 1 day. In particular, in this experiment, the input length +Lx is set to 1 day and the output length Ly is chosen from +{1 day (1D), 1 week (1W), 2 weeks (2W), 1 month (1M)}. We +inspect forecasting performances of different methods on +ETTh1 and ETTm1 datasets. The results are reported in Ta- +ble III. As depicted, Conformer still achieves the best (or +competitive) performance, which suggests the high capacity +of Conformer in perceiving long-term signals. +C. Performance Comparisons Under Univariate LTTF +Table IV reports prediction performances of different meth- +ods under the univariate LTTF setting. Conformer achieves +the best (or competitive) MSE and MAE scores under vari- +ous predict-length settings. In particular, satisfactory predic- +tion improvements can be observed on Exchange, ECL and +Weather datasets. For instance, compared to the second best +results, Conformer achieves 45.8% (0.3889→0.2107) MSE +reduction under predict-192 on Exchange dataset, and 15.9% +8 + +Published as a conference paper at ICDE 2023 +TABLE V: Ablation study of the input representation. +Dataset +ECL +ETTm1 +Predict Length +48 +96 +192 +384 +768 +96 +192 +384 +768 +X in = X v + ¯ΓS (refer to Eq. (6)) +MSE +0.1921 +0.2124 +0.2378 +0.2643 +0.3396 +0.6954 +0.7856 +0.9298 +0.9835 +MAE +0.3034 +0.3193 +0.3456 +0.3620 +0.4092 +0.5901 +0.6387 +0.6988 +0.7193 +X in +−Γ +def += X v +MSE +0.1995 +0.2614 +0.2787 +0.2932 +0.3406 +0.8342 +0.9878 +1.0936 +1.1578 +MAE +0.3123 +0.3659 +0.3766 +0.3791 +0.4083 +0.6623 +0.7353 +0.7786 +0.8066 +X in +−R +def += Wv ⊙ X + bv + ¯ΓS +MSE +0.1896 +0.2178 +0.2421 +0.2674 +0.3425 +0.7216 +0.8313 +0.9429 +0.9794 +MAE +0.3002 +0.3257 +0.3512 +0.3654 +0.4109 +0.6107 +0.6593 +0.7031 +0.7152 +X in +−R−Γ +def += Wv ⊙ X + bv +MSE +0.2010 +0.2735 +0.2749 +0.3078 +0.3391 +0.8853 +0.9754 +1.1112 +1.1538 +MAE +0.3125 +0.3770 +0.3710 +0.3899 +0.4076 +0.6846 +0.7387 +0.7842 +0.7996 +X in +−X +def += Wv ⊙ WRX + bv + ¯ΓS +MSE +0.2774 +0.2622 +0.2789 +0.3040 +0.3065 +0.8342 +0.9878 +1.0936 +1.1578 +MAE +0.3638 +0.3557 +0.3732 +0.3955 +0.3889 +0.6623 +0.7353 +0.7786 +0.8066 +X in +−X−Γ +def += Wv ⊙ WRX + bv +MSE +0.2493 +0.2631 +0.2649 +0.2931 +0.3217 +0.7344 +0.8455 +1.0541 +1.0540 +MAE +0.3404 +0.3473 +0.3561 +0.3822 +0.4057 +0.6221 +0.6636 +0.7540 +0.7474 +TABLE VI: Ablation study of the Stationary and Instant Recurrent Network (on Wind dataset). +Setting +Multivariate Time-Series Forecasting +Univariate Time-Series Forecasting +Predict Length +48 +96 +192 +48 +96 +192 +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +Conformer (with full SIRN) +0.9479 0.6539 1.1725 0.7641 1.3291 0.8464 2.6124 1.1886 3.1175 1.3198 3.3957 1.3623 +Conformer (with Auto-Corr [13]) 1.0253 0.7109 1.2878 0.8191 1.4263 0.8742 2.7366 1.2251 3.2173 1.3381 3.4182 1.3816 +Conformer (with Prob-Attn [15]) 1.0182 0.7069 1.2817 0.8144 1.4246 0.8734 2.7557 1.2229 3.2231 1.3425 3.4423 1.3801 +Conformer (with LSH-Attn [12]) 1.0223 0.7086 1.2778 0.8136 1.4209 0.8730 2.7454 1.2249 3.1930 1.3405 3.4140 1.3793 +Conformer (with Log-Attn [14]) +1.0393 0.7157 1.2866 0.8165 1.4272 0.8755 2.7449 1.2365 3.2116 1.3476 3.4148 1.3831 +Conformer (with Full-Attn [26]) +1.0165 0.7070 1.2756 0.8117 1.4195 0.8715 2.7356 1.2229 3.1964 1.3477 3.4165 1.3809 +TABLE VII: Ablation Study of Normalizing Flow for LTTF on the Wind dataset. +Setting +Multivariate Time-series Forecasting +Univariate Time-series Forecasting +Predict Length +48 +96 +192 +48 +96 +192 +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +MSE +MAE +Conformer +0.9479 0.6539 1.1725 0.7641 1.3291 0.8464 2.6124 1.1886 3.1175 1.3198 3.3957 1.3623 +Conformer ze+zd +−NF +1.0082 0.7015 1.2488 0.8017 1.4095 0.8686 2.7961 1.2492 3.3128 1.3807 3.4604 1.4155 +Conformer ze +−NF +0.9866 0.6953 1.2163 0.7960 1.3632 0.8599 2.7514 1.2363 3.2797 1.3814 3.4432 1.4176 +Conformer zd +−NF +0.9956 0.6949 1.2167 0.7954 1.3682 0.8473 2.7977 1.2651 3.3767 1.4256 3.5208 1.4302 +Conformer −NF +0.9796 0.6927 1.2184 0.8015 1.3455 0.8517 2.7974 1.2614 3.5117 1.4469 3.4421 1.4159 +(0.4352→0.3659) on ECL dataset and 7.4% (0.0914→0.0846) +on Weather dataset under predict-384 and predict-48, respec- +tively. Moreover, Conformer still achieves the best scores on +the AirDelay dataset, which further demonstrates the effective- +ness of Conformer in extracting complex temporal patterns. +In addition, under the univariate LTTF setting, we find that +RNN-based methods achieve competitive prediction results on +Weather and Wind datasets, which can validate the advantages +of RNN in extracting temporal dynamics of the time-series +data with low entropy and regular patterns. +D. Ablation Study +We conduct the ablation study under multivariate TF setting6. +1) Multivariate Correlation and Multiscale Dynamics: +We compare Conformer with its tailored variants w.r.t. the +multivariate correlation and multiscale dynamics, and report +their prediction performances on ECL and ETTm1 datasets. +From Table V, we can obtain several insightful clues on +how to embed the input series for LTTF. 1) X in v. X in +−R: +Multivariate correlation contributes less when the dimensions +of series is higher (#dims. of ECL data is much larger than +ETTm1 data) or the predict-length is prolonged. 2) X in v. +6Hereinafter, all the experiments are carried out under the multivariate TF +setting by default. +X in +−X−Γ: Temporal dependency is more important for the series +with lower dimensions, and for high dimensional time-series, +the effectiveness of temporal dependency can be replaced by +the inter-series correlation when Ly climbs up. 3) X in +−R v. +X in +−X and X in +−R−Γ v. X in +−X−Γ: Multiscale dynamics delivers +better performance when being guided by the raw series, +which holds regardless of #dims. of time-series. Besides, +multivariate correlation contributes more than the raw data for +low dimensional time-series. 4) X in +−R v. X in +−R−Γ and X in +−X v. +X in +−X−Γ: Multiscale dynamics could harm the performance for +LTTF when being equipped with the multivariate correlation +if the raw time-series is absent. +2) Stationary and Instant Recurrent Network (SIRN): +For the ablation study of the proposed SIRN, we compare +Conformer to its different variants by tailoring the encoder- +decoder architecture on the Wind dataset, which can be +found in Table VI. Specifically, we replace the sliding-window +attention and the RNNs with other self-attention mechanisms +to verify the effectiveness of SIRN. From Table VI, we can see +that SIRN achieves best performance under different settings, +which validate the effectiveness of information utilization of +combining the local and global patterns. +3) Normalizing Flow: To verify the effectiveness of nor- +malizing flow block in Conformer for LTTF task, we compare +9 + +Published as a conference paper at ICDE 2023 +(a) Input length. +(b) Window size w. +(c) Trade-off parameter λ. +(d) Number of transformations +in Normalizing Flow. +Fig. 4: Parameter sensitivity analysis of Conformer. +(a) Time complexity. +(b) Memory cost. +Fig. 5: Computation efficiency analysis. The input length is +set as 96 and all the experiments are conducted on the Wind +dataset under the multivariate forecasting setting. +the original Conformer with its several variants. In particular, +we tailor the normalizing flow block in Conformer by real- +izing a generative forecast method with the help of Gaussian +probabilistic model as follows: +• Conformer ze +−NF : The outcome distribution zt (yielded by +normalizing flow) is replaced by ze (obtained by Eq. (15)). +• Conformer +zd +−NF : The outcome distribution zt (yielded +by normalizing flow) is replaced by zd. In particular, we +replace he with hd in Eq. (15) and generate zd accordingly. +• Conformer ze+zd +−NF : The outcome distribution zt (yielded by +normalizing flow) is replaced by z0 (obtained by Eq. (16)). +• Conformer −NF : We implement a tailored Conformer by +removing the normalizing flow framework. +The prediction results on Wind dataset are reported in Ta- +ble VII. We can observe that: 1) The contribution of normal- +izing flow is indispensable for LTTF regardless of forecast +setting and predict length, and 2) the way that we adapt the +normalizing flow to the LTTF task is effective. +E. Parameter Sensitivity Analysis +We report parameter sensitivity analysis in Fig. 4, which is +conducted on the Wind dataset. To be specific, we inspect four +hyper parameters including the input length Lx, the window +size w of sliding-window attention, the trade-off parameter +λ and the number of transformations in Normalizing Flow. +Generally, we can observe that the performance of Conformer +is quite stable most of the time w.r.t. the varying of different +hyper-parameters. In particular, as shown in Fig. 4a, long-term +time-series forecasting setting (e.g., Ly = 384) seems to be +more capable of handling longer input, though the volatility +of performance is small. +F. Computational Efficiency Analysis +We conduct execution time consumption and memory usage +comparisons between Conformer (with sliding-window atten- +tion) and other attention mechanisms. We replace the stan- +dard self-attention mechanism in Transformer with different +variants and carry out the prediction with the corresponding +method for 103 times (taking the sequences in different time +spans as inputs), then the averaged running time per forecast +is reported. For the memory cost comparisons, the maximum +memory usage is recorded. The time consumption and memory +usage of different attentions are demonstrated in Fig. 5. +Conformer performs with better efficiency in both short- and +long-term time-series forecasting. +G. Model Analysis +1) Fusing Inter-Series and Across-Time Dependencies: +As introduced in Section IV-A, the series data is embedded +and fused by taking the multivariate correlation and multiscale +dynamics into account. To further assess the effectiveness +of input representation module in Conformer, we realize +different ways of fusing multivariate correlation and multiscale +dynamics below (let WΓ = Softmax(¯ΓS)): +• Method 1: X in = Wv ⊙ (WΓWRX + X) + bv +• Method 2: X in = Wv ⊙ (WRX + WΓX) + bv +• Method 3: X in = Wv ⊙ (WRX + WΓX + X) + bv +• Method 4: X in = [Wv ⊙ (WRX + X) + bv]WΓ +The results are reported in Table VIII. We can see that how +to fuse the multivariate correlation and temporal dependency +is important for the LTTF task. This impact weighs more +for low dimensional time-series data since the self-attention +mechanism in encoder-decoder architecture can better explore +intricate dependencies when the dimensionality grows. +2) Uncertainty-Aware Forecasting: The outcome vari- +ance of the Normalizing Flow block can suggest the fluctuation +range of the forecasting results. We randomly select a case in +ETTm1 dataset under the multivariate setting and demonstrate +the forecasting results with uncertainty quantification for dif- +ferent output lengths in Fig. 6. We can see that Conformer +tends to make a conservative forecast and the uncertainty +quantification can cover the extreme ground truth values if +the NF block can be weighted more. +3) How Far The Message Should Be Cascaded in Nor- +malizing Flow: We inspect how the normalizing flow works +for LTTF by varying the number of transformations on two +10 + +Execution time (ms) +ProbSparseAttentionfrom Informer +630 +Sliding-Window AttentionfromComformer +Auto-CorrelationfromAutoformer +530 +Full-AttentionfromTransformer +430 +330 +230 +130 +30 +1000 +Prediction lengthProbSparseAttentionfromInformer +Sliding-Window Attentionfrom Comformer +22. +Auto-CorrelationfromAutoformer +Full-AttentionfromTransformer +21 +20. +21 +38. +768 +1000 +Prediction lengthLy=48 +1.8 +96=^7 +Ly=192 +1.6 +Ly=384 +-v=768 +E +S +1.4 +M +1.2 +1.0 +0.8 +Input length (Lx)Ly=48 +96=^7 +1.8 +Ly=192 +Ly=384 +1.6 +Ly=768 +ISE +1.4 +M +1.2 +1.0 +0.8 +-2 +4 +6 +8 +12 +Sliding-window size (w)Ly=48 +1.8 +96=^7 +Ly=192 +1.6 +Ly=384 +y=768 +E +S +1.4 +M +1.2 +1.0 +0.8 +0.2 +0.4 +0.6 +0.8 +1.0 +Trade-off parameter ()Ly=48 +1.8 +96=^7 +Ly=192 +1.6. +Ly=384 +Ly=768 +ISE +1.4 +1.2 +1.0 +0.8 +L +2 +-5 +3 +4 +#transformations in NFPublished as a conference paper at ICDE 2023 +TABLE VIII: Comparisons of fusing inter-series correlation and time dependency for LTTF. +Dataset +ECL +Exchange +Predict Length +48 +96 +192 +384 +768 +48 +96 +192 +384 +Conformer +MSE +0.1921 +0.2124 +0.2378 +0.2643 +0.3396 +0.0764 +0.1193 +0.2900 +0.4730 +MAE +0.3034 +0.3193 +0.3456 +0.3620 +0.4092 +0.2093 +0.2607 +0.4187 +0.5369 +Conformer (Method 1) +MSE +0.2003 +0.2713 +0.2826 +0.2898 +0.3441 +0.1839 +0.2938 +0.4347 +1.0596 +MAE +0.3117 +0.3784 +0.3790 +0.3775 +0.4150 +0.3371 +0.4313 +0.5190 +0.8072 +Conformer (Method 2) +MSE +0.1965 +0.2354 +0.2632 +0.2987 +0.3437 +0.1593 +0.3433 +0.4321 +0.5486 +MAE +0.3007 +0.3323 +0.3587 +0.3821 +0.4191 +0.3144 +0.4530 +0.5197 +0.6007 +Conformer (Method 3) +MSE +0.1997 +0.2791 +0.2771 +0.3061 +0.3433 +0.2443 +0.3310 +0.4089 +0.9034 +MAE +0.3117 +0.3854 +0.3744 +0.3881 +0.4135 +0.3795 +0.4597 +0.4965 +0.7444 +Conformer (Method 4) +MSE +0.2010 +0.2735 +0.2749 +0.3078 +0.3391 +0.1135 +0.1534 +0.2344 +0.5701 +MAE +0.3135 +0.3770 +0.3710 +0.3899 +0.4076 +0.2597 +0.3055 +0.3805 +0.5978 +TABLE IX: Comparisons of feeding hidden states to the normalizing flow block. The best scores are in boldface and the 2nd +best scores are in underlines. +Dataset +ECL +Exchange +Predict Length +48 +96 +192 +384 +768 +48 +96 +192 +384 +Conformer +MSE +0.1921 +0.2124 +0.2378 +0.2643 +0.3396 +0.0764 +0.1193 +0.2900 +0.4730 +MAE +0.3034 +0.3193 +0.3456 +0.3620 +0.4092 +0.2093 +0.2607 +0.4187 +0.5369 +Conformer (h(e) +k , h(d) +k ) +MSE +0.1901 +0.2300 +0.2814 +0.3057 +0.3387 +0.1150 +0.1506 +0.2787 +0.5593 +MAE +0.3010 +0.3322 +0.3776 +0.3920 +0.4168 +0.2643 +0.3013 +0.4108 +0.5872 +Conformer (h(e) +1 , h(d) +k ) +MSE +0.2004 +0.2283 +0.2554 +0.2896 +0.3398 +0.1156 +0.1476 +0.2577 +0.6053 +MAE +0.3112 +0.3321 +0.3588 +0.3782 +0.4121 +0.2655 +0.2983 +0.3911 +0.6086 +Conformer (h(e) +1 , h(d) +1 ) +MSE +0.1984 +0.2265 +0.2507 +0.2751 +0.3565 +0.1181 +0.1669 +0.2498 +0.5300 +MAE +0.3083 +0.3304 +0.3543 +0.3732 +0.4226 +0.2676 +0.3157 +0.3925 +0.5609 +Conformer (h(e) +k , h(d) +1 ) +MSE +0.1904 +0.2202 +0.2512 +0.2799 +0.3547 +0.1155 +0.1497 +0.2846 +0.5203 +MAE +0.3018 +0.3260 +0.3586 +0.3796 +0.4259 +0.2654 +0.3004 +0.4140 +0.5549 +(a) Predict Length = 96. +(b) Predict Length = 192. +(c) Predict Length = 384. +(d) Predict Length = 768. +Fig. 6: With the help of Normalizing Flow, Conformer can generate the prediction results with uncertainty quantification for +LTTF. Four illustrative cases are demonstrated on the ETTm1 dataset under the multivariate setting. λ denotes the contributions +of the encoder-decoder, that is, 1 − λ represents the impacts of the normalizing flow block. +cases in ECL and ETTm1 datasets, respectively, in Fig. 7. We +can see that the further the latent variable being transformed +the better the outcome series performs. Therefore, the power of +normalizing flow in Conformer for LTTF should be explored +more dedicatedly. +4) How to Feed Hidden States to The Normalizing Flow +Block in Conformer: As shown in Fig. 1, in both encoder +and decoder, the first outcome hidden state of the last SIRN +layer is fed to the normalizing flow. To assess the effect +of feeding hidden states to normalizing flow, we implement +Conformer by combining the outcome hidden states in the +first/last SIRN layer of the encoder/decoder, which results in +Conformer (h(e) +k , h(d) +k ), Conformer (h(e) +1 , h(d) +k ), Conformer +(h(e) +1 , h(d) +1 ) and Conformer (h(e) +k , h(d) +1 ) where k denotes the +last SIRN layer. We report the prediction results in Table IX. +As can be seen, the impact of feeding different hidden states +to normalizing flow is generally marginal though, the low +dimensional time-series forecasting is more sensitive to the +way of absorbing hidden states for normalizing flow. +H. Multivariate Time-series Forecasting Showcase +We additionally plot the prediction and the ground truth +of the target value. The qualitative comparisons between +Conformer and other baselines on ETTm1 dataset are demon- +strated in Fig. 8. We can see that, our model obviously achieves +the best performance among different methods. +I. Discussion +Windowed Attention: Conformer v. Swin Transformer. +The windowed attention mechanism is applied in many appli- +cations thanks to its linear complexity, such that the powerful +self-attention can be scaled up to large data. The very recent +Swin Transformer [50] and its variant [51] adopt the windowed +attention and devise a shifted window attention to implement +a general purpose backbone for computer vision tasks. Basi- +cally, both Conformer and Swin Transformer exploit the self- +attention within neighbored/partitioned windows regarding the +11 + +Point Estimate +Ground Truth +入=0.95 +入=0.9 +^=0.8Point Estimate +Ground Truth +A=0.95 +^=0.9 +^=0.8Point Estimate +Ground Truth +入=0.95 +入=0.9 +^=0.8Point Estimate +Ground Truth +^=0.95 +入=0.9 +^=0.8Published as a conference paper at ICDE 2023 +(a) #transformations = 1. +(b) #transformations = 2. +(c) #transformations = 3. +(d) #transformations = 4. +(e) #transformations = 1. +(f) #transformations = 2. +(g) #transformations = 3. +(h) #transformations = 4. +Fig. 7: Uncertainty-aware LTTF with varying #transforms. To evaluate the performance of normalizing flow more clearly, we +omit the contribution of SIRN by setting λ = 0 in Eq. (18). (a)–(d) and (e)–(h) demonstrate two cases in ECL and ETTm1 +datasets, respectively. +(a) Conformer +(b) Longformer +(c) Reformer +(d) Informer +(e) Autoformer +(f) N-Beats +(g) LSTNet +(h) GRU +Fig. 8: Prediction cases on the ETTm1 dataset under the input-96-predict-192 setting. +computational efficiency. Besides the locality, connectivity is +another merit one can not neglect. To achieve connectivity, a +shifted window mechanism is proposed for Swin Transformer, +while we propose SIRN for Conformer so as to absorb long- +range dependencies in the time-series data. +Comparisons of Computational Complexity. The win- +dowed attention contributes most to the complexity reduc- +tion of Conformer. Hence, we take different SOTA attention +mechanisms as competitors to conduct the computational +complexity analysis in Section V-F. The computational costs +of other components in Conformer are not elaborated, which +will be provided in our future work. +VI. CONCLUSION +In this paper, we proposed a transformer-based model, +namely Conformer, to address the long-term time-series fore- +casting (LTTF) problem. Specifically, Conformer first embeds +the input time series with the multivariate correlation modeling +and multiscale dynamics extraction to fuel the downstream +self-attention mechanism. Then, to reduce the computation +complexity of self-attention and fully distill the series-level +temporal dependencies without sacrificing information utiliza- +tion for LTTF, sliding-window attention, as well as a proposed +stationary and instant recurrent network (SIRN), are equipped +to the Conformer. Moreover, a normalizing flow framework +is employed to further absorb the latent states in the SIRN, +such that the underlying distribution can be learned and the +target series can be directly reconstructed in a generative way. +Extensive empirical studies on six real-world datasets validate +that Conformer achieves state-of-the-art performance on long- +term time-series forecasting under multivariate and univariate +prediction settings. In addition, with the help of normalizing +flow, Conformer can generate the prediction results with +uncertainty quantification. +ACKNOWLEDGMENT +We thank China Longyuan Power Group Corp. Ltd. for sup- +porting this work. Besides, this work was supported in part by +National Key R&D Progamm of China (No. 2021ZD0110303). +12 + +0.5 +Point Estimate +Ground Truth +Value +0.0 +Target +0.5 +-1.0 +-1.5 +0 +20 +40 +60 +80 +Time PointPoint Estimate +0.25 +Ground Truth +0.00 +lue +0.25 +Val +0.50 +0.75 +-1.00 +-1.25 +-1.50 +-1.75 +0 +20 +40 +60 +80 +Time Point0.5 +Point Estimate +Ground Truth +Value +0.0 +Target +0.5 +-1.0 +1.5 +0 +20 +40 +60 +80 +Time PointPoint Estimate +0.25 +Ground Truth +0.00 +lue +0.25 +Val +0.50 +get +-0.75 +-1.00 +1.25 +-1.50 +0 +20 +40 +60 +80 +Time Point1.25 +Point Estimate +1.00 +Ground Truth +0.75 +lue +Val +0.50 +0.25 +0.00 +0.25 +-0.50 +-0.75 +0 +255075100125150175200 +Time Point2.00 +Point Estimate +1.75 +Ground Truth +1.50 +an +Val +.25 +L.00 +let +00.75 +0.50 +0.25 +0.00 +0 +255075100125150175200 +TimePointPoint Estimate +0.8 +Ground Truth +0.6 +lue +0.4 +Vall +0.2 +get +0.0 +-0.2 +-0.4 +-0.6 +0 +255075100125150175200 +Time Point0.8 +Point Estimate +Ground Truth +0.6 +lue +0.4 +Val +get +0.2 +0.0 +-0.2 +-0.4 +0 +255075100125150175200 +Time PointPrediction +0.3 - +Ground Truth +0.2 +0.1 : +0.0 +0.1 : +-0.2 +-0.3 : +0 +25 +50 +75 +100 +125 +150 +175 +200Prediction +0.3 +Ground Truth +0.2 +0.1 : +0.0 - +0.1 : +-0.2 - +0.3 : +0 +25 +50 +75 +100 +125 +150 +175 +2000.4 +Prediction +0.2 +GroundTruth +0.0 +0.2 +-0.4- +-0.6 +0.8 +-1.0: +0 +25 +50 +75 +100 +125 +150 +175 +2000.4 - +Prediction +Ground Truth +0.2 +0.0: +-0.2 +-0.4 +0 +25 +50 +75 +100 +125 +150 +175 +200Prediction +0.3 - +Ground Truth +0.2 +0.1 : +0.0 : +-0.1 +-0.2 +-0.3 +-0.4 - +0 +25 +50 +75 +100 +125 +150 +175 +200Prediction +Ground Truth +2.0 +1.5 +1.0 +0.5 +0.0 +0 +25 +50 +75 +100 +125 +150 +175 +200Prediction +0.3 : +Ground Truth +0.2 +0.1 : +0.0 - +-0.1 : +0 +25 +50 +75 +100 +125 +150 +175 +200Prediction +0.3 : +Ground Truth +0.2 +0.1 : +0.0 - +-0.1 : +0 +25 +50 +75 +100 +125 +150 +175 +200Published as a conference paper at ICDE 2023 +REFERENCES +[1] G. Lai, W.-C. Chang, Y. Yang, and H. Liu, “Modeling long-and +short-term temporal patterns with deep neural networks,” in The 41st +International ACM SIGIR Conference on Research & Development in +Information Retrieval, 2018, pp. 95–104. +[2] Y. Wang, R. Zou, F. Liu, L. Zhang, and Q. Liu, “A review of wind +speed and wind power forecasting with deep neural networks,” Applied +Energy, vol. 304, p. 117766, 2021. +[3] J. Han, H. Liu, H. Zhu, H. Xiong, and D. Dou, “Joint air quality and +weather prediction based on multi-adversarial spatiotemporal networks,” +in Proceedings of the 35th AAAI Conference on Artificial Intelligence, +2021. +[4] Y. Matsubara, Y. Sakurai, W. G. Van Panhuis, and C. Faloutsos, “Funnel: +automatic mining of spatially coevolving epidemics,” in Proceedings +of the 20th ACM SIGKDD international conference on Knowledge +discovery and data mining, 2014, pp. 105–114. +[5] A. A. Ariyo, A. O. Adewumi, and C. K. Ayo, “Stock price prediction +using the arima model,” in 2014 UKSim-AMSS 16th International +Conference on Computer Modelling and Simulation. +IEEE, 2014, pp. +106–112. +[6] K. Gregor, I. Danihelka, A. Mnih, C. Blundell, and D. Wierstra, +“Deep autoregressive networks,” in International Conference on Ma- +chine Learning. +PMLR, 2014, pp. 1242–1250. +[7] I. Melnyk and A. Banerjee, “Estimating structured vector autoregressive +models,” in International Conference on Machine Learning. +PMLR, +2016, pp. 830–839. +[8] K.-j. Kim, “Financial time series forecasting using support vector +machines,” Neurocomputing, vol. 55, no. 1-2, pp. 307–319, 2003. +[9] D. Salinas, V. Flunkert, J. Gasthaus, and T. Januschowski, “Deepar: +Probabilistic forecasting with autoregressive recurrent networks,” Inter- +national Journal of Forecasting, vol. 36, no. 3, pp. 1181–1191, 2020. +[10] A. Graves, “Generating sequences with recurrent neural networks,” arXiv +preprint arXiv:1308.0850, 2013. +[11] I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning +with neural networks,” in Advances in neural information processing +systems, 2014, pp. 3104–3112. +[12] N. Kitaev, Ł. Kaiser, and A. Levskaya, “Reformer: The efficient trans- +former,” arXiv preprint arXiv:2001.04451, 2020. +[13] J. Xu, J. Wang, M. Long et al., “Autoformer: Decomposition transform- +ers with auto-correlation for long-term series forecasting,” Advances in +Neural Information Processing Systems, vol. 34, 2021. +[14] S. Li, X. Jin, Y. Xuan, X. Zhou, W. Chen, Y. Wang, and X. Yan, +“Enhancing the locality and breaking the memory bottleneck of +transformer on time series forecasting,” CoRR, vol. abs/1907.00235, +2019. [Online]. Available: http://arxiv.org/abs/1907.00235 +[15] H. Zhou, S. Zhang, J. Peng, S. Zhang, J. Li, H. Xiong, and W. Zhang, +“Informer: Beyond efficient transformer for long sequence time-series +forecasting,” in Proceedings of the AAAI Conference on Artificial +Intelligence, vol. 35, no. 12, 2021, pp. 11 106–11 115. +[16] I. Beltagy, M. E. Peters, and A. Cohan, “Longformer: The long- +document transformer,” CoRR, vol. abs/2004.05150, 2020. [Online]. +Available: https://arxiv.org/abs/2004.05150 +[17] M. Zaheer, G. Guruganesh, A. Dubey, J. Ainslie, C. Alberti, S. Onta˜n´on, +P. Pham, A. Ravula, Q. Wang, L. Yang, and A. Ahmed, “Big bird: +Transformers for longer sequences,” CoRR, vol. abs/2007.14062, 2020. +[Online]. Available: https://arxiv.org/abs/2007.14062 +[18] J. Gawlikowski, C. R. N. Tassi, M. Ali, J. Lee, M. Humt, J. Feng, +A. Kruspe, R. Triebel, P. Jung, R. Roscher et al., “A survey of uncertainty +in deep neural networks,” arXiv preprint arXiv:2107.03342, 2021. +[19] L.-J. Cao and F. E. H. Tay, “Support vector machine with adaptive +parameters in financial time series forecasting,” IEEE Transactions on +neural networks, vol. 14, no. 6, pp. 1506–1518, 2003. +[20] S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural +computation, vol. 9, no. 8, pp. 1735–1780, 1997. +[21] R. Dey and F. M. Salem, “Gate-variants of gated recurrent unit (gru) +neural networks,” in 2017 IEEE 60th international midwest symposium +on circuits and systems (MWSCAS). +IEEE, 2017, pp. 1597–1600. +[22] A. Borovykh, S. Bohte, and C. W. Oosterlee, “Conditional time se- +ries forecasting with convolutional neural networks,” arXiv preprint +arXiv:1703.04691, 2017. +[23] R. Mittelman, “Time-series modeling with undecimated fully convolu- +tional neural networks,” arXiv preprint arXiv:1508.00317, 2015. +[24] A. Borovykh, S. Bohte, and C. W. Oosterlee, “Dilated convolutional +neural networks for time series forecasting,” Journal of Computational +Finance, Forthcoming, 2018. +[25] M. Benhaddi and J. Ouarzazi, “Multivariate time series forecasting with +dilated residual convolutional neural networks for urban air quality +prediction,” Arabian Journal for Science and Engineering, vol. 46, no. 4, +pp. 3423–3442, 2021. +[26] A. +Vaswani, +N. +Shazeer, +N. +Parmar, +J. +Uszkoreit, +L. +Jones, +A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all +you need,” CoRR, vol. abs/1706.03762, 2017. [Online]. Available: +http://arxiv.org/abs/1706.03762 +[27] R. Child, S. Gray, A. Radford, and I. Sutskever, “Generating long +sequences with sparse transformers,” CoRR, vol. abs/1904.10509, 2019. +[Online]. Available: http://arxiv.org/abs/1904.10509 +[28] D. A. Reynolds, “Gaussian mixture models.” Encyclopedia of biomet- +rics, vol. 741, pp. 659–663, 2009. +[29] Y. Wu, J. Ni, W. Cheng, B. Zong, D. Song, Z. Chen, Y. Liu, X. Zhang, +H. Chen, and S. Davidson, “Dynamic gaussian mixture based deep +generative model for robust forecasting on sparse multivariate time +series,” arXiv preprint arXiv:2103.02164, 2021. +[30] S. S. Rangapuram, L. D. Werner, K. Benidis, P. Mercado, J. Gasthaus, +and T. Januschowski, “End-to-end learning of coherent probabilistic +forecasts for hierarchical time series,” in Proceedings of the 38th +International Conference on Machine Learning, ser. Proceedings of +Machine Learning Research, M. Meila and T. Zhang, Eds., vol. +139. +PMLR, 18–24 Jul 2021, pp. 8832–8843. [Online]. Available: +https://proceedings.mlr.press/v139/rangapuram21a.html +[31] K. Rasul, C. Seward, I. Schuster, and R. Vollgraf, “Autoregressive +denoising diffusion models for multivariate probabilistic time series +forecasting,” CoRR, vol. abs/2101.12072, 2021. [Online]. Available: +https://arxiv.org/abs/2101.12072 +[32] I. Kobyzev, S. Prince, and M. Brubaker, “Normalizing flows: An +introduction and review of current methods,” IEEE Transactions on +Pattern Analysis and Machine Intelligence, 2020. +[33] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, +S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” +Advances in neural information processing systems, vol. 27, 2014. +[34] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” arXiv +preprint arXiv:1312.6114, 2013. +[35] H. J. Nussbaumer, “The fast fourier transform,” in Fast Fourier Trans- +form and Convolution Algorithms. +Springer, 1981, pp. 80–111. +[36] W. T. Cochran, J. W. Cooley, D. L. Favin, H. D. Helms, R. A. Kaenel, +W. W. Lang, G. C. Maling, D. E. Nelson, C. M. Rader, and P. D. Welch, +“What is the fast fourier transform?” Proceedings of the IEEE, vol. 55, +no. 10, pp. 1664–1674, 1967. +[37] H. Musbah, M. El-Hawary, and H. Aly, “Identifying seasonality in time +series by applying fast fourier transform,” in 2019 IEEE Electrical Power +and Energy Conference (EPEC), 2019, pp. 1–4. +[38] G. E. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung, Time series +analysis: forecasting and control. +John Wiley & Sons, 2015. +[39] M. Zaheer, A. Ahmed, and A. J. Smola, “Latent lstm allocation: +Joint clustering and non-linear dynamic modeling of sequence data,” +in International Conference on Machine Learning. +PMLR, 2017, pp. +3967–3976. +[40] O. Kovaleva, A. Romanov, A. Rogers, and A. Rumshisky, “Revealing +the dark secrets of BERT,” CoRR, vol. abs/1908.08593, 2019. [Online]. +Available: http://arxiv.org/abs/1908.08593 +[41] K. Cho, B. Van Merri¨enboer, C. Gulcehre, D. Bahdanau, F. Bougares, +H. Schwenk, and Y. Bengio, “Learning phrase representations using +rnn encoder-decoder for statistical machine translation,” arXiv preprint +arXiv:1406.1078, 2014. +[42] Y. Luo, Z. Chen, and T. Yoshioka, “Dual-path rnn: efficient long +sequence modeling for time-domain single-channel speech separation,” +in ICASSP 2020-2020 IEEE International Conference on Acoustics, +Speech and Signal Processing (ICASSP). +IEEE, 2020, pp. 46–50. +[43] F. Wang, Z. Xuan, Z. Zhen, K. Li, T. Wang, and M. Shi, “A day-ahead pv +power forecasting method based on lstm-rnn model and time correlation +modification under partial daily pattern prediction framework,” Energy +Conversion and Management, vol. 212, p. 112766, 2020. +[44] M. Monti, J. Fiorentino, E. Milanetti, G. Gosti, and G. G. Tartaglia, +“Prediction of time series gene expression and structural analysis of gene +regulatory networks using recurrent neural networks,” Entropy, vol. 24, +no. 2, p. 141, 2022. +13 + +Published as a conference paper at ICDE 2023 +[45] C. Robert, C. William, and T. Irma, “Stl: A seasonal-trend decomposition +procedure based on loess,” Journal of official statistics, vol. 6, no. 1, +pp. 3–73, 1990. +[46] N. Nguyen and B. Quanz, “Temporal latent auto-encoder: A method for +probabilistic multivariate time series forecasting,” in Proceedings of the +AAAI Conference on Artificial Intelligence, vol. 35, no. 10, 2021, pp. +9117–9125. +[47] D. P. Kingma, T. Salimans, R. Jozefowicz, X. Chen, I. Sutskever, and +M. Welling, “Improved variational inference with inverse autoregressive +flow,” Advances in neural information processing systems, vol. 29, pp. +4743–4751, 2016. +[48] B. N. Oreshkin, D. Carpov, N. Chapados, and Y. Bengio, “N-beats: +Neural basis expansion analysis for interpretable time series forecasting,” +in International Conference on Learning Representations, 2019. +[49] Z. Yue, Y. Wang, J. Duan, T. Yang, C. Huang, Y. Tong, and B. Xu, +“Ts2vec: Towards universal representation of time series,” in Proceed- +ings of the AAAI Conference on Artificial Intelligence, vol. 36, no. 8, +2022, pp. 8980–8987. +[50] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and +B. Guo, “Swin transformer: Hierarchical vision transformer using shifted +windows,” in Proceedings of the IEEE/CVF International Conference on +Computer Vision (ICCV), 2021. +[51] Z. Liu, H. Hu, Y. Lin, Z. Yao, Z. Xie, Y. Wei, J. Ning, Y. Cao, Z. Zhang, +L. Dong, F. Wei, and B. Guo, “Swin transformer v2: Scaling up capacity +and resolution,” in International Conference on Computer Vision and +Pattern Recognition (CVPR), 2022. +14 + diff --git a/gtA0T4oBgHgl3EQfH_9T/content/tmp_files/load_file.txt b/gtA0T4oBgHgl3EQfH_9T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..212fd2c05684b5dccd3699de5b42163507f41f3d --- /dev/null +++ b/gtA0T4oBgHgl3EQfH_9T/content/tmp_files/load_file.txt @@ -0,0 +1,2608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf,len=2607 +page_content='Published as a conference paper at ICDE 2023 Towards Long-Term Time-Series Forecasting: Feature, Pattern, and Distribution Yan Li§ † ∗, Xinjiang Lu† �, Haoyi Xiong†, Jian Tang¶ ♮, Jiantao Su♮, Bo Jin♯, Dejing Dou† †Baidu Research, §Zhejiang University, ¶Tsinghua University, ♮China Longyuan Power Group Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', ♯Dalian University of Technology ly21121@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='cn, {luxinjiang,xionghaoyi}@baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='com, {12101779,12091329}@chnenergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='cn, jinbo@dlut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='cn, dejingdou@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='com Abstract—Long-term time-series forecasting (LTTF) has be- come a pressing demand in many applications, such as wind power supply planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Transformer models have been adopted to deliver high prediction capacity because of the high compu- tational self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Though one could lower the complexity of Transformers by inducing the sparsity in point-wise self-attentions for LTTF, the limited information utilization pro- hibits the model from exploring the complex dependencies com- prehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To this end, we propose an efficient Transformer- based model, named Conformer, which differentiates itself from existing methods for LTTF in three aspects: (i) an encoder- decoder architecture incorporating a linear complexity without sacrificing information utilization is proposed on top of sliding- window attention and Stationary and Instant Recurrent Network (SIRN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (ii) a module derived from the normalizing flow is devised to further improve the information utilization by inferring the outputs with the latent variables in SIRN directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (iii) the inter-series correlation and temporal dynamics in time-series data are modeled explicitly to fuel the downstream self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Extensive experiments on seven real-world datasets demonstrate that Conformer outperforms the state-of-the-art methods on LTTF and generates reliable prediction results with uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Index Terms—Long-term time-series forecasting, Transformer, Normalizing Flow I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' INTRODUCTION Time-series data evolve over time, which can result in perplexing time evolution patterns over the short- and long- term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The time evolution nature of time-series data is of great interest to many downstream tasks including time-series classi- fication, outlier detection, and time-series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Among these tasks, time-series forecasting (TF) has attracted many researchers and practitioners in a wide range of application domains, such as transportation and urban planning [1], energy and smart grid management [2], as well as weather [3] and disease propagation analysis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In many real-world application scenarios, given a substantial amount of time-series data recorded, there is a necessity to make a decision in advance, such that, with long-term prediction, the benefits can be maximized while the potential risks can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Therefore, in this work, we study the problem of forecasting time series that looks far into the future, namely long-term time-series forecasting (LTTF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' ∗This work was done when the first author was an intern at Baidu Research under the supervision of the second author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' While tons of TF methods [5]–[8] have been proposed with statistical learners, the use of domain knowledge however seems indispensable to model the temporal dependencies for TF but also limits the potential in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Recently, deep models [9]–[13] have been proposed for TF, which can be categorized into two types: the RNN-based and the Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' RNN-based methods capture and utilize long- and short-term temporal dependencies to make the prediction, but fail to deliver good performance in long- term time-series forecasting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Transformer-based models have achieved promising performance in extracting temporal patterns for LTTF because of the usage of self-attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' However, such “full” attention mechanisms bring quadratic computation complexity for TF tasks, which thus becomes the main bottleneck for Transformer-based models to solve the long-term time-series forecasting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Several works have been devoted to improving the computa- tion efficiency of self-attention mechanisms and lowering the complexity of handling a length-L sequence to (O(L log L) or O(L √ L)), such as Logtrans [14], Reformer [12], In- former [15] and Autoformer [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In the NLP field, some pioneering works have been proposed to reduce the complexity of self-attention to linear (O(L)), including Longformer [16] and BigBird [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' However, these deep models with a linear complexity might limit the information utilization and strain the performance of LTTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lowering the computational com- plexity to O(L) without sacrificing information utilization is a big challenge for LTTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition to the complexity, as the input length climbs up, the intricate time-series could exhibit obscure and confusing temporal patterns, which may lead to unstable prediction for self-attention-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, multivariate long- term time-series often embody multiple temporal patterns at different temporal resolutions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', seconds, minutes, hours, or days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' On the other hand, the intricate and prevailing multi- dimensional characteristics of the time-series data exhibit multi-faceted complex correlations among different series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Therefore, how to make the prediction for LTTF more stable and disaggregate multiscale dynamics and multivariate depen- dencies in time-series data are two more challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To this end, our work devotes to the above three challenges and proposes a novel model based on Transformer for LTTF, namely Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, Conformer first explicitly ex- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='02068v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='LG] 5 Jan 2023 Published as a conference paper at ICDE 2023 plores the inter-series correlations and temporal dependencies with Fast Fourier Transform (FFT) plus multiscale dynamics extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, to address the LTTF problem in a sequence- to-sequence manner with linear computational complexity, an encoder-decoder architecture is employed on top of the sliding-window self-attention mechanism and the proposed stationary and instant recurrent network (namely, SIRN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' More specifically, the sliding-window attention allows each point to attend to its local neighbors for reference, such that the self- attention dedicated to a length-L time-series requires the O(L) complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides, to explore global signals in time-series data without violating the linear complexity, we renovate the cycle structure of the recurrent neural network (RNN) and distill stationary and instant patterns in long-term time-series with the series decomposition model in a recurrent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, to relieve the fluctuation effect caused by the aleatoric uncertainty [18] of time series data and improve the prediction reliability for LTTF, we further put efforts to model the underlying distribution of time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To be specific, we devise a normalizing flow block to absorb latent states yielded in the SIRN model and generate the distribution of future series directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' More specifically, we leverage the outcome latent state of the encoder, as well as the latent state of the decoder, as input to initiate the normalizing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Afterward, the latent state of the decoder can be cascaded to infer the distribution of the target series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Along this line, the information utilization for LTTF can be further enhanced and the time-series forecasting can be implemented in a generative fashion, which is more noise-resistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Extensive experiments on seven real-world datasets validate that Conformer outperforms the state-of-the-art (SOTA) base- lines with satisfactory margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To sum up, our contributions can be highlighted as follows: We reduce the complexity of self-attention to O(L) without sacrificing prediction capacity with the help of windowed attention and the renovated recurrent network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We design a normalizing flow block to infer target series from hidden states directly, which can further improve the prediction and equip the output with uncertainty awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Extensive experiments on five benchmark datasets and two collected datasets validate the superior long-term time- series forecasting performance of Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Methods for Time-Series Forecasting Many statistical methods have achieved big success in time- series forecasting (TF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For instance, ARIMA [5] is flexible to subsume multiple types of time-series but the limited scal- ability strains its further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Vector Autoregression (VAR) [6], [7] makes significant progress in multivariate TF by discovering dependencies between high-dimensional variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides, there exist other traditional methods for the TF problem, such as SVR [8], SVM [19], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', which also play important roles in different fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Another line of studies focuses on deep learning methods for TF, including RNN- and CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For example, LSTM [20] and GRU [21] show their strengths in extracting the long- and short-term dependencies, LSTNet [1] combines the CNN and RNN to capture temporal dependencies in the time-series data, DeepAR [9] utilizes the autoregressive model, as well as the RNN, to model the distribution of future time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' There are also some works focusing on CNN models [22]–[25], which can capture inner patterns of the time-series data through convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The Transformer [26] has shown its great superiority in NLP problems because of its effective self-attention mechanism, and it has been extended to many different fields successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' There are many attempts to apply the Transformer to TF tasks, and the main idea lies in aiming to break the bottleneck of efficiency by focusing on the sparsity of the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The LogSparse Transformer [14] allows each point to attend to itself and its previous points with exponential step size, Reformer [12] explores the hashing self-attention, Informer [15] utilizes probability estimation to reduce the time and memory complexities, Autoformer [13] studies the auto-correlation mechanism in place of self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' All the above models reduce the complexity of self-attention to O(L log L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The Sparse Transformer [27] reduces the com- plexity to O(L √ L) with attention matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The very recent Longformer [16] and BigBird [17] adopt a number of attention patterns and can further reduce the complexity to O(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' However, the above reduction of complexity is often at the expense of sacrificing information utilization and the self-attention mechanism might not be reliable when temporal patterns are intricate in the LTTF task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Generative Models There are works attempting to learn the distribution of future time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gaussian mixture model (GMM) [28] can learn the complex probability distribution with the EM algorithm, but it fails to suit dynamic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [29] proposed a generative model for TF by using the dynamic Gaussian mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [30] devises an end-to-end model to make coherent and probabilistic forecasts by generating the distribu- tion of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition, the authors of [31] proposed an autoregressive model to learn the distribution of the data and make the probabilistic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The variational inference was proposed for generative mod- eling and introduced latent variables to explain the observed data [32], which provides more flexibility in the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Both GAN [33] and VAE [34] show their impressive perfor- mances in distribution inference, but the cumbersome training process plus the limited generalization to new data hinder them for wider applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Normalizing Flows (NFs) are a family of generative models, an NF is the transformation of a simple distribution that results in a more complex distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' NF models have been applied in many fields successfully to learn intractable distribution, including image generation, noise modeling, video generation, audio generation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer employs the NF as an inner block for LTTF to absorb latent states in the encoder-decoder architecture, which differentiates itself from prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2 Published as a conference paper at ICDE 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1: The framework overview of Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, the encoder extracts local patterns with sliding-window multi-head attention (MHA) and explores long-term trends and instant patterns with the proposed SIRN module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The decoder then receives long sequence inputs with the target elements being padded into zeros, measures the weighted composition of multi-faceted temporal patterns, and generates the prediction for target elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' At last, the normalizing flow block absorbs latent states yielded in the encoder-decoder architecture and predicts target elements with a chain of invertible transformations directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' PROBLEM STATEMENT We introduce the problem definition in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Given a length-L time-series X = {x1, x2, · · · xL| xi ∈ Rdx} where xi is not limited to the univariate case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', dx ≥ 1), the time series forecasting problem takes a length-Lx time-series X = {xm+1, · · · , xm+Lx} as input to predict the future length-Ly time series Y = {xn+1, · · · , xn+Ly} (n = m+Lx and m = 1, · · · , L − Ly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For the sake of clarity, we denote Y = {yn+1, · · · , yn+Ly | yj ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Long-term time-series forecasting is to predict the future time-series with larger Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' METHODOLOGY The framework overview of Conformer is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer mainly consists of three parts: the input representa- tion block, encoder-decoder architecture, and normalizing flow block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' First, the input representation block preprocesses and embeds the input time series accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, the encoder- decoder architecture explores the local temporal patterns with windowed attention from time-series representations and ex- amines long-term intricate dynamics from both stationary and instant perspectives with the help of recurrent network and time-series decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, to improve information utilization, the normalizing flow block leverages latent states in the recurrent network and generates target series from the latent states directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The technical details of these three components will be introduced in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Input Representation The time series data exhibits intricate patterns since multi- faceted underlying signals are often complex and varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Given a length-L time series X, X = {x1, x2, · · · , xL|xi ∈ Rdx} (dx ≥ 1), we investigate the underlying multi-faceted relatedness in X from two perspectives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', the “vertical” feature perspective, and the “horizontal” temporal perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1) Multivariate Correlation: Complex relatedness among different variables in a multivariate time series hinders the effectiveness of distinguishing and harnessing important sig- nals for future series prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' On the one hand, the impacts of different variables on forecasting future series differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For instance, the heatmaps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2 illustrate rhythms of differ- ent variables in various time-series datasets, it is clear that (a) Exchange rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (b) Wind power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2: Different variables of time-series data evolve at varying rhythms and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The details of these datasets can be found in Section V-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' different variables exhibit distinct relatedness to the target variable, which can also vary over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' On the other hand, the well-leveraged dependencies among variables can benefit time-series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fast Fourier Transform (FFT) [35] has been proven to be effective in discovering the correlations for time series data [36]–[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Inspired by this, we adopt FFT to represent implicit multivariate correlations of a length-L time series by exploring the auto-correlation as follows: MRXX = f −1(f(X)f ∗(X)) , (1) where f and f −1 denote FFT and inverse FFT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The asterisk represents a conjugate operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides, we em- ploy Softmax to highlight informative variables accordingly: WR = Softmax(MRXX ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (2) 2) Multiscale Dynamics: Temporal patterns are helpful in solving the long-term time-series forecasting problem [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We further examine the temporal patterns by means of multiscale representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Specifically, a time series can present distinct temporal patterns at different temporal resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In other words, more attention should be paid to informative dynamics extracted at certain temporal resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To implement the temporal pattern extraction at different scales, we first devise a temporal resolution set S ⫅ {second, minute, hour, day, week, month, year} for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then the 3 Input Representation Multivariate Correlation Windowed MHA Windowed MHA Initial Distribution Encoder Input Decoder Input Construction Multiscale Dynamics 0 0 0 0 SIRN SIRN Chain of Transformations Multivariate Correlation Windowed MHA Windowed MHA Multiscale Dynamics SIRN SIRN Outputscounty1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 Dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 0 500 1000 1500 2000 2500 Time Point 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 pred 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 Dimension pred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 pred 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 pred 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 0 500 1000 1500 2000 2500 Time PointPublished as a conference paper at ICDE 2023 sampled time-series set ΓS = {ΓS1, · · · , ΓSK} is obtained, where K denotes the number of temporal resolutions and ΓSk is the sequence of sampled timestamps at corresponding tem- poral resolution Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Afterward, each series in ΓS is embedded into a latent space with d×L dimensionality, such that different series in ΓS are additive: ˜ΓS = E(ΓS) = {E(ΓS1), · · · , E(ΓSK)} = {˜ΓS1, · · · , ˜ΓSK} , (3) where E denotes an embedding operation and ˜ΓSk ∈ Rd×L represents the embedded series at a certain temporal resolution Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then the multiscale temporal patterns can be modeled as: ¯ΓS = WS Concat(˜ΓS) + (bS)′ = K � k=1 WS k (˜ΓSk)′ + (bS)′ , (4) where WS ∈ RL×L×K and bS ∈ Rd×L are trainable weights and bias, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The prime symbol denotes the matrix transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides, WS k ∈ RL×L denotes the k-th sliced matrix of WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3) Fusing Multivariate and Temporal Dependencies: Moreover, to make different variables in multivariate time series more distinguishable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' their importance for future series, we further apply the convolution to take temporal dependencies into account, which is defined as follows: X v = Wv ⊙ (WR X + X) + bv , (5) where ⊙ denotes the convolution operation, and Wv ∈ Rdx×d and bv ∈ Rd×L denote weights and bias, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Finally, by combining the above multivariate correlations and multiscale dynamics with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (2) and (5), the outcome time-series representation can be obtained as follows: X in = X v + ¯ΓS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (6) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Encoder-Decoder Architecture Our proposed Conformer adopts the encoder-decoder archi- tecture for long-term time-series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1) Attention Mechanism: The standard attention mech- anism [26] takes a three-tuple (query, key, value) as input and employs the scaled dot product and Softmax to cal- culate the weights against the value as: Attn(Q, K, V ) = Softmax( QKT √dk )V , where Q ∈ RL×dk, K ∈ RL×dk, and V ∈ RL×dv represent query, key and value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, the multi-head attention (MHA) [26] employs projections for the original query, key, and value N times, and the i-th projected query, key, and value can be obtained by Qi = QW Q i , Ki = KW K i , and Vi = V W V i , where W Q i ∈ Rdk×dk/N, W K i ∈ Rdk×dK/N, and W V i ∈ Rdv×dv/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Afterward, the attention can be applied to these queries, keys, and values in parallel, and the outcome is further concatenated and projected as follows: hai =Attn(Qi, Ki, Vi), i = 1, 2, · · · , N MHA(Q, K, V ) = Concat(ha1, ha2, · · · , haN)W o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (7) Sliding-Window Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Duplicated messages exist across different heads in full self-attention [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' A time series often shows a strong locality of reference, thus a great deal of information about a point can be derived from its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Hence, the full attention message might be too redundant for future series prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Given the importance of locality for TF, the sliding-window attention (with fixed window size w) allows each point attends to its 1 2w neighbors on each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Thus, the time complexity of this pattern is O(w × L), which scales linearly with input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Therefore, we adopt this windowed attention to realize self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2) Stationary and Instant Recurrent Network: Although the windowed attention can reduce the complexity to O(L), the information utilization could be sacrificed for LTTF due to point-wise sparse connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' RNNs have achieved big successes in many sequential data applications [41]–[44] at- tributed to their capabilities of capturing dynamics in se- quences via cycles in the network of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To enhance information utilization without increasing time and memory complexities, we, therefore, renovate the recurrent network accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, we not only distill the stationary (trend) and instant (seasonal) temporal patterns from input series but also integrate the distilled long-term patterns, as well as the aforementioned local temporal patterns, into the time-series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The architecture of the proposed Stationary and Instant Recurrent Network (SIRN) is demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Specifically, we feed the input representation to the first RNN block (followed by a Softmax) to initialize the global representation and add it to the local representation, as well as the original input representation, as follows: X in =SoftMax(RNN(X in)) × X in + MHAW (X in) + X in , (8) where MHAW (·) denotes the sliding-window attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Note that the RNN block (followed by Softmax) in the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (8) aims to capture the global temporal dependency, which can supplement the local dependency captured by the windowed attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Though intricate and diverse, the complex temporal patterns in different time-series data can be roughly divided into (coarse-grained) stationary trends and (fine-grained) instant patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Along this line, we employ the series decomposition introduced in [13], [45] to distill stationary and instant patterns by capturing trend and seasonal parts of the time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Similar to [13], we adopt the moving average to capture long- term trends and the residual of the original series subtracting the moving average as seasonal patterns: Xt = AvgPool(Padding(X in)), Xs = X in − Xt, (9) where Xt, Xs ∈ RL×dx denote the trend and seasonal parts of X in, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, we use a convolution layer to embed the seasonal pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' And, we feed the embedded rep- resentation, coupled with the local representation, to another decomposition block for distilling more seasonal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' This distillation process can be implemented in a recurrent way: 4 Published as a conference paper at ICDE 2023 (a) Stationary and instant recurrent network (SIRN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (b) Normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3: The architecture of SIRN and the normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (a) The first RNN block embeds the global information of input time-series and the second RNN block represents the aggregated trend information extracted by the decomposition block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The decomposition procedure following the initial decomposition can be repeated multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The latent state yielded by the first RNN will be utilized in the normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (b) After initiating the flow of transformations with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (15) and (16), the latent state of decoder is adopted to generate the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X (l) t , X (l) s = Decomp(Conv(X (l−1) s ) + MHAW (X in)), l = 1, · · · , η , (10) where Decomp denotes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (9), X (0) s = Xs and X (0) t = Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' On the other hand, the trend parts generated by different decompositions are merged and fed to the second RNN block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Finally, the distilled multi-faceted temporal dynamics are fused to generate the outcome representation: X out = W(X (η) s + RNN( η � l=0 X (l) t )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (11) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Time Series Prediction with Normalizing Flow The aforementioned SIRN framework adopts RNN to ex- tract global signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition, the hidden states yielded by RNN are beneficial for understanding the distribution of time- series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Specifically, we design a normalizing-flow block to learn the distribution of hidden states to increase the reliability of prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1) Background of Normalizing Flow: A time series X = {x1, · · · , xL} can be reconstructed by maximizing the marginal log-likelihood: log p(X) = �L i=1 log p(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Due to the intractability of such log-likelihood, a parametric inference model over the latent variables z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', q(z|x), was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, one can optimize the variational lower bound on the marginal log-likelihood of each observation x as follows: log p(x) ⩾Eq(z|x)[log p(x, z) − log q(z, x)] = log p(x) − DKL(q(z|x) || p(z|x)) =L(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' θ) , (12) where DKL(·) denotes the Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' When the dimensionality of z climbs up, the diagonal posterior distribution is often adopted, which is, however, not flexible enough to match the complex true posterior distributions [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To solve this, the Normalizing Flow [47] was proposed to build flexible posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Basically, one can start off with an initial random variable z0 (with a simple distribution, coupled with a known density function), and then apply a chain of invertible transformations ft, such that the outcome zT has a more flexible distribution: z0 ∽ q(z0|x), zt = ft(zt−1), t = 1, · · · , T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (13) Besides, as long as the Jacobian determinant det ��� dzt dzt−1 ��� is available, the transformation can take the following definition: ft(zt−1) = zt−1 + u g(wT zt−1 + b) , (14) where u, w and b are parameters, and g(·) denotes a nonlinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2) Normalizing Flow for LTTF: The proposed architec- ture of normalizing flow in Conformer is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Let h denote the hidden state yielded by the first RNN block in SIRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, draw a random variable from a Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', ϵ ∽ N(0, I), and the distribution of the hidden state in the encoder can be obtained as: ze = FCN(e) µ (he) + FCN(e) σ (he) · ϵ , (15) where FCN(e) µ and FCN(e) σ are two fully connected networks, he denotes the hidden state in encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Afterward, we take the latent representation ze and the decoder latent state hd as input to initiate the normalizing flow: z0 = FCN(d) µ (hd) + FCN(d) σ (hd) · ze .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (16) Now that the normalizing flow can be iterated as follows: zt = FCN(t) µ (hd, zt−1) + FCN(t) σ (hd, zt−1) · zt−1, t = 1, · · · , T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (17) Here, we utilize the decoder latent state to cascade the mes- sage, such that the future series can be generated directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Loss Function In order to coordinate with the other parts of Conformer, the commonly used log-likelihood is substituted for the MSE (mean squared error) loss function for learning the normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, the random variable sampled from the outcome distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', zt, is deemed as the point estimation of the target series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, we adopt MSE loss 5 Decomp Conv SoftMax + Decomp RNN + RNNFCN FCN FCN FCN N(O, I)Published as a conference paper at ICDE 2023 functions on prediction w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' the target series for both encoder- decoder architecture and normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Finally, the loss function is defined as follows: L = λ · MSE(Yout, Y) + (1 − λ) · MSE(Zout, Y) (18) where Yout and Zout denote the output of decoder and normalizing flow, respectively, and λ is a trade-off hyper- parameter balancing the relative contributions of encoder- decoder and normalizing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Experiment Settings 1) Datasets: We conduct experiments on seven datasets including five benchmark datasets and two collected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Table I describes some basic statistics of these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' ECL1 was collected in 15-minute intervals from 2011 to 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We select the records from 2012 to 2014 since many zero values exist in 2011 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The processed dataset contains the hourly electricity consumption of 321 clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We use ’MT 321’ as the target, and the train/val/test is 12/2/2 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Weather2 was recorded in 10-minute intervals from 07/2020 to 07/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' There exist 21 meteorological indicators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', the amount of rain, humidity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We choose temperature as the target, and the train/val/test is 10/1/1 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Exchange [1] records the daily exchange rates of eight countries from 1990 to 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We use the exchange rates of Singapore as the target, The train/val/test is 16/2/2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' ETT [15] records the electricity transformer temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Every data point consists of six power load features and the target value is “oil temperture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' This dataset is separated into {ETTh1, ETTh2} and {ETTm1, ETTm2} for 1-hour-level and 15-minute-level observations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We use ETTh1 and ETTm1 as our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The train/val/test are 12/2/2 and 12/1/1 months for ETTh1 and ETTm1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wind (Wind Power)3 records the generated wind power of a wind farm in 15-minute intervals from 01/2020 to 07/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The train/val/test is 12/1/1 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' AirDelay was collected from the “On-Time” database in the TranStas data library4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We extracted the flights arrived at the airports in Texas and examined arrival delays in the first month of the year 2022, and the canceled flights were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Note that the time interval of this dataset is varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' This dataset was split into train/val/test as 7:1:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2) Baselines: We compare Conformer with 9 baselines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', 5 Transformer methods (Autoformer, Informer, Reformer, Longformer, and LogTrans), 2 RNN methods (GRU and LSTNet), and 2 other deep methods (TS2Vec and N-Beats).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' GRU [21]: GRU employs the gating mechanism such that each recurrent unit adaptively captures temporal signals in the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In this work, we adopt a 2-layer GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='edu/ml/ datasets/ElectricityLoadDiagrams20112014 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='bgc-jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='de/wetter/ 3We collect this dataset and publish it at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='com/PaddlePaddle/ PaddleSpatial/tree/main/paddlespatial/datasets/WindPower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='transtats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='bts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The processed dataset is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='com/PaddlePaddle/PaddleSpatial/tree/main/paddlespatial/ datasets/AirDelay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' TABLE I: Statistical descriptions of the time-series datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Datasets # Dims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Time Span # Points Target Variable Interval ECL 321 01/2012 - 12/2014 26304 MT 321 1 hour Weather 21 01/2020 - 06/2021 36761 Temperature 10 mins Exchange 8 01/1990 - 12/2016 7588 Country8 1 day ETTh1 7 07/2016 - 07/2018 17420 OT 1 hour ETTm1 7 07/2016 - 07/2018 69680 OT 15 mins Wind 7 01/2020 - 05/2021 45550 Wind Power 15 mins AirDelay 6 01/01 - 01/31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2022 54451 ArrDelay – LSTNet [1]: LSTNet combines the convolution and recur- rent networks to extract short-term dependencies among variables and long-term trends in the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Note that, to simplify the parameter tuning, the highway and skip connection mechanisms are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' N-Beats [48]: N-Beats was proposed to address time-series forecasting via a deep model on top of the backward and forward residual links and a very deep stack of fully- connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We implement N-Beats for multivariate LTTF with suggested settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Reformer [12]: Reformer uses locality-sensitive hashing (LSH) attention and reversible residual layers to reduce the computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We implement Reformer by setting the bucket length and the number of rounds for LSH attention as 24 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Longformer [16]: Longformer combines the windowed attention with a task motivated global attention to scale up linearly as the sequence length grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' LogTrans [14]: LogTrans breaks the memory bottleneck of Transformer for LTTF via producing queries and keys with the help of causal convolutional self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The number of the LogTransformer blocks is set to 2 and the sub len of the sparse-attention is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Informer [15]: Informer proposes the ProbSparse slef- attention to reduce time and memory complexities, and handles the long-term sequence with self-attention distill- ing operation and generative style decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Autoformer [13]: Autoformer renovates the series decom- position with the help of auto-correlation mechanism, and put the series decomposition as a basic inner block of the deep model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' TS2Vec [49]: TS2Vec is a universal framework for learning representations of time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' It performs contrastive learn- ing in a hierarchical way over augmented context views, which leads to the robust contextual representation for each timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We implement TS2Vec for univariate LTTF with the suggested settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' All baselines employ the one-step prediction strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For the RNN-based methods, the number of hidden states is chosen from {16, 24, 32, 64}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For the Transformer-based methods, the number of heads of the self-attention is 8 and the di- mensionality is set as 512 for all attention mechanisms in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, the sampling factor of the self- attention is set to 1 for both Informer and Autoformer, other settings are the same as suggested by [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' All Transformer- based baselines (except Autoformer) use the same embedding 6 Published as a conference paper at ICDE 2023 TABLE II: Comparisons of multivariate LTTF results (the best and 2nd best scores are boldfaced and underlined, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Model Transformer-based RNN-based Others Conformer Longformer [16] Autoformer [13] Informer [15] Reformer [12] LSTNet [1] GRU [21] N-beats [48] Metric MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE ECL 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='3193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='3156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='3939 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='6044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='7750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5981 method applied to the Informer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' As suggested by [13], we omit the position embedding and keep the value embedding and timestamp embedding for Autoformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3) Implementation Details: Conformer5 includes a 2-layer encoder and a 1-layer decoder, as well as a 2-layer normalizing flow block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The window size of the sliding-window attention is 2, and λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (18) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We use an Adam optimizer, and the initial learning rate is 1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The batch-size is 32 and the training process employs early stopping within 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition, we use MAE (mean absolute error) and MSE (mean squared error) as the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' An input-Lx-predict-Ly window is applied to roll the train, validation and test sets with stride one time step, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' This setting is adopted for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The input length Lx is 96 and the predict length Ly is chosen from {48, 96, 192, 384, 768} on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The averaged results in 5 runs are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' All models are implemented in PyTorch and trained/tested on a Linux machine with one A100 40GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' All of the RNN blocks in Conformer are implemented with GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Under the multivariate LTTF setting, we adopt 1-layer GRU and 2-layer GRU for encoder and decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Under the univariate LTTF setting, both the encoder and decoder adopt 1-layer GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 5The source code of Conformer is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='com/ PaddlePaddle/PaddleSpatial/tree/main/research/Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Prediction Results of Multivariate LTTF We compare Conformer to other baselines in terms of MSE and MAE under the multivariate time-series forecasting setting, and the results are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We can observe that Conformer outperforms SOTA Transformer-based models, as well as other competitive methods, under differ- ent predict-length settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For example, under the predict- 96 setting, compared to the second best results, Conformer achieves 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2021→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2700 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2073 dependencies in high-dimensional time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, MSE and MAE scores of Conformer grows slower as the predict length prolongs than other baselines indicating better stability of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For the datasets w/ periodicity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', Weather, ECL) and w/o periodicity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', Exchange), Conformer consistently delivers good performance, which suggests the promising generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition, for the dataset with irregular time intervals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', AirDelay), Conformer still achieves the best performance consistently, while the improvements are less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' This suggests that the temporal patterns in less-structured time-series data are more challenging for deep models to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Forecasting with Time-Determined Lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We further evaluate the performance of multivariate LTTF when the input and output lengths are configured as time-determined intervals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, in this experiment, the input length Lx is set to 1 day and the output length Ly is chosen from {1 day (1D), 1 week (1W), 2 weeks (2W), 1 month (1M)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We inspect forecasting performances of different methods on ETTh1 and ETTm1 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The results are reported in Ta- ble III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' As depicted, Conformer still achieves the best (or competitive) performance, which suggests the high capacity of Conformer in perceiving long-term signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Performance Comparisons Under Univariate LTTF Table IV reports prediction performances of different meth- ods under the univariate LTTF setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer achieves the best (or competitive) MSE and MAE scores under vari- ous predict-length settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, satisfactory predic- tion improvements can be observed on Exchange, ECL and Weather datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For instance, compared to the second best results, Conformer achieves 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='3889→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2107) MSE reduction under predict-192 on Exchange dataset, and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='9% 8 Published as a conference paper at ICDE 2023 TABLE V: Ablation study of the input representation.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='4352→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='3659) on ECL dataset and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='4% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0914→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0846) on Weather dataset under predict-384 and predict-48, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, Conformer still achieves the best scores on the AirDelay dataset, which further demonstrates the effective- ness of Conformer in extracting complex temporal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition, under the univariate LTTF setting, we find that RNN-based methods achieve competitive prediction results on Weather and Wind datasets, which can validate the advantages of RNN in extracting temporal dynamics of the time-series data with low entropy and regular patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ablation Study We conduct the ablation study under multivariate TF setting6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1) Multivariate Correlation and Multiscale Dynamics: We compare Conformer with its tailored variants w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' the multivariate correlation and multiscale dynamics, and report their prediction performances on ECL and ETTm1 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' From Table V, we can obtain several insightful clues on how to embed the input series for LTTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1) X in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X in −R: Multivariate correlation contributes less when the dimensions of series is higher (#dims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' of ECL data is much larger than ETTm1 data) or the predict-length is prolonged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2) X in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 6Hereinafter, all the experiments are carried out under the multivariate TF setting by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X in −X−Γ: Temporal dependency is more important for the series with lower dimensions, and for high dimensional time-series, the effectiveness of temporal dependency can be replaced by the inter-series correlation when Ly climbs up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3) X in −R v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X in −X and X in −R−Γ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X in −X−Γ: Multiscale dynamics delivers better performance when being guided by the raw series, which holds regardless of #dims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' of time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides, multivariate correlation contributes more than the raw data for low dimensional time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4) X in −R v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X in −R−Γ and X in −X v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' X in −X−Γ: Multiscale dynamics could harm the performance for LTTF when being equipped with the multivariate correlation if the raw time-series is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2) Stationary and Instant Recurrent Network (SIRN): For the ablation study of the proposed SIRN, we compare Conformer to its different variants by tailoring the encoder- decoder architecture on the Wind dataset, which can be found in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Specifically, we replace the sliding-window attention and the RNNs with other self-attention mechanisms to verify the effectiveness of SIRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' From Table VI, we can see that SIRN achieves best performance under different settings, which validate the effectiveness of information utilization of combining the local and global patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3) Normalizing Flow: To verify the effectiveness of nor- malizing flow block in Conformer for LTTF task, we compare 9 Published as a conference paper at ICDE 2023 (a) Input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (b) Window size w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (c) Trade-off parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (d) Number of transformations in Normalizing Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4: Parameter sensitivity analysis of Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (a) Time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (b) Memory cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 5: Computation efficiency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The input length is set as 96 and all the experiments are conducted on the Wind dataset under the multivariate forecasting setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' the original Conformer with its several variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, we tailor the normalizing flow block in Conformer by real- izing a generative forecast method with the help of Gaussian probabilistic model as follows: Conformer ze −NF : The outcome distribution zt (yielded by normalizing flow) is replaced by ze (obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer zd −NF : The outcome distribution zt (yielded by normalizing flow) is replaced by zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, we replace he with hd in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (15) and generate zd accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer ze+zd −NF : The outcome distribution zt (yielded by normalizing flow) is replaced by z0 (obtained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer −NF : We implement a tailored Conformer by removing the normalizing flow framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The prediction results on Wind dataset are reported in Ta- ble VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We can observe that: 1) The contribution of normal- izing flow is indispensable for LTTF regardless of forecast setting and predict length, and 2) the way that we adapt the normalizing flow to the LTTF task is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Parameter Sensitivity Analysis We report parameter sensitivity analysis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4, which is conducted on the Wind dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To be specific, we inspect four hyper parameters including the input length Lx, the window size w of sliding-window attention, the trade-off parameter λ and the number of transformations in Normalizing Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Generally, we can observe that the performance of Conformer is quite stable most of the time w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' the varying of different hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In particular, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4a, long-term time-series forecasting setting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', Ly = 384) seems to be more capable of handling longer input, though the volatility of performance is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Computational Efficiency Analysis We conduct execution time consumption and memory usage comparisons between Conformer (with sliding-window atten- tion) and other attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We replace the stan- dard self-attention mechanism in Transformer with different variants and carry out the prediction with the corresponding method for 103 times (taking the sequences in different time spans as inputs), then the averaged running time per forecast is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' For the memory cost comparisons, the maximum memory usage is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The time consumption and memory usage of different attentions are demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Conformer performs with better efficiency in both short- and long-term time-series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Model Analysis 1) Fusing Inter-Series and Across-Time Dependencies: As introduced in Section IV-A, the series data is embedded and fused by taking the multivariate correlation and multiscale dynamics into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To further assess the effectiveness of input representation module in Conformer, we realize different ways of fusing multivariate correlation and multiscale dynamics below (let WΓ = Softmax(¯ΓS)): Method 1: X in = Wv ⊙ (WΓWRX + X) + bv Method 2: X in = Wv ⊙ (WRX + WΓX) + bv Method 3: X in = Wv ⊙ (WRX + WΓX + X) + bv Method 4: X in = [Wv ⊙ (WRX + X) + bv]WΓ The results are reported in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We can see that how to fuse the multivariate correlation and temporal dependency is important for the LTTF task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' This impact weighs more for low dimensional time-series data since the self-attention mechanism in encoder-decoder architecture can better explore intricate dependencies when the dimensionality grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2) Uncertainty-Aware Forecasting: The outcome vari- ance of the Normalizing Flow block can suggest the fluctuation range of the forecasting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We randomly select a case in ETTm1 dataset under the multivariate setting and demonstrate the forecasting results with uncertainty quantification for dif- ferent output lengths in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We can see that Conformer tends to make a conservative forecast and the uncertainty quantification can cover the extreme ground truth values if the NF block can be weighted more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3) How Far The Message Should Be Cascaded in Nor- malizing Flow: We inspect how the normalizing flow works for LTTF by varying the number of transformations on two 10 Execution time (ms) ProbSparseAttentionfrom Informer 630 Sliding-Window AttentionfromComformer Auto-CorrelationfromAutoformer 530 Full-AttentionfromTransformer 430 330 230 130 30 1000 Prediction lengthProbSparseAttentionfromInformer Sliding-Window Attentionfrom Comformer 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Auto-CorrelationfromAutoformer Full-AttentionfromTransformer 21 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 21 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 768 1000 Prediction lengthLy=48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 96=^7 Ly=192 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='6 Ly=384 v=768 E S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='4 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 Input length (Lx)Ly=48 96=^7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 Ly=192 Ly=384 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='6 Ly=768 ISE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='4 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 2 4 6 8 12 Sliding-window size (w)Ly=48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 96=^7 Ly=192 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='6 Ly=384 y=768 E S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='4 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2 0.' metadata={'source': 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8 L 2 5 3 4 #transformations in NFPublished as a conference paper at ICDE 2023 TABLE VIII: Comparisons of fusing inter-series correlation and time dependency for LTTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Dataset ECL Exchange Predict Length 48 96 192 384 768 48 96 192 384 Conformer MSE 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (d) Predict Length = 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 6: With the help of Normalizing Flow, Conformer can generate the prediction results with uncertainty quantification for LTTF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Four illustrative cases are demonstrated on the ETTm1 dataset under the multivariate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' λ denotes the contributions of the encoder-decoder, that is, 1 − λ represents the impacts of the normalizing flow block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' cases in ECL and ETTm1 datasets, respectively, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We can see that the further the latent variable being transformed the better the outcome series performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Therefore, the power of normalizing flow in Conformer for LTTF should be explored more dedicatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4) How to Feed Hidden States to The Normalizing Flow Block in Conformer: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1, in both encoder and decoder, the first outcome hidden state of the last SIRN layer is fed to the normalizing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To assess the effect of feeding hidden states to normalizing flow, we implement Conformer by combining the outcome hidden states in the first/last SIRN layer of the encoder/decoder, which results in Conformer (h(e) k , h(d) k ), Conformer (h(e) 1 , h(d) k ), Conformer (h(e) 1 , h(d) 1 ) and Conformer (h(e) k , h(d) 1 ) where k denotes the last SIRN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We report the prediction results in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' As can be seen, the impact of feeding different hidden states to normalizing flow is generally marginal though, the low dimensional time-series forecasting is more sensitive to the way of absorbing hidden states for normalizing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Multivariate Time-series Forecasting Showcase We additionally plot the prediction and the ground truth of the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The qualitative comparisons between Conformer and other baselines on ETTm1 dataset are demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' We can see that, our model obviously achieves the best performance among different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Discussion Windowed Attention: Conformer v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Swin Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The windowed attention mechanism is applied in many appli- cations thanks to its linear complexity, such that the powerful self-attention can be scaled up to large data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The very recent Swin Transformer [50] and its variant [51] adopt the windowed attention and devise a shifted window attention to implement a general purpose backbone for computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Basi- cally, both Conformer and Swin Transformer exploit the self- attention within neighbored/partitioned windows regarding the 11 Point Estimate Ground Truth 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='95 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='9 ^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8Point Estimate Ground Truth A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='95 ^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='9 ^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8Point Estimate Ground Truth 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='95 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='9 ^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8Point Estimate Ground Truth ^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='95 入=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='9 ^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='8Published as a conference paper at ICDE 2023 (a) #transformations = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (b) #transformations = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (c) #transformations = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (d) #transformations = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (e) #transformations = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (f) #transformations = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (g) #transformations = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (h) #transformations = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 7: Uncertainty-aware LTTF with varying #transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To evaluate the performance of normalizing flow more clearly, we omit the contribution of SIRN by setting λ = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (a)–(d) and (e)–(h) demonstrate two cases in ECL and ETTm1 datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' (a) Conformer (b) Longformer (c) Reformer (d) Informer (e) Autoformer (f) N-Beats (g) LSTNet (h) GRU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 8: Prediction cases on the ETTm1 dataset under the input-96-predict-192 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides the locality, connectivity is another merit one can not neglect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' To achieve connectivity, a shifted window mechanism is proposed for Swin Transformer, while we propose SIRN for Conformer so as to absorb long- range dependencies in the time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Comparisons of Computational Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The win- dowed attention contributes most to the complexity reduc- tion of Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Hence, we take different SOTA attention mechanisms as competitors to conduct the computational complexity analysis in Section V-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' The computational costs of other components in Conformer are not elaborated, which will be provided in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a transformer-based model, namely Conformer, to address the long-term time-series fore- casting (LTTF) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Specifically, Conformer first embeds the input time series with the multivariate correlation modeling and multiscale dynamics extraction to fuel the downstream self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Then, to reduce the computation complexity of self-attention and fully distill the series-level temporal dependencies without sacrificing information utiliza- tion for LTTF, sliding-window attention, as well as a proposed stationary and instant recurrent network (SIRN), are equipped to the Conformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Moreover, a normalizing flow framework is employed to further absorb the latent states in the SIRN, such that the underlying distribution can be learned and the target series can be directly reconstructed in a generative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Extensive empirical studies on six real-world datasets validate that Conformer achieves state-of-the-art performance on long- term time-series forecasting under multivariate and univariate prediction settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' In addition, with the help of normalizing flow, Conformer can generate the prediction results with uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank China Longyuan Power Group Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' for sup- porting this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Besides, this work was supported in part by National Key R&D Progamm of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2021ZD0110303).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 Point Estimate Ground Truth Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 Target 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='5 0 20 40 60 80 Time PointPoint Estimate 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, “Modeling long-and short-term temporal patterns with deep neural networks,” in The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 95–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, “A review of wind speed and wind power forecasting with deep neural networks,” Applied Energy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 304, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 117766, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Han, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xiong, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Dou, “Joint air quality and weather prediction based on multi-adversarial spatiotemporal networks,” in Proceedings of the 35th AAAI Conference on Artificial Intelligence, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Matsubara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Sakurai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Van Panhuis, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Faloutsos, “Funnel: automatic mining of spatially coevolving epidemics,” in Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 105–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ariyo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Adewumi, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ayo, “Stock price prediction using the arima model,” in 2014 UKSim-AMSS 16th International Conference on Computer Modelling and Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' IEEE, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 106–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gregor, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Danihelka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Mnih, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Blundell, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wierstra, “Deep autoregressive networks,” in International Conference on Ma- chine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' PMLR, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1242–1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Melnyk and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Banerjee, “Estimating structured vector autoregressive models,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' PMLR, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 830–839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kim, “Financial time series forecasting using support vector machines,” Neurocomputing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 307–319, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Salinas, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Flunkert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gasthaus, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Januschowski, “Deepar: Probabilistic forecasting with autoregressive recurrent networks,” Inter- national Journal of Forecasting, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1181–1191, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Graves, “Generating sequences with recurrent neural networks,” arXiv preprint arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='0850, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [11] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Sutskever, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Vinyals, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Le, “Sequence to sequence learning with neural networks,” in Advances in neural information processing systems, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3104–3112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kitaev, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kaiser, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Levskaya, “Reformer: The efficient trans- former,” arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='04451, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', “Autoformer: Decomposition transform- ers with auto-correlation for long-term series forecasting,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Jin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xuan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yan, “Enhancing the locality and breaking the memory bottleneck of transformer on time series forecasting,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='00235, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='00235 [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xiong, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, “Informer: Beyond efficient transformer for long sequence time-series forecasting,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 12, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 11 106–11 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Beltagy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Peters, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cohan, “Longformer: The long- document transformer,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='05150, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='05150 [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zaheer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Guruganesh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Dubey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ainslie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Alberti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Onta˜n´on, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Pham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ravula, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ahmed, “Big bird: Transformers for longer sequences,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='14062, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='14062 [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gawlikowski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Tassi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ali, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Humt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Feng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kruspe, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Triebel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Jung, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Roscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', “A survey of uncertainty in deep neural networks,” arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='03342, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cao and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Tay, “Support vector machine with adaptive parameters in financial time series forecasting,” IEEE Transactions on neural networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1506–1518, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Hochreiter and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Schmidhuber, “Long short-term memory,” Neural computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1735–1780, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Dey and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Salem, “Gate-variants of gated recurrent unit (gru) neural networks,” in 2017 IEEE 60th international midwest symposium on circuits and systems (MWSCAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1597–1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Borovykh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bohte, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Oosterlee, “Conditional time se- ries forecasting with convolutional neural networks,” arXiv preprint arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='04691, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Mittelman, “Time-series modeling with undecimated fully convolu- tional neural networks,” arXiv preprint arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='00317, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Borovykh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bohte, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Oosterlee, “Dilated convolutional neural networks for time series forecasting,” Journal of Computational Finance, Forthcoming, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Benhaddi and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ouarzazi, “Multivariate time series forecasting with dilated residual convolutional neural networks for urban air quality prediction,” Arabian Journal for Science and Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3423–3442, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Vaswani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Shazeer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Parmar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Uszkoreit, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Jones, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gomez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kaiser, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Polosukhin, “Attention is all you need,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='03762, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='03762 [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Child, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Radford, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Sutskever, “Generating long sequences with sparse transformers,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='10509, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='10509 [28] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Reynolds, “Gaussian mixture models.” Encyclopedia of biomet- rics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 741, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 659–663, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ni, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Song, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Davidson, “Dynamic gaussian mixture based deep generative model for robust forecasting on sparse multivariate time series,” arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='02164, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Rangapuram, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Werner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Benidis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Mercado, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gasthaus, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Januschowski, “End-to-end learning of coherent probabilistic forecasts for hierarchical time series,” in Proceedings of the 38th International Conference on Machine Learning, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Proceedings of Machine Learning Research, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Meila and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' PMLR, 18–24 Jul 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 8832–8843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='mlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='press/v139/rangapuram21a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='html [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Rasul, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Seward, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Schuster, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Vollgraf, “Autoregressive denoising diffusion models for multivariate probabilistic time series forecasting,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='12072, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='12072 [32] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kobyzev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Prince, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Brubaker, “Normalizing flows: An introduction and review of current methods,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [33] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Pouget-Abadie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Mirza, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Warde-Farley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ozair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Courville, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bengio, “Generative adversarial nets,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 27, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kingma and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Welling, “Auto-encoding variational bayes,” arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='6114, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Nussbaumer, “The fast fourier transform,” in Fast Fourier Trans- form and Convolution Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Springer, 1981, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 80–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [36] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cochran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cooley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Favin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Helms, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kaenel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Maling, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Nelson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Rader, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Welch, “What is the fast fourier transform?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1664–1674, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Musbah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' El-Hawary, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Aly, “Identifying seasonality in time series by applying fast fourier transform,” in 2019 IEEE Electrical Power and Energy Conference (EPEC), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [38] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Box, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Jenkins, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Reinsel, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ljung, Time series analysis: forecasting and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' John Wiley & Sons, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zaheer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ahmed, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Smola, “Latent lstm allocation: Joint clustering and non-linear dynamic modeling of sequence data,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' PMLR, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3967–3976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [40] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kovaleva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Romanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Rogers, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Rumshisky, “Revealing the dark secrets of BERT,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' abs/1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='08593, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='org/abs/1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='08593 [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cho, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Van Merri¨enboer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gulcehre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bahdanau, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bougares, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Schwenk, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bengio, “Learning phrase representations using rnn encoder-decoder for statistical machine translation,” arXiv preprint arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content='1078, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [42] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Luo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chen, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yoshioka, “Dual-path rnn: efficient long sequence modeling for time-domain single-channel speech separation,” in ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 46–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [43] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xuan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Shi, “A day-ahead pv power forecasting method based on lstm-rnn model and time correlation modification under partial daily pattern prediction framework,” Energy Conversion and Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 212, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 112766, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Monti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Fiorentino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Milanetti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Gosti, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Tartaglia, “Prediction of time series gene expression and structural analysis of gene regulatory networks using recurrent neural networks,” Entropy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 141, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 13 Published as a conference paper at ICDE 2023 [45] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Robert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' William, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Irma, “Stl: A seasonal-trend decomposition procedure based on loess,” Journal of official statistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 3–73, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [46] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Nguyen and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Quanz, “Temporal latent auto-encoder: A method for probabilistic multivariate time series forecasting,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 10, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 9117–9125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Kingma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Salimans, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Jozefowicz, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Sutskever, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Welling, “Improved variational inference with inverse autoregressive flow,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 4743–4751, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [48] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Oreshkin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Carpov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Chapados, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Bengio, “N-beats: Neural basis expansion analysis for interpretable time series forecasting,” in International Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [49] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yue, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Duan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Tong, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xu, “Ts2vec: Towards universal representation of time series,” in Proceed- ings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 8, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 8980–8987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [50] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Guo, “Swin transformer: Hierarchical vision transformer using shifted windows,” in Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' [51] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Yao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Ning, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Cao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Dong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Wei, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' Guo, “Swin transformer v2: Scaling up capacity and resolution,” in International Conference on Computer Vision and Pattern Recognition (CVPR), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} +page_content=' 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtA0T4oBgHgl3EQfH_9T/content/2301.02068v1.pdf'} diff --git a/kNE_T4oBgHgl3EQf5Rz9/content/tmp_files/2301.08358v1.pdf.txt b/kNE_T4oBgHgl3EQf5Rz9/content/tmp_files/2301.08358v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cbbc122cdc706aca4c187a9891c98e51cc5695b --- /dev/null +++ b/kNE_T4oBgHgl3EQf5Rz9/content/tmp_files/2301.08358v1.pdf.txt @@ -0,0 +1,4476 @@ +On thermodynamically compatible finite volume schemes for +continuum mechanics +S. Busto1, M. Dumbser2, I. Peshkov3, E. Romenski4 +(1) Department of Applied Mathematics I, Universidade de Vigo, Campus As Lagoas, 36310 Vigo, Spain +(2,3) Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, +38123 Trento, Italy +(4) Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, 630090 Novosibirsk, Russia +Abstract +In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for +overdetermined, hyperbolic and thermodynamically compatible PDE systems. In the following we will +denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible +gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum me- +chanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic +solids at large deformations as well as viscous fluids as two special cases of a more general first order +hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies +in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual +total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of +a thermodynamically compatible discretization of all the other equations. For this, we first construct a +discrete framework for the compressible Euler equations that mimics the continuous framework of Go- +dunov’s seminal paper An interesting class of quasilinear systems of 1961 exactly at the discrete level. +All other terms in the governing equations of the more general GPR model, including non-conservative +products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the +exact conservation of total energy density as a direct consequence of all the other equations. As a result, +the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy +a discrete entropy inequality by construction. We show some computational results obtained with HTC +schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the +governing partial differential equations. +Keywords: thermodynamically compatible finite volume schemes; semi-discrete and fully-discrete Go- +dunov formalism; vanishing viscosity limit; entropy inequality; hyperbolic thermodynamically compatible +PDE systems; overdetermined hyperbolic PDE systems; unified GPR model for solid mechanics and fluid +mechanics. +1 +Introduction +In his groundbreaking work An interesting class of quasilinear systems [29] published 60 years ago in +1961 Godunov discovered the connection between symmetric hyperbolicity in the sense of Friedrichs [24] +and thermodynamic compatibility, 10 years before the work of Friedrichs & Lax on the same subject [25]. +In subsequent work by Godunov & Romenski and collaborators, the theory of symmetric hyperbolic and +thermodynamic compatible (SHTC) systems was extended to a wide class of mathematical models in con- +tinuum physics, ranging from the magnetohydrodynamics (MHD) equations over nonlinear hyperelasticity +to compressible multi-phase flows and relativistic gasdynamics, see e.g. [30, 31, 32, 33, 51, 49, 28, 50]. All +SHTC systems can be rigorously derived from an underlying variational principle. A connection between +SHTC systems and Hamiltonian mechanics was established in [45], emphasizing a peculiar role of the +1saray.busto@uvigo.es +2michael.dumbser@unitn.it +3ilya.peshkov@unitn.it +4evrom@math.nsc.ru +1 +arXiv:2301.08358v1 [math.NA] 19 Jan 2023 + +energy potential (Hamiltonian), while an extension to continuum mechanics with torsion was provided +in [47]. +Notwithstanding the mathematical elegance of the SHTC framework, to the best knowledge of the +authors it was up to now never directly carried over to the discrete level. +Most existing papers on +thermodynamically compatible schemes are based on the ideas of the seminal work of Tadmor [54], in +which a discrete compatibility with the entropy equation is sought, rather than a discrete compatibility +with the total energy conservation, as suggested by the SHTC framework. A fully discrete entropy-stable +scheme has been recently forwarded in [48], while the convergence of entropy-stable schemes was proven +in [17]. For high order entropy-compatible schemes the reader is referred to [26, 18, 38, 19, 35] and +references therein. In [23, 2] entropy compatible schemes were applied to non-conservative hyperbolic +equations. Last, but not least, we also would like to mention the general framework for the construction +of numerical methods that satisfy additional extra conservation laws recently introduced by Abgrall in +[1]. A first attempt to achieve discrete energy conservation as a consequence of all other equations was +made in [10] for a novel hyperbolic model of unsteady turbulent shallow water flows. For compatible +schemes in the context of Lagrangian hydrodynamics, where total energy conservation is obtained as a +consequence of the discrete mass, momentum and internal energy equations, see the interesting papers +[15, 3], while a fully-discrete compatible kinetic energy preserving scheme was forwarded in [53]. However, +all aforementioned schemes address only the compressible Euler equations and not the full GPR model +of continuum mechanics. +The main contribution of this paper is thus a new thermodynamically compatible finite volume scheme +for the GPR model of continuum mechanics [51, 46, 20] in which the discrete energy conservation law +is obtained as a consequence of a compatible discretization of all the other equations. +To the best +knowledge of the authors, this is the first time that such a provably thermodynamically compatible +scheme is proposed for the PDE system (1), which is able to describe solid mechanics and fluid mechanics +at the same time. We stress that the main objective of this paper is not to introduce a better or more +efficient numerical scheme compared to existing methods, but to introduce a radically new concept: direct +discretization of the entropy inequality in order to obtain the discrete total energy conservation law as +a consequence. We also would like to clearly indicate the three main shortcomings of the new method +introduced in this paper: +i) the numerical fluxes are only known implicitly via path integrals of the physical flux in phase space; +however, also other numerical methods are based on path integrals, like the Osher-Solomon flux +[40], the entropy-consistent scheme of Tadmor [54] and the family of path-conservative schemes of +Castro and Par´es [16, 41]; +ii) currently, in our new framework, a numerical scheme that provably satisfies total energy conserva- +tion at the fully-discrete level can only be achieved at the aid of a special implicit time integrator; +iii) in the case of the semi-discrete scheme, total energy conservation is in general lost at the fully- +discrete level once a standard, nonsymplectic Runge-Kutta time discretization is employed. +The rest of this paper is organized as follows. In Section 2 we present the unified first order hyperbolic +model of continuum mechanics (GPR model) under consideration. In Sections 3 and 5 the construction +of thermodynamically compatible semi-discrete and fully-discrete finite volume schemes is explained for +the one-dimensional case. In Section 4 an extension to the general multi-dimensional case is presented, +together with a proof of nonlinear stability in the energy norm and a proof of the entropy inequality +satisfied by the scheme. Numerical results are shown in Section 6 for the fluid and the solid limits of the +governing PDE system. The paper closes with some concluding remarks and an outlook to future work +in Section 7. +2 +Mathematical model and its structure +We consider the following first order hyperbolic model of continuum mechanics regularized with vanishing +viscosity terms and which goes back to the work of Godunov [29], Godunov & Romenski [31, 51, 33] and +2 + +Peshkov & Romenski, see [46, 20]: +∂ρ +∂t + ∂(ρvk) +∂xk +− ∂ +∂xm +� +ϵ ∂ρ +∂xm +� += 0, +(1a) +∂ρvi +∂t ++ ∂ (ρvivk + p δik + σik + ωik) +∂xk +− ∂ +∂xm +� +ϵ∂ρvi +∂xm +� += 0, +(1b) +∂ρS +∂t + ∂ (ρSvk + βk) +∂xk +− ∂ +∂xm +� +ϵ ∂ρS +∂xm +� += Π + αikαik +θ1(τ1)T + +βiβi +θ2(τ2)T ≥ 0, +(1c) +∂Aik +∂t ++ ∂(Aimvm) +∂xk ++ vm +�∂Aik +∂xm +− ∂Aim +∂xk +� +− ∂ +∂xm +� +ϵ∂Aik +∂xm +� += − αik +θ1(τ1), +(1d) +∂Jk +∂t + ∂ (Jmvm + T) +∂xk ++ vm +� ∂Jk +∂xm +− ∂Jm +∂xk +� +− ∂ +∂xm +� +ϵ ∂Jk +∂xm +� += − +βk +θ2(τ2), +(1e) +∂E +∂t + ∂ (vk (E1+E2+E3 + E4) + vi (p δik+σik + ωik)+hk) +∂xk +− ∂ +∂xm +� +ϵ ∂E +∂xm +� += 0. +(1f) +In the overdetermined system above q = {qi} = (ρ, ρvi, ρS, Aik, Jk)T denotes the state vector, the total +energy potential is E = ρE = E1 + E2 + E3 + E4 with Ei = ρEi, ϵ > 0 is a vanishing viscosity and the +nonnegative entropy production term due to the viscous terms is given by +Π = ϵ +T ∂xmqi ∂2 +qiqjE ∂xmqj ≥ 0, +(2) +since ϵ > 0 and we assume that the temperature T > 0 and that the Hessian of the total energy potential is +at least positive semi-definite, Hij := ∂2 +qiqjE ≥ 0. Throughout this paper, we use the notations ∂p = ∂/∂p +and ∂2 +pq = ∂2/(∂p∂q) for the first and second partial derivatives w.r.t. generic coordinates or quantities +p and q, which may also be vectors or components of a vector. Furthermore, we make use of the Einstein +summation convention over repeated indices. Last but not least, in some occasions we also use bold face +symbols in order to denote vectors and matrices, e.g. q = {qi} and A = {Aik}, and so on. In the above +model the four contributions to the total energy density are +E1 = +ργ +γ − 1eS/cv, +E2 = 1 +2ρvivi, +E3 = 1 +4ρc2 +s ˚ +Gij ˚ +Gij, +E4 = 1 +2c2 +hρJiJi, +(3) +with the metric tensor G components and its trace-free part ˚ +G given by Gik = AjiAjk, and ˚ +Gik = +Gik − 1 +3 Gmmδik. The vector of thermodynamic dual variables reads p = ∂qE = {pi} = (r, vi, T, αik, βk)T +with +r = ∂ρE, +vi = ∂ρviE, +T = ∂ρSE, +αik = ∂AikE, +βk = ∂JkE. +(4) +The pressure is defined as p = ρ ∂ρE + ρvi ∂ρviE + ρS ∂ρSE − E = ρ2∂ρE, the stress tensors due to shear +stress and thermal stress are, respectively, +σik = Aji∂AjkE = Ajiαjk = ρc2 +sGij ˚ +Gjk, +ωik = Ji∂JkE = Jiβk = ρc2 +hJiJk, +(5) +while the heat flux vector is given by +hk = ∂ρSE ∂JkE = Tβk = ρc2 +hTJk. +(6) +Note that for our convenience, we use the opposite sign in the definition of the stress tensor compared to +the generally accepted notation. Furthermore, θ1(τ1) > 0 and θ2(τ2) > 0 are two algebraic functions of +the state vector q and the positive relaxation times τ1 > 0 and τ2 > 0: +θ1 = 1 +3ρz1τ1 c2 +s |A|− 5 +3 , +θ2 = ρz2τ2 c2 +h, +z1 = ρ0 +ρ , +z2 = ρ0T0 +ρ T , +(7) +3 + +with ρ0 and T0 being some reference density and temperature. It is easy to check that (1f) is a consequence +of (1a)-(1e), i.e. +(1f) = r · (1a) + vi · (1b) + T · (1c) + αik · (1d) + βk · (1e). +(8) +In [20] a formal asymptotic analysis of the model (1a)-(1f) was carried out, revealing that in the stiff +limit the stress tensor σik and the heat flux hk tend to +σik = −1 +6ρ0c2 +sτ1 +� +∂kvi + ∂ivk − 2 +3 (∂mvm) δik +� +, +hk = −ρ0T0c2 +hτ2∂kT, +(9) +i.e. when the relaxation times τ1, τ2 → 0, the Navier-Stokes-Fourier equations are retrieved with effective +shear viscosity µ = 1 +6ρ0c2 +sτ1 and heat conductivity κ = ρ0T0c2 +hτ2. +3 +Thermodynamically compatible semi-discrete finite volume +scheme for the complete model in one space dimension +In this section, we derive the thermodynamically compatible semi-discrete finite volume scheme for model +(1) in one space dimension. To this end, we start analysing the black terms on the system which enter +into the original Godunov formalism [29]. Once compatibility of these terms is established for the Euler +subsystem, we can include dissipative terms which require the consideration of the non-negative entropy +production term, coloured in blue. The third step is the study of the red terms of (1b)-(1f) corresponding +to the discretization of the distortion field and the thermal impulse. Finally, also the relaxation terms, +in green, are addressed. +Throughout the discretization, we will employ lower case subscripts, i, j, k, for tensor indices while +lower case superscripts, ℓ, refer to the spatial discretization index. +Accordingly, we denote by Ωℓ = +[xℓ− 1 +2 , xℓ+ 1 +2 ] a spatial control volume in one space dimension. +The Godunov form [29] of the Euler +subsystem (black terms in (1)) reads +(∂pL)t + ∂x (∂p(v1L)) = 0, +(10) +q = ∂pL, +p = ∂qE, +f = ∂p(v1L), +F = p · f − v1L, +(11) +with the generating potential L = p·q−E, which is the Legendre transform of the total energy potential +E. The semi-discrete finite volume discretization of (10) reads +d +dtqℓ = −f ℓ+ 1 +2 − f ℓ− 1 +2 +∆x += − +� +f ℓ+ 1 +2 − f ℓ� +− +� +f ℓ− 1 +2 − f ℓ� +∆x +(12) +with f ℓ = f(qℓ) and f(q) = (ρv1, ρviv1 + pδi1, ρSv1, 0, 0)T , containing only the fluxes of the Euler +subsystem, i.e. the black terms in (1), and F being the corresponding energy flux. Just like on the +continuous level, our first objective is to get a discrete form of the energy conservation as consequence +of the discrete form of equations (1a)-(1c), see (8). +Therefore we proceed alike we would do on the +continuous level and we perform the dot product of the discrete dual variables, pℓ = ∂qE(qℓ), with the +discrete equations, obtaining +pℓ · d +dtqℓ = d +dtEℓ = −pℓ · (f ℓ+ 1 +2 − f ℓ) + (f ℓ − f ℓ− 1 +2 ) +∆x +. +(13) +We now introduce the fluctuations D +ℓ+ 1 +2 ,− +E += pℓ · (f ℓ+ 1 +2 − f ℓ), D +ℓ− 1 +2 ,+ +E += pℓ · (f ℓ − f ℓ− 1 +2 ). In order to +achieve a flux conservative expression for the discrete formulation of (1f), we must be able to rewrite the +fluctuations related to an interface as a flux difference +D +ℓ+ 1 +2 ,− +E ++ D +ℓ+ 1 +2 ,+ +E += F ℓ+1 − F ℓ, +(14) +4 + +with F ℓ a consistent approximation of the total energy flux F. This condition (14) is mandatory in order +to guarantee total energy conservation for vanishing energy flux at the boundary via the telescopic-sum +property +� +ℓ +∆x d +dtEℓ = − +� +ℓ +� +D +ℓ+ 1 +2 ,− +E ++ D +ℓ− 1 +2 ,+ +E +� += − +� +ℓ +� +F ℓ+1 − F ℓ� += 0. +(15) +Substitution of the fluctuations in the former definition gives +pℓ · (f ℓ+ 1 +2 − f ℓ) + pℓ+1 · (f ℓ+1 − f ℓ+ 1 +2 ) += −f ℓ+ 1 +2 · +� +pℓ+1 − pℓ� ++ pℓ+1 · f ℓ+1 − pℓ · f ℓ += +F ℓ+1 − F ℓ +(16) +and, taking into account definition (11) for f and F, we conclude +−∂p(v1L)ℓ+ 1 +2 · +� +pℓ+1 − pℓ� ++ pℓ+1 · f ℓ+1 − pℓ · f ℓ += +pℓ+1 · f ℓ+1 − (v1L)ℓ+1 − pℓ · f ℓ + (v1L)ℓ, +(17) +where F ℓ = pℓ ·f ℓ −(v1L)ℓ. Accordingly, the numerical flux f ℓ+ 1 +2 = ∂p(v1L)ℓ+ 1 +2 must verify the Roe-type +property, +f ℓ+ 1 +2 · +� +pℓ+1 − pℓ� += ∂p(v1L)ℓ+ 1 +2 · +� +pℓ+1 − pℓ� += (v1L)ℓ+1 − (v1L)ℓ. +(18) +Next, we make use of the key idea on which path conservative schemes are based, see [16, 41], and +construct a path integral in phase-space by recalling the fundamental theorem of calculus +(v1L)ℓ+1 − (v1L)ℓ = +pℓ+1 +� +pℓ +∂p(v1L) · dp = +1 +� +0 +∂p(v1L) · ∂ψ +∂s ds. +(19) +Note that a similar methodology has already been successfully used in the construction of entropy- +conservative fluxes [54]. +Since the path, ψ(s), s ∈ [0, 1], can be freely chosen, we can select any +parametrization convenient for our purposes. As a path connecting pℓ and pℓ+1 we choose the sim- +ple straight line segment path in p variables: +ψ(s) = pℓ + s +� +pℓ+1 − pℓ� +, +∂ψ +∂s = pℓ+1 − pℓ, +0 ≤ s ≤ 1. +(20) +So (19) together with (20) leads to +(v1L)ℓ+1 − (v1L)ℓ = +� +� +1 +� +0 +f(ψ(s))ds +� +� · +� +pℓ+1 − pℓ� +. +(21) +Therefore, the corresponding thermodynamically compatible numerical flux, +f +ℓ+ 1 +2 +p += +1 +� +0 +f(ψ(s))ds = +� +f +ℓ+ 1 +2 +ρ +, f +ℓ+ 1 +2 +ρv , f +ℓ+ 1 +2 +ρS , 0, 0 +�T +, +(22) +guarantees (18) by construction. The subscript p refers to the segment path in p variables (p-scheme). +For a different choice of path in terms of q variables (q-scheme) the reader is referred to [10]. From the +numerical point of view all path integrals appearing in this paper are approximated using a sufficiently +accurate numerical quadrature rule, see e.g. [22]. If not stated otherwise, throughout this paper, we use +a standard Gauss-Legendre quadrature rule with nGP = 3 points in order to compute the path integral +appearing in (22). +For a quantitative study of the influence of the quadrature rule on total energy +conservation, see Section 6. +5 + +3.1 +Compatible scheme with dissipation terms +So far we have presented a compatible discretization for the black terms in (1). To derive a dissipative +scheme, we still need to include a compatible numerical dissipation. Let us enlarge (12) with an additional +dissipative flux and corresponding production terms: +d +dtqℓ + f ℓ+ 1 +2 − f ℓ− 1 +2 +∆x += gℓ+ 1 +2 − gℓ− 1 +2 +∆x ++ Pℓ. +(23) +We first focus on the numerical flux +gℓ+ 1 +2 = ϵℓ+ 1 +2 ∆qℓ+ 1 +2 +∆x +, +∆qℓ+ 1 +2 = qℓ+1 − qℓ, +(24) +whose scalar numerical dissipation is either chosen to be constant, ϵℓ+ 1 +2 = ϵ, or taken of the form +ϵℓ+ 1 +2 = 1 +2 +� +1 − φℓ+ 1 +2 +� +∆x s +ℓ+ 1 +2 +max ≥ 0. +(25) +In the former expression, we have denoted by s +ℓ+ 1 +2 +max the maximum signal speed at the cell interface and +introduced φℓ+ 1 +2 which allows the use of a flux limiter, hence a reduction of the numerical dissipation in +smooth regions. In particular, we consider the minbee flux limiter given by +φℓ+ 1 +2 = min +� +φ +ℓ+ 1 +2 +− +, φ +ℓ+ 1 +2 ++ +� +, +with +φ +ℓ+ 1 +2 +± += max +� +0, min +� +1, h +ℓ+ 1 +2 +± +�� +, +(26) +where +h +ℓ+ 1 +2 +− += Eℓ − Eℓ−1 +Eℓ+1 − Eℓ , +and +h +ℓ+ 1 +2 ++ += Eℓ+2 − Eℓ+1 +Eℓ+1 − Eℓ +(27) +are the ratios of the total energy potential slopes, see the SLIC scheme presented in [57] for further details +on flux limiting strategies. Note that an alternative approach to the use of flux limiters is the definition +of a fixed numerical dissipation. +The dot product of pℓ by (23) yields +dEℓ +dt + 1 +∆x +� +F ℓ+ 1 +2 − F ℓ− 1 +2 +� += +1 +∆xpℓ · +� +gℓ+ 1 +2 − gℓ− 1 +2 +� ++ pℓ · Pℓ, +(28) +where the left hand side has already been studied in the Godunov formalism. We therefore focus on the +right hand side of the former equation obtaining +pℓ · Pℓ + pℓ · gℓ+ 1 +2 − gℓ− 1 +2 +∆x += pℓ · Pℓ + 1 +∆x +�1 +2pℓ · gℓ+ 1 +2 + 1 +2pℓ+1 · gℓ+ 1 +2 + 1 +2pℓ · gℓ+ 1 +2 − 1 +2pℓ+1 · gℓ+ 1 +2 +� +− 1 +∆x +�1 +2pℓ · gℓ− 1 +2 + 1 +2pℓ−1 · gℓ− 1 +2 + 1 +2pℓ · gℓ− 1 +2 − 1 +2pℓ−1 · gℓ− 1 +2 +� += pℓ · Pℓ + 1 +2 +pℓ+1 + pℓ +∆x +· ϵℓ+ 1 +2 ∆qℓ+ 1 +2 +∆x +− 1 +2 +pℓ + pℓ−1 +∆x +· ϵℓ− 1 +2 ∆qℓ− 1 +2 +∆x +−1 +2 +pℓ+1 − pℓ +∆x +· ϵℓ+ 1 +2 ∆qℓ+ 1 +2 +∆x +− 1 +2 +pℓ − pℓ−1 +∆x +· ϵℓ− 1 +2 ∆qℓ− 1 +2 +∆x +. +(29) +Besides, applying path integration yields +qℓ+1 +� +qℓ +p · dq = +qℓ+1 +� +qℓ +∂qE · dq = Eℓ+1 − Eℓ = ∆Eℓ+ 1 +2 . +(30) +6 + +So 1 +2(pℓ+1 + pℓ) · ∆qℓ+ 1 +2 can be seen as an approximation of ∆Eℓ+ 1 +2 . As a consequence of (29) and (30), +the energy flux including convective and diffusive terms is +F +ℓ+ 1 +2 +d += F ℓ+ 1 +2 − 1 +2(pℓ+1 + pℓ) · ϵℓ+ 1 +2 ∆qℓ+ 1 +2 +∆x +≈ F ℓ+ 1 +2 − ϵℓ+ 1 +2 ∆Eℓ+ 1 +2 +∆x +. +(31) +To transform the jumps in p variables into jumps in q variables, we need to introduce a Roe-type matrix +∂2 +qq ˜Eℓ+ 1 +2 verifying the Roe property +∂2 +qq ˜Eℓ+ 1 +2 · (qℓ+1 − qℓ) = pℓ+1 − pℓ. +(32) +For its calculation, we introduce another segment path ˜ψ, written in terms of q +˜ψ(s) = qℓ + s +� +qℓ+1 − qℓ� +, +0 ≤ s ≤ 1, +(33) +allowing to compute the sought Roe matrix as +˜H +ℓ+ 1 +2 = ∂2 +qq ˜Eℓ+ 1 +2 = +1 +� +0 +∂2 +qqE +� +˜ψ(s) +� +ds =: +� +∂2 +pp ˜Lℓ+ 1 +2 +�−1 +, +(34) +which satisfies (32) by construction. Substituting the obtained flux in (28) and taking into account (32) +gives +d +dtEℓ + F +ℓ+ 1 +2 +d +− F +ℓ− 1 +2 +d +∆x += pℓ · Pℓ +(35) +−1 +2ϵℓ+ 1 +2 qℓ+1 − qℓ +∆x +· ˜H +ℓ+ 1 +2 qℓ+1 − qℓ +∆x +− 1 +2ϵℓ− 1 +2 qℓ − qℓ−1 +∆x +· ˜H +ℓ− 1 +2 qℓ − qℓ−1 +∆x +. +Thus, defining the production term Pℓ = (0, 0, Πℓ, 0, 0)T +pℓ · Pℓ = T ℓΠℓ = 1 +2ϵℓ+ 1 +2 ∆qℓ+ 1 +2 +∆x +· ˜H +ℓ+ 1 +2 ∆qℓ+ 1 +2 +∆x ++ 1 +2ϵℓ− 1 +2 ∆qℓ− 1 +2 +∆x +· ˜H +ℓ− 1 +2 ∆qℓ− 1 +2 +∆x +, +(36) +we obtain the sought semi-discrete total energy conservation law +d +dtEℓ + F +ℓ+ 1 +2 +d +− F +ℓ− 1 +2 +d +∆x += 0. +(37) +Note that, as expected, the above definition provides a zero production term for all equations but for +(1c). The final compatible flux including convective and diffusive terms reads +f +ℓ+ 1 +2 +p,d += +1 +� +0 +f(ψ(s))ds − ϵℓ+ 1 +2 +∆x +� +qℓ+1 − qℓ� +. +(38) +3.2 +Compatible discretization of the terms related to the distortion field +The momentum flux in (1b) gathers four terms. The first two, in black, belong to the Euler subsystem +and have already been studied in the previous sections. The third term, σik, is related to the distortion +field and thus its compatibility must be analysed together with the distortion transport equations, (1d), +and the terms E3 and σik in the energy equation (1f). Let us consider the red terms in (1d) (except for +the convective term v1∂xAik), +∂(Aimvm) +∂x +− vm +∂Aim +∂x += Aim +∂vm +∂x , +(39) +7 + +and the following chosen discretization +∆xAim∂x∂vm ≈ A +ℓ+ 1 +2 +im +� +vℓ+1 +m +− vℓ +m +� +with +A +ℓ+ 1 +2 +im += 1 +2 +� +Aℓ+1 +im + Aℓ +im +� +. +(40) +Multiplication of equations (1b), (1d) by ∂ρviE = vi, ∂AikE = αik and assuming a compatible discretiza- +tion with the term ∂x (viσik) in (1f) leads to +vℓ +i +� +σ +ℓ+ 1 +2 +ik +− σℓ +ik +� ++ vℓ+1 +i +� +σℓ+1 +ik +− σ +ℓ+ 1 +2 +ik +� ++ αℓ +ik +1 +2A +ℓ+ 1 +2 +im +� +vℓ+1 +m +− vℓ +m +� ++αℓ+1 +ik +1 +2A +ℓ+ 1 +2 +im +� +vℓ+1 +m +− vℓ +m +� += vℓ+1 +i +σℓ+1 +ik +− vℓ +iσℓ +ik. +(41) +We therefore obtain the following discretization for σ +ℓ+ 1 +2 +ik +: +σ +ℓ+ 1 +2 +ik += 1 +2 +� +αℓ+1 +mk + αℓ +mk +� +A +ℓ+ 1 +2 +mi . +(42) +We now focus on the remaining flux term v1∂xAik. We multiply the continuity equation (1a) by the +dual variable ∂ρE3 = E3 and (1d) by ∂AikE = αik and impose the compatibility condition with the energy +conservation equation yielding +Eℓ +3 +� +ρv +ℓ+ 1 +2 +1 +− ρvℓ +1 +� ++ Eℓ+1 +3 +� +ρvℓ+1 +1 +− ρv +ℓ+ 1 +2 +1 +� ++ αℓ +ik +1 +2 ˜v +ℓ+ 1 +2 +A 1 +� +Aℓ+1 +ik +− Aℓ +ik +� ++αℓ+1 +ik +1 +2 ˜v +ℓ+ 1 +2 +A 1 +� +Aℓ+1 +ik +− Aℓ +ik +� += ρvℓ+1 +1 +Eℓ+1 +3 +− ρvℓ +1Eℓ +3, +(43) +where the approximation of the averaged velocity ˜v +ℓ+ 1 +2 +A 1 still needs to be defined. Collecting terms, we get +−ρv +ℓ+ 1 +2 +1 +� +Eℓ+1 +3 +− Eℓ +3 +� ++ 1 +2 ˜v +ℓ+ 1 +2 +A 1 +� +αℓ+1 +ik ++ αℓ +ik +� � +Aℓ+1 +ik +− Aℓ +ik +� += 0. +(44) +Hence, the average velocity must be discretised as +˜v +ℓ+ 1 +2 +A 1 = +ρv +ℓ+ 1 +2 +1 +� +Eℓ+1 +3 +− Eℓ +3 +� +1 +2 +� +αℓ+1 +ik ++ αℓ +ik +� � +Aℓ+1 +ik +− Aℓ +ik +� +(45) +if the denominator in (45) is non-zero, otherwise we set ˜v +ℓ+ 1 +2 +A 1 = 1 +2 +� +vℓ+1 +1 ++ vℓ +1 +� +. Finally, if Eℓ+1 +3 +− Eℓ +3 = 0, +from (44), we get ˜v +ℓ+ 1 +2 +A 1 = 0. +3.3 +Compatible discretization of the terms related to the thermal impulse +Similarly to what has been done for the distortion field, in this section we derive the discretization +of the red terms in (1b), (1e), (1f) related to the heat flux. First, we focus on terms ∂xωi1 in (1b), +Jm∂xvm = ∂x(Jmvm) − vm∂xJm in (1e) and ∂x (viωi1) in (1f). Multiplying the momentum equation by +∂ρviE = vi, the thermal impulse equation by ∂J1E = β1 and requiring compatibility with the energy +equation we get +vℓ +i +� +ω +ℓ+ 1 +2 +i1 +− ωℓ +i1 +� ++ vℓ+1 +i +� +ωℓ+1 +i1 +− ω +ℓ+ 1 +2 +i1 +� ++ βℓ +1 +1 +2J +ℓ+ 1 +2 +m +� +vℓ+1 +m +− vℓ +m +� ++βℓ+1 +1 +1 +2J +ℓ+ 1 +2 +m +� +vℓ+1 +m +− vℓ +m +� += vℓ+1 +i +ωℓ+1 +i1 +− vℓ +iωℓ +i1. +(46) +Relabeling the repeated index m and defining J +ℓ+ 1 +2 +i += 1 +2 +� +Jℓ+1 +i ++ Jℓ +i +� +, yields +−ω +ℓ+ 1 +2 +i1 +� +vℓ+1 +i +− vℓ +i +� ++ 1 +2 +� +βℓ+1 +1 ++ βℓ +1 +� +J +ℓ+ 1 +2 +i +� +vℓ+1 +i +− vℓ +i +� += 0. +(47) +8 + +Thus choosing ω +ℓ+ 1 +2 +i1 += 1 +2 +� +βℓ+1 +1 ++ βℓ +1 +� +J +ℓ+ 1 +2 +i +gives the sought compatibility. +Next, we need to compute the discretization related to the term v1∂xJk in (1e). Multiplication of +(1a) by ∂ρE4 = E4 and addition of (1e) multiplied by ∂JkE = βk yields +Eℓ +4 +� +ρv +ℓ+ 1 +2 +1 +− ρvℓ +1 +� ++ Eℓ+1 +4 +� +ρvℓ+1 +1 +− ρv +ℓ+ 1 +2 +1 +� ++ 1 +2 ˜v +ℓ+ 1 +2 +J 1 βℓ +k +� +Jℓ+1 +k +− Jℓ +k +� ++1 +2 ˜v +ℓ+ 1 +2 +J 1 βℓ+1 +k +� +Jℓ+1 +k +− Jℓ +k +� += ρvℓ+1 +1 +Eℓ+1 +4 +− ρvℓ +1Eℓ +4. +(48) +Hence, +˜v +ℓ+ 1 +2 +J 1 += +ρv +ℓ+ 1 +2 +1 +� +Eℓ+1 +4 +− Eℓ +4 +� +1 +2 +� +βℓ+1 +k ++ βℓ +k +� � +Jℓ+1 +k +− Jℓ +k +� +(49) +is the compatible discretization for the advection speed related to the thermal impulse. Analogous to the +previous section, for null denominator and Eℓ+1 +4 +− Eℓ +4 ̸= 0 we define ˜v +ℓ+ 1 +2 +J 1 +as the arithmetic average of +the velocity in the two related cells. +It now just remains to establish the discrete compatibility between the term βk in equation (1c), T in +equation (1e) and hk in equation (1f). Let us assume we have the following given discretization for the +gradient of T: +∆x∂xT ≈ 1 +2 +� +T ℓ+1 − T ℓ� +. +(50) +Then, multiplication of the thermal impulse equation (1e) by ∂JkE = βk and the entropy relation by +∂ρSE = T gives +T ℓ � +β +ℓ+ 1 +2 +k +− βℓ +k +� ++ T ℓ+1 � +βℓ+1 +k +− β +ℓ+ 1 +2 +k +� ++ βℓ +k +1 +2 +� +T ℓ+1 − T ℓ� ++βℓ+1 +k +1 +2 +� +T ℓ+1 − T ℓ� += βℓ+1 +k +T ℓ+1 − βℓ +kT ℓ. +(51) +Hence, by simply defining β +ℓ+ 1 +2 +k += 1 +2 +� +βℓ+1 +k ++ βℓ +k +� +we get a compatible discretization of the equations. +3.4 +Compatible discretization of relaxation terms +Finally, it is easy to see that the relaxation terms, in green in (1c)-(1e), cancel. Multiplication of ∂ρSE = T, +∂AikE = αik and ∂JkE = βk by the green terms in (1c)-(1e), respectively, and adding the result gives +T αikαik +θ1(τ1)T + T +βiβi +θ2(τ2)T − αik +αik +θ1(τ1) − βk +βk +θ2(τ2) = 0. +(52) +Thus, the compatibility is proven by construction. +4 +Thermodynamically compatible semi-discrete finite volume +scheme for the complete model in two space dimensions +The derivation of the thermodynamically compatible semi-discrete finite volume scheme for the complete +model in two space dimensions can be done following the steps described in the previous section. Here +we summarize the final scheme and provide the mathematical proofs of the marginal nonlinear stability +in the energy norm and of the semi-discrete cell entropy inequality. Let us consider the spatial control +volume Ωℓ with circumcenter xℓ, one of its neighbors Ωr and the common edge ∂Ωℓr, n = (n1, n2)T being +the outward unit normal vector to the face ∂Ωℓr and Nℓ being the set of neighbors of cell Ωℓ. The final +semi-discrete finite volume scheme reads +∂ρℓ +∂t = − +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Dℓr,− +ρ ++ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +ρ, n, +(53a) +9 + +∂(ρvℓ +i) +∂t += − +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Dℓr,− +ρvi − 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� σℓr, − +ik +nk +− 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� ωℓr, − +ik +nk+ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +ρvi, n, +(53b) +∂(ρSℓ) +∂t += − +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Dℓr,− +ρS − 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� � +βℓr +k − βℓ +k +� +nk ++ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +ρS, n+ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Πℓr,− +n ++ αℓ +ikαℓ +ik +θℓ +1 (τ1) T ℓ + +βℓ +i βℓ +i +θℓ +2 (τ2) T ℓ , +(53c) +∂Aℓ +ik +∂t +=− 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� 1 +2Aℓr +im +� +vr +m − vℓ +m +� +nk +− 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� 1 +2 ˜uℓr +A, n +� +Ar +ik − Aℓ +ik +� ++ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +Aik, n− αℓ +ik +θℓ +1 (τ1), +(53d) +∂Jℓ +k +∂t =− 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� 1 +2Jℓr +i +� +vr +m − vℓ +m +� +nk− 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� 1 +2T ℓr,−nk +− 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� 1 +2 ˜uℓr +J, n +� +Jr +k − Jℓ +k +� ++ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +Jk, n− +βℓ +i +θℓ +2 (τ2) +(53e) +with +Dℓr,− +q += +� +f ℓr +q, k − f ℓ +q, k +� +nk, +(54) +gℓr +q, n = ϵℓr qr − qℓ +δℓr += ϵℓr ∆qℓr +δℓr , +δℓr = +��xr − xℓ�� = ∆xn1 + ∆yn2, +(55) +σℓr,− +jk += σℓr +jk − σℓ +jk, +σℓr +jk = 1 +2Aℓr +ij +� +αℓ +ik + αr +ik +� +, +(56) +ωℓr,− +jk += ωℓr +jk − ωℓ +jk, +ωℓr +jk = 1 +2 +� +βℓ +k + βr +k +� +Jℓr +i , +(57) +Πℓr,− +n += 1 +2ϵℓr ∆qℓr +T ℓ +· ∂2 +qqEℓr ∆qℓr +δℓr , +T ℓ = +� +ρℓ�γ−1 +(γ − 1) cv +e +Sℓ +cv , +(58) +Aℓr +im = 1 +2 +� +Aℓ +im + Ar +im +� +, +˜uℓr +A, n = ˜vℓr +A, jnj = +f ℓr +ρ, jnj +� +Er +3 − Eℓ +3 +� +1 +2 +� +αℓ +ik + αr +ik +� � +Ar +ik − Aℓ +ik +�, +(59) +Jℓr +i = 1 +2 +� +Jℓ +i + Jr +i +� +, ˜uℓr +J, n =˜vℓr +J, jnj = +f ℓr +ρ, jnj +� +Er +4 − Eℓ +4 +� +1 +2 +� +βℓ +k + βr +k +� � +Jr +k − Jℓ +k +�, +T ℓr,− =T r − T ℓ. +(60) +Theorem 1. The thermodynamically compatible semi-discrete finite volume +scheme (53) admits the semi-discrete energy conservation law +∂Eℓ +∂t = − 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Dℓr,− +E ++ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +E, n +(61) +with +Dℓr,− +E ++ Dℓr,+ +E += Dℓr,− +E ++ Drℓ,− +E += F r − F ℓ. +(62) +Assuming that the jumps on the boundary vanish, the scheme is nonlinearly marginally stable in the +energy norm, i.e. the scheme satisfies the identity +� +Ω +∂Eℓ +∂t dV = +� +ℓ +|Ωℓ|∂Eℓ +∂t = 0. +(63) +10 + +Proof. We start considering the contributions of the dot product of vector +pℓ = ∂qEℓ = +� +∂ρEℓ, ∂ρviEℓ, ∂ρSEℓ, ∂AikEℓ, ∂JkEℓ�T +with the time derivative terms in (53): +∂ρEℓ ∂ρℓ +∂t + ∂ρviEℓ ∂ρvℓ +i +∂t ++ ∂ρSEℓ ∂ρSℓ +∂t ++ ∂AikEℓ ∂Aℓ +ik +∂t ++ ∂JkEℓ ∂Jℓ +k +∂t = ∂qEℓ ∂qℓ +∂t = ∂Eℓ +∂t . +(64) +We now define the fluctuations associated to the total energy equation as +Dℓr,− +E += +∂ρEℓ +1Dℓr,− +ρ ++ ∂ρEℓ +2Dℓr,− +ρ ++∂ρEℓ +3Dℓr,− +ρ ++∂ρEℓ +4Dℓr,− +ρ ++ ∂ρviEℓDℓr,− +ρvi ++∂ρviEℓ � +σℓr,− +ik +nk + ωℓr,− +ik +nk +� ++ ∂ρSEℓDℓr,− +ρS +∂ρSEℓ � +βℓr +k − βℓ +k +� +nk ++∂AikEℓ 1 +2δℓr Aℓr +im +� +vr +m − vℓ +m +� +nk+∂AikEℓ 1 +2 ˜uℓr +A, n +� +Ar +ik − Aℓ +ik +� ++∂JkEℓ 1 +2δℓr Jℓr +i +� +vr +m − vℓ +m +� +nk+∂JkEℓ 1 +2 ˜uℓr +J, n +� +Jr +k − Jℓ +k +� ++ ∂JkEℓ 1 +2T ℓr,−nk. +(65) +On the other hand, from (55) and applying relations analogous to the ones introduced in (29) and +(32), we have +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� � +pℓ · Pℓr,− +n ++ pℓ · gℓr +n +� += +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +pℓ · Pℓr,− +n ++ pℓ · ϵℓr ∆qℓr +δℓr +� += +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +pℓ· Pℓr,− +n ++ 1 +2pℓ· ϵℓr ∆qℓr +δℓr + 1 +2pr· ϵℓr ∆qℓr +δℓr + 1 +2pℓ· ϵℓr ∆qℓr +δℓr − 1 +2pr· ϵℓr ∆qℓr +δℓr +� += +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +pℓ · Pℓr,− +n ++ 1 +2 +� +pℓ + pr� +· ϵℓr ∆qℓr +δℓr +− 1 +2 +� +pr − pℓ� +· ϵℓr ∆qℓr +δℓr +� += +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +pℓ · Pℓr,− +n ++ ϵℓr ∆Eℓr +δℓr +− 1 +2ϵℓr ∆qℓr +δℓr ∂2 +qqEℓr∆qℓr +� +. +(66) +Substitution of Pℓr,− +n += +� +0, 0, Πℓr,− +n +, 0, 0 +� +combined with (58) yields +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� � +pℓ · gℓr +n + pℓ · Pℓr,− +n +� += +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� ϵℓr ∆Eℓr +δℓr += +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +E, n. +(67) +Finally, taking into account the dot product of pℓ by the diffusion terms in (53) and applying (67), we +get +pℓ · +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +n + ∂ρSEℓ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Πℓr +n +(68) += +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +pℓ · ϵℓr ∆qℓr +δℓr ++ ∂ρSEℓ 1 +4ϵℓr ∆qℓr +T ℓ ∂2 +qqEℓr ∆qℓr +δℓr +� += +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +E, n. +From (64), (65), (68) and noting that the dot product of ∂qEℓ by the green terms in (53a)-(53e) is +zero, we conclude +∂Eℓ +∂t = − 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Dℓr,− +E ++ 1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� gℓr +E, n. +The second part of the proof concerns marginal stability. +Integration of equation (61) over the +computational domain Ω gives +� +Ω +∂Eℓ +∂t dV = +� +ℓ +��Ωℓ�� ∂Eℓ +∂t = − +� +ℓ +� +r∈Nℓ +��∂Ωℓr�� Dℓr,− +E ++ +� +ℓ +� +r∈Nℓ +��∂Ωℓr�� gℓr +E, n. +11 + +Assuming that the solution on the boundaries of the domain tends to a constant value, the jumps on +q become zero at ∂Ω and fluctuations and dissipative terms vanish. Besides, the remaining dissipative +terms can be seen as a telescopic sum which cancels. Reordering of the first summation in the right hand +side of the former equation to cluster the contributions at each face we obtain +� +Ω +∂Eℓ +∂t dV = +� +ℓ +��Ωℓ�� ∂Eℓ +∂t = − +� +ℓr +��∂Ωℓr�� +� +Dℓr,− +E ++ Drℓ,− +E +� +. +Consequently, marginal stability is proven given that the contributions of fluctuations in the interior cell +boundaries cancel. Let us focus on a face ∂Ωℓr. We start analysing the terms corresponding with the +Godunov formalism (black terms) in (65): +∂ρEℓ +1Dℓr,− +ρ ++ ∂ρEℓ +2Dℓr,− +ρ ++ ∂ρviEℓDℓr,− +ρvi + ∂ρSEℓDℓr,− +ρS ++∂ρEr +1Drℓ,− +ρ ++ ∂ρEr +2Drℓ,− +ρ ++ ∂ρviErDrℓ,− +ρvi + ∂ρSErDrℓ,− +ρS += − +� +∂ρEr +1 − ∂ρEℓ +1 +� +f ℓr +ρ, knk − +� +∂ρEr +2 − ∂ρEℓ +2 +� +f ℓr +ρ, knk − +� +∂ρviEr − ∂ρviEℓ� +f ℓr +ρvi, knk +− +� +∂ρSEr − ∂ρSEℓ� +f ℓr +ρS, knk − ∂ρEℓ +1f ℓ +ρ, knk − ∂ρEℓ +2f ℓ +ρ, knk − ∂ρviEℓf ℓ +ρvi, knk +−∂ρSEℓf ℓ +ρS, knk + ∂ρEr +1f r +ρ, knk + ∂ρEr +2f r +ρ, knk + ∂ρviErf r +ρvi, knk + ∂ρSErf r +ρS, knk += − +� +pr − pℓ� +· f ℓr +q, knk + pr · f r +q, knk − pℓ · f ℓ +q, knk += +� +pr · f r +q, k − (vkL)r� +nk − +� +pℓ · f ℓ +q, k − (vkL)ℓ� +nk = F r +G − F ℓ +G, +(69) +with FG standing for the black terms in the energy flux in (1f). Regarding the red terms in (65), we have +∂ρEℓ +3Dℓr,− +ρ ++∂ρEℓ +4Dℓr,− +ρ ++∂ρviEℓ � +σℓr,− +ik +nk + ωℓr,− +ik +nk +� ++∂ρSEℓ � +βℓr +k − βℓ +k +� +nk ++∂AikEℓ 1 +2δℓr Aℓr +im +� +vr +m − vℓ +m +� +nk+∂AikEℓ 1 +2 ˜uℓr +A, n +� +Ar +ik − Aℓ +ik +� ++∂JkEℓ 1 +2δℓr Jℓr +i +� +vr +m − vℓ +m +� +nk+∂JkEℓ 1 +2 ˜uℓr +J, n +� +Jr +k − Jℓ +k +� ++∂JkEℓ 1 +2T ℓr,−nk ++∂ρEr +3Drℓ,− +ρ ++∂ρEr +4Drℓ,− +ρ +−∂ρviEr � +σrℓ,− +ik +nk + ωrℓ,− +ik +nk +� +−∂ρSEr � +βrℓ +k − βr +k +� +nk +−∂AikEr 1 +2Arℓ +im +� +vℓ +m − vr +m +� +nk+∂AikEr 1 +2 ˜urℓ +A, n +� +Aℓ +ik − Ar +ik +� +−∂JkEr 1 +2Jrℓ +i +� +vℓ +m − vr +m +� +nk+∂JkEr 1 +2 ˜urℓ +J, n +� +Jℓ +k − Jr +k +� +−∂JkEr 1 +2T rℓ,−nk. +Substitution of ∂qE by its expression in state variables, q, together with (54) yields +Eℓ +3 +� +f ℓr +ρ, k −f ℓ +ρ, k +� +nk−Er +3 +� +f ℓr +ρ, k −f r +ρ, k +� +nk+αℓ +ik +1 +2 ˜uℓr +A, n +� +Ar +ik −Aℓ +ik +� ++αr +ik +1 +2 ˜urℓ +A, n +� +Aℓ +ik −Ar +ik +� ++Eℓ +4 +� +f ℓr +ρ, k − f ℓ +ρ, k +� +nk−Er +4 +� +f ℓr +ρ, k − f r +ρ, k +� +nk+βℓ +k +1 +2 ˜uℓr +J, n +� +Jr +k − Jℓ +k +� ++βr +k +1 +2 ˜urℓ +J, n +� +Jℓ +k − Jr +k +� ++ +� +vℓ +i +� +σℓr +ik −σℓ +ik +� +− vr +i +� +σℓr +ik −σr +ik +� ++ αℓ +ik +1 +2Aℓr +im +� +vr +m −vℓ +m +� +− αr +ik +1 +2Arℓ +im +� +vℓ +m −vr +m +�� +nk ++ +� +vℓ +i +� +ωℓr +ik − ωℓ +ik +� +− vr +i +� +ωℓr +ik − ωr +ik +� ++ βℓ +k +1 +2Jℓr +i +� +vr +m − vℓ +m +� +− βr +k +1 +2Jrℓ +i +� +vℓ +m − vr +m +�� +nk ++ +� +T ℓ � +βℓr +k − βℓ +k +� +− T r � +βℓr +k − βr +k +� ++ βℓ +k +1 +2T ℓr,− − βr +k +1 +2T rℓ,− +� +nk. +Taking into account (56)-(60) and collecting terms gives +(ρvr +kEr +3 + ρvr +kEr +4 + vr +iσr +ik + vr +iωr +ik + βr +kT r) nk +− +� +ρvℓ +kEℓ +3 + ρvℓ +kEℓ +4 + vr +iσr +ik + vr +iωr +ik + βℓ +kT ℓ� +nk. +(70) +12 + +Gathering (69) and (70), we obtain +Dℓr,− +E ++ Drℓ,− +E += +F r +G + (ρvr +kEr +3 + ρvr +kEr +4 + vr +iσr +ik + vr +iωr +ik + βr +kT r) nk − +F ℓ +G − +� +ρvℓ +kEℓ +3 + ρvℓ +kEℓ +4 + vℓ +iσℓ +ik + vℓ +iωℓ +ik + βℓ +kT ℓ� +nk = F r − F ℓ. +(71) +Thus the fluctuations can be seen as the difference between fluxes which will cancel out when adding the +contributions of all cells, and hence the scheme is marginally stable in the energy norm, as claimed: +� +Ω +∂Eℓ +∂t dV = +� +ℓ +|Ωℓ|∂Eℓ +∂t = − +� +ℓr +� +Dℓr,− +E ++ Dℓr,+ +E +� += − +� +ℓr +� +F r − F ℓ� += 0. +(72) +Theorem 2. Assuming T ℓ > 0 and Hℓr = ∂2 +qqEℓr ≥ 0 the semi-discrete finite volume scheme (53) with +production term (58) satisfies the semi-discrete cell entropy inequality +∂ρSℓ +∂t ++ +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| Dℓr,− +ρS ++ +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +βℓr +k − βℓ +k +� +nk− +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| gℓr +ρS, n ≥ 0. +(73) +Proof. The proof is an immediate consequence of the discretization (53c) with (58): +∂ρSℓ +∂t ++ +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| Dℓr,− +ρS ++ +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| +� +βℓr +k − βℓ +k +� +nk− +� +r∈Nℓ +��∂Ωℓr�� +|Ωℓ| gℓr +ρS, n += +1 +|Ωℓ| +� +r∈Nℓ +��∂Ωℓr�� Πℓr,− +n ++ αℓ +ikαℓ +ik +θℓ +1 (τ1) T ℓ + +βℓ +i βℓ +i +θℓ +2 (τ2) T ℓ ≥ 0, +(74) +since Πℓr,− +n += 1 +2ϵℓr ∆qℓr +T ℓ ∂2 +qqEℓr ∆qℓr +δℓr +≥ 0 due to ∂2 +qqE ≥ 0 and θℓ +1,2 > 0 as well as T ℓ > 0. +5 +Thermodynamically compatible fully-discrete finite volume +scheme for the Euler subsystem +In this section we present a fully-discrete finite volume scheme for the Euler subsystem of (1), i.e. for the +black and blue terms. For simplicity, we restrict the considerations to one space dimension. As before, +the spatial control volumes are denoted by Ωℓ = [xℓ− 1 +2 , xℓ+ 1 +2 ]. The scheme reads +qn+1,ℓ − qn,ℓ +∆t += − +(f +ℓ+ 1 +2 +˜p +− f ℓ) − (f +ℓ− 1 +2 +˜p +− f ℓ) +∆x ++ +g +ℓ+ 1 +2 +˜p +− g +ℓ+ 1 +2 +˜p +∆x ++ ˜Pℓ, +(75) +Again, the subscript ˜p refers to the fact that the flux is evaluated using the segment path in p variables +defined below, similar to (20)-(22). In order to construct a thermodynamically compatible fully-discrete +scheme, where the total energy conservation law (1f) is a consequence of the discrete equations (75), we +introduce a new average quantity ˜pℓ. Since by construction one has +qn+1,ℓ +� +qn,ℓ +∂qE · dq = En+1,ℓ − En,ℓ, +(76) +for any path connecting qn,ℓ with qn+1,ℓ, we define the quantity ˜pℓ as +˜pℓ = +1 +� +0 +∂qE(τ(s))ds, +(77) +13 + +with the straight-line segment path +τ = τ(s) = qn,ℓ + s +� +qn+1,ℓ − qn,ℓ� +. +(78) +Therefore, ˜pℓ satisfies the Roe-type property +˜pℓ · +� +qn+1,ℓ − qn,ℓ� += En+1,ℓ − En,ℓ, +(79) +which is fundamental for the construction of our thermodynamically compatible fully-discrete scheme. +We now multiply (75) with ˜pℓ from the left and neglecting the viscous fluxes gℓ± 1 +2 leads to +˜pℓ · qn+1,ℓ − qn,ℓ +∆t += En+1,ℓ − En,ℓ +∆t += −˜pℓ · +(f +ℓ+ 1 +2 +˜p +− f ℓ) + (f ℓ − f +ℓ− 1 +2 +˜p +) +∆x +. +(80) +To obtain a conservative form of the fully discrete energy conservation law, we require +˜pℓ · (f +ℓ+ 1 +2 +˜p +− f ℓ) + ˜pℓ+1 · (f ℓ+1 − f +ℓ+ 1 +2 +˜p +) = ˜F ℓ+1 − ˜F ℓ. +(81) +Using the parametrization (10) and the associated relations (11) we get +−∂p(� +v1L)ℓ+ 1 +2 · +�˜pℓ+1 − ˜pℓ� ++ ˜pℓ+1 · f ℓ+1 − ˜pℓ · f ℓ = ˜pℓ+1 · f ℓ+1 − � +v1L +ℓ+1 − ˜pℓ · f ℓ + � +v1L +ℓ. +(82) +Hence, the numerical flux f +ℓ+ 1 +2 +˜p += ∂p(� +v1L)ℓ+ 1 +2 must satisfy the following jump condition: +f +ℓ+ 1 +2 +˜p +· +�˜pℓ+1 − ˜pℓ� += ∂p(� +v1L)ℓ+ 1 +2 · +�˜pℓ+1 − ˜pℓ� += � +v1L +ℓ+1 − � +v1L +ℓ. +(83) +We choose again a simple straight line segment path, this time in the ˜p variables: +ψ(s) = ψ(˜pℓ, ˜pℓ+1, s) = ˜pℓ + s +�˜pℓ+1 − ˜pℓ� +, +0 ≤ s ≤ 1, +(84) +Using the same reasoning as for the semi-discrete scheme (19)-(38), we find the thermodynamically +compatible numerical flux of the fully discrete p-scheme as +f +ℓ+ 1 +2 +˜p += +� +f +ℓ+ 1 +2 +ρ +, f +ℓ+ 1 +2 +ρvi , f +ℓ+ 1 +2 +ρS +�T += +1 +� +0 +f(ψ(˜pℓ, ˜pℓ+1, s))ds, +(85) +with the jump ∆˜pℓ+ 1 +2 = ˜pℓ+1 − ˜pℓ and the numerical viscosity flux defined as +g +ℓ+ 1 +2 +˜p += +� +g +ℓ+ 1 +2 +ρ +, g +ℓ+ 1 +2 +ρvi , g +ℓ+ 1 +2 +ρS +�T += ϵℓ+ 1 +2 ∂2 +pp ˜Lℓ+ 1 +2 ˜pℓ+1 − ˜pℓ +∆x +. +(86) +The corresponding production term reads ˜Pℓ = (0, 0, ˜Πℓ)T with +˜pℓ · ˜Pℓ = ˜T ℓ ˜Πℓ = 1 +2ϵℓ+ 1 +2 ∆˜pℓ+ 1 +2 +∆x +· ∂2 +pp ˜Lℓ+ 1 +2 ∆˜pℓ+ 1 +2 +∆x ++ 1 +2ϵℓ− 1 +2 ∆˜pℓ− 1 +2 +∆x +· ∂2 +pp ˜Lℓ− 1 +2 ∆˜pℓ− 1 +2 +∆x +. +(87) +The disadvantage of the p-scheme is that it requires the expression of the physical flux f in terms +of the p variables, or, equivalently, it requires the variable transformation q = q(p), which in general +may be quite cumbersome. +However, for the Euler subsystem at least this conversion is simple and +analytic. Recall that ∂2 +pp ˜Lℓ− 1 +2 = (∂2 +qq ˜Eℓ− 1 +2 )−1, see (34). +Note that the proposed fully-discrete scheme +is implicit, since ˜pℓ is a function of qn,ℓ and qn+1,ℓ, see (77) and (78). In order to obtain a simple +and straightforward implementation of the fully-discrete scheme, we propose the following predictor- +corrector approach, based on a Picard-type iteration, similar to the iterative procedure employed in the +fully-discrete kinetic energy-preserving scheme proposed for the Euler equations in [53]: +qn+1,ℓ +k+1 += qn,ℓ − ∆t +∆x +� +f +ℓ+ 1 +2 +˜pk +− f +ℓ− 1 +2 +˜pk +� ++ ∆t +∆x +� +g +ℓ+ 1 +2 +˜pk +− g +ℓ− 1 +2 +˜pk +� ++ ˜Pℓ +k, +(88) +14 + +with the quantity ˜pℓ +k defined as +˜pℓ +k = +1 +� +0 +∂qE(τ)ds, +with +τ = qn,ℓ + s +� +qn+1,ℓ +k +− qn,ℓ� +. +(89) +As initial guess for the iterative scheme we set qn+1,ℓ +0 += qn,ℓ and the iterations are stopped when the +following condition is satisfied: +� +ℓ +� +En+1,ℓ +k +− E +� +qn+1,ℓ +k+1 +��2 +< δ2. +(90) +with δ > 0 an arbitrarily small tolerance, typically of the order of the machine precision. +Recall that +En+1,ℓ +k += En,ℓ +k +˜pℓ +k· +� +qn+1,ℓ +k +− qn,ℓ� +, see (79). This completes the description of the fully-discrete Godunov +formalism for the inviscid Euler subsystem. +Theorem 3. The thermodynamically compatible fully-discrete finite volume +scheme +qn+1,ℓ − qn,ℓ +∆t += − +(f +ℓ+ 1 +2 +˜p +− f ℓ) − (f +ℓ− 1 +2 +˜p +− f ℓ) +∆x ++ +g +ℓ+ 1 +2 +˜p +− g +ℓ− 1 +2 +˜p +∆x ++ ˜Pℓ. +(91) +with production term ˜Pℓ according to (87) and fluxes (85) and (86) verifies the fully discrete energy +conservation law +En+1,ℓ − En,ℓ +∆t += − 1 +∆x +� +˜D +ℓ+ 1 +2 ,− +E ++ ˜D +ℓ− 1 +2 ,+ +E +� ++ g +ℓ+ 1 +2 +E +− g +ℓ− 1 +2 +E +∆x +. +(92) +The fluctuations above are defined as +˜D +ℓ+ 1 +2 ,− +E += ˜pℓ · (f +ℓ+ 1 +2 +˜p +− f ℓ), +˜D +ℓ− 1 +2 ,+ +E += ˜pℓ · (f ℓ − f +ℓ− 1 +2 +˜p +) +(93) +and satisfy +˜D +ℓ+ 1 +2 ,− +E ++ ˜D +ℓ+ 1 +2 ,+ +E += ˜F ℓ+1 − ˜F ℓ = F(˜pℓ+1) − F(˜pℓ). +(94) +The numerical viscosity flux in (92) reads +g +ℓ+ 1 +2 +E += 1 +2 +ϵℓ+ 1 +2 +∆x +�˜pℓ + ˜pℓ+1� +· ∂pp ˜Lℓ+ 1 +2 �˜pℓ+1 − ˜pℓ� +. +(95) +As a consequence, for vanishing jumps on the boundary, the scheme is nonlinearly marginally stable in +the energy norm. +Proof. Multiplying (91) with ˜p defined according to (77) and using the Roe property (79) yields +˜pℓ · qn+1,ℓ − qn,ℓ +∆t += En+1,ℓ − En,ℓ +∆t +. +(96) +Furthermore, using (93) one has immediately +˜pℓ · +� +f +ℓ+ 1 +2 +˜p,d − f ℓ� ++ ˜pℓ · +� +f ℓ − f +ℓ+ 1 +2 +˜p,d +� += ˜D +ℓ+ 1 +2 ,− +E ++ ˜D +ℓ− 1 +2 ,+ +E +. +(97) +By construction, (81)-(85), the fluctuations satisfy (94). For the numerical viscosity we get after some +calculations, see (29) and (66) for the semi-discrete case, +˜pℓ · +g +ℓ+ 1 +2 +˜p +− g +ℓ− 1 +2 +˜p +∆x ++ ˜pℓ · ˜P +15 + += ϵℓ+ 1 +2 +∆x ˜pℓ · ∂2 +pp ˜Lℓ+ 1 +2 �˜pℓ+1 − ˜pℓ� +− ϵℓ− 1 +2 +∆x ˜pℓ · ∂2 +pp ˜Lℓ− 1 +2 �˜pℓ − ˜pℓ−1� ++ ˜pℓ · ˜P += 1 +2 +˜pℓ+1 + ˜pℓ +∆x +· ϵℓ+ 1 +2 ∂2 +pp ˜Lℓ+ 1 +2 ∆˜pℓ+ 1 +2 +∆x +− 1 +2 +˜pℓ + ˜pℓ−1 +∆x +· ϵℓ− 1 +2 ∂2 +pp ˜Lℓ− 1 +2 ∆˜pℓ− 1 +2 +∆x +−1 +2 +∆˜pℓ+ 1 +2 +∆x +· ϵℓ+ 1 +2 ∂2 +pp ˜Lℓ+ 1 +2 ∆˜pℓ+ 1 +2 +∆x +− 1 +2 +∆˜pℓ− 1 +2 +∆x +· ϵℓ− 1 +2 ∂2 +pp ˜Lℓ− 1 +2 ∆˜pℓ− 1 +2 +∆x ++ ˜pℓ · ˜P += g +ℓ+ 1 +2 +E +− g +ℓ− 1 +2 +E +∆x +(98) +due to the definition (95) and the production term that satisfies (87). Multiplication of (92) by ∆t∆x +and summation over Ωℓ yields +� +ℓ +∆x +� +En+1,ℓ − En,ℓ� += − +� +ℓ +∆t +� +˜D +ℓ+ 1 +2 ,− +E ++ ˜D +ℓ− 1 +2 ,+ +E ++ g +ℓ+ 1 +2 +E +− g +ℓ− 1 +2 +E +� += 0. +(99) +The terms on the right hand side of (99) are a telescopic sum that vanishes because the fluctuations +satisfy (94) and since the jumps vanish at the boundary. +Theorem 4. The fully-discrete finite volume scheme (91) with production term ˜Pℓ according to (87) and +fluxes (85) and (86) satisfies the fully discrete cell entropy inequality +(ρS)n+1,ℓ ≥ (ρS)n,ℓ − ∆t +∆x +� +f +ℓ+ 1 +2 +ρS +− f +ℓ− 1 +2 +ρS +� ++ ∆t +∆x +� +g +ℓ+ 1 +2 +ρS +− g +ℓ− 1 +2 +ρS +� +, +(100) +assuming that ˜T ℓ > 0 and ∂2 +pp ˜Lℓ± 1 +2 > 0. +Proof. The fully-discrete form of the entropy equation (1c) according to the scheme (91) reads +(ρS)n+1,ℓ = (ρS)n,ℓ − ∆t +∆x +� +f +ℓ+ 1 +2 +ρS +− f +ℓ− 1 +2 +ρS +� ++ ∆t +∆x +� +g +ℓ+ 1 +2 +ρS +− g +ℓ− 1 +2 +ρS +� ++ ∆t ˜Πℓ, +(101) +with +˜Πℓ = 1 +˜T ℓ +� +1 +2ϵℓ+ 1 +2 ∆˜pℓ+ 1 +2 +∆x +· ∂2 +pp ˜Lℓ+ 1 +2 ∆˜pℓ+ 1 +2 +∆x ++ 1 +2ϵℓ− 1 +2 ∆˜pℓ− 1 +2 +∆x +· ∂2 +pp ˜Lℓ− 1 +2 ∆˜pℓ− 1 +2 +∆x +� +≥ 0, +(102) +since we assume ˜T ℓ > 0 and ∂2 +pp ˜Lℓ± 1 +2 > 0, hence one directly obtains the inequality (100). +6 +Numerical results +The new schemes for hyperbolic and thermodynamically compatible PDE systems (HTC schemes) pro- +posed in this paper do not discretize the energy conservation law (1f) explicitly, but consider the entropy +inequality (1c) instead. Semi-discrete / fully-discrete energy conservation is obtained as a mere conse- +quence of the thermodynamically compatible discretization of the PDEs (1a)-(1e). As such, the proposed +approach is different from most existing finite volume discretizations. The main aim of the following +numerical test problems is therefore to show that the scheme is able to compute correct solutions to +problems with shock waves, as predicted by Theorems 1 and 3 on the semi-discrete and fully-discrete +energy conservation, respectively. Furthermore, we check numerically whether the relaxation limit of +the model (Navier-Stokes limit) is properly captured, i.e. when for sufficiently small relaxation times +τ1 and τ2 the behaviour of the medium becomes the one of a viscous heat-conducting Newtonian fluid. +For more advanced applications of the GPR model, the reader is referred to [20, 21, 9, 56, 44, 50]. In +the following tests, when a viscosity coefficient µ is specified together with a shear sound speed cs, the +corresponding relaxation time τ1 is calculated as τ1 = 6µ/(ρ0c2 +s), according to (9) and the results of the +asymptotic analysis carried out in [20]. In all tests of this section, the semi-discrete HTC schemes are +integrated in time using an explicit third order TVD Runge-Kutta scheme, see [52, 34]. For more efficient +16 + +Table 1: Numerical convergence results for the isentropic vortex problem, obtained with the semi-discrete +HTC scheme proposed in this paper. The reported L2 error norms refer to a final time of t = 0.25. +Nx = Ny +∥ρ∥2 +∥ρv1∥2 +∥ρS∥2 +O(ρ) +O(ρv1) +O(ρS) +32 +6.1094E-03 +9.1324E-03 +4.7896E-04 +64 +1.5602E-03 +2.3633E-03 +1.3256E-04 +2.0 +2.0 +1.9 +128 +3.9230E-04 +5.9585E-04 +3.3972E-05 +2.0 +2.0 +2.0 +256 +9.8232E-05 +1.4928E-04 +8.5455E-06 +2.0 +2.0 +2.0 +512 +2.4626E-05 +3.7369E-05 +2.1397E-06 +2.0 +2.0 +2.0 +Table 2: Numerical convergence results for the isentropic vortex problem, obtained with the fully-discrete +HTC scheme proposed in this paper. The reported L2 error norms refer to a final time of t = 0.25. +Nx = Ny +∥ρ∥2 +∥ρv1∥2 +∥ρS∥2 +O(ρ) +O(ρv1) +O(ρS) +32 +6.2046E-03 +9.1891E-03 +4.8312E-04 +64 +1.5749E-03 +2.3742E-03 +1.3384E-04 +2.0 +2.0 +1.9 +128 +3.9424E-04 +5.9737E-04 +3.4170E-05 +2.0 +2.0 +2.0 +256 +9.8481E-05 +1.4948E-04 +8.5718E-06 +2.0 +2.0 +2.0 +512 +2.4658E-05 +3.7394E-05 +2.1430E-06 +2.0 +2.0 +2.0 +IMEX Runge-Kutta schemes in the case of stiff relaxation source terms, see the work of Pareschi & Russo +[42, 43] as well as [39, 14, 8, 37]. In all numerical tests carried out with the semi-discrete scheme, we +assume the time step ∆t to be small enough so that time discretization errors can be neglected concening +the conservation of total energy. In the following, if not stated otherwise, the numerical viscosity ϵℓ+ 1 +2 is +chosen according to (25). When explicit values of ϵ are provided, the numerical dissipation is chosen as +a constant, ϵℓ+ 1 +2 = ϵ. +6.1 +Numerical convergence study +In this section, we solve the isentropic vortex problem forwarded in [36] in order to verify the accuracy of +the proposed HTC schemes. We apply the schemes to the pure inviscid Euler equations, i.e. to the black +terms in (1a)-(1c), setting γ = 1.4, cs = 0, ch = 0 and ϵ = 0. The computational domain is Ω = [0, 10]2 +with periodic boundaries everywhere. The initial conditions for the perturbations are given in [36, 20] +and are not repeated here to save space. The background velocity is chosen as v0 = 0 so that a stationary +vortex is obtained. In this situation, the exact solution is given by the initial condition for all times. +Simulations are run with the semi-discrete HTC scheme until a final time of t = 0.25 using an equidistant +Cartesian grid composed of Nx × Ny control volumes. The L2 errors obtained with the semi-discrete +HTC schemes at the final time for the density ρ, the momentum density ρv1 and the entropy density ρS +are shown in Table 1 together with the corresponding convergence rates. The results for the fully discrete +HTC scheme are reported in Table 2. One can observe that all proposed HTC schemes are of second +order of accuracy. +6.2 +Simple shear motion in solids and fluids +We first apply the new HTC schemes to simple shear motion in solids and fluids. The one-dimensional +computational domain is Ω = [−0.5, +0.5] and the initial condition of the problem, which is also prescribed +at the boundaries of Ω, is given by ρ = 1, v1 = v3 = 0, p = 1, A = I, J = 0, while the velocity component +v2 is v2 = −v0 for x < 0 and v2 = +v0 for x ≥ 0, with v0 = 0.1. The remaining parameters of this test +are γ = 1.4, cv = 1, ρ0 = 1, cs = 1 and ch = 0. The calculations are carried out with the new HTC +17 + +schemes on a grid composed of 1024 control volumes up to a final time of t = 0.4. In the Navier-Stokes +limit of the GPR model, a reference solution can be obtained by the exact solution of the incompressible +Navier-Stokes equations for the first problem of Stokes, see e.g. [20, 7, 6, 12]. For the solid limit of the +GPR model (τ1 → ∞), this initial condition leads to two shear waves traveling to the left and right, +respectively, with speed cs. A reference solution can be obtained using a classical second order MUSCL- +Hancock scheme [57] on a fine mesh of 32000 cells. We stress that for all cases with µ > 0 the HTC +scheme has been run without any numerical viscosity, i.e. setting ϵ = 0. The comparison between the +numerical solutions obtained with the new HTC schemes and the aforementioned reference solutions is +presented in Fig. 1, where one can observe an excellent agreement for all cases. +x +v +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.125 +-0.1 +-0.075 +-0.05 +-0.025 +0 +0.025 +0.05 +0.075 +0.1 +0.125 +Reference solution +HTC scheme +x +v +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.125 +-0.1 +-0.075 +-0.05 +-0.025 +0 +0.025 +0.05 +0.075 +0.1 +0.125 +Reference solution +HTC scheme +x +v +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.125 +-0.1 +-0.075 +-0.05 +-0.025 +0 +0.025 +0.05 +0.075 +0.1 +0.125 +Reference solution +HTC scheme +x +v +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.125 +-0.1 +-0.075 +-0.05 +-0.025 +0 +0.025 +0.05 +0.075 +0.1 +0.125 +Reference solution +HTC scheme +Figure 1: Numerical solution at time t = 0.4 obtained with the new thermodynamically compatible HTC +schemes for the GPR model applied to a simple shear flow in fluids and in an elastic solid. Results for +the solid (top left) and for fluids with different viscosities: µ = 10−2 (top right), µ = 10−3 (bottom left) +and µ = 10−4 (bottom right). For fluids, this test corresponds to the first problem of Stokes, which has +an exact analytical solution. +6.3 +Riemann problems +In this section, we solve a set of Riemann problems with initial data according to Table 3, for both the +Euler equations of compressible gasdynamics, which are a subset of the GPR model (black terms in (1)), +and for the full GPR model in both its fluid and solid limit. The initial discontinuity is located in xc. +For the Euler equations, we consider semi-discrete as well as fully-discrete schemes and the exact solution +of the Riemann problem has been provided in [57], while for the GPR model we consider two types of +completely independent numerical reference solutions. The first reference solution is obtained by using a +classical MUSCL-Hancock finite volume scheme on a fine mesh of 128000 elements, discretizing the total +energy conservation law (1f) instead of the entropy inequality (1c). An alternative reference solution is +18 + +obtained by solving the GPR model (1a)-(1c) with the entropy inequality in its vanishing viscosity limit, +using a fourth order ADER-DG scheme on a fine mesh composed of 14400 order elements, including also +the quadratic entropy production term in (1c). In this case, thermodynamic compatibility is achieved +simply at the aid of a fully resolved simulation employing sufficiently fine meshes in combination with +high order of accuracy in space and time, see [10]. The numerical results obtained with the semi-discrete +and fully-discrete HTC schemes for the compressible Euler equations are shown in Figs. 2 and 3, while the +numerical results obtained with the semi-discrete HTC scheme applied to the fluid and solid limits of the +GPR model are presented in Fig. 4 and 5, respectively, together with the reference solution obtained with +the MUSCL-Hancock scheme solving the energy conservation law (1f), as well as the reference solution +obtained with the high order ADER-DG scheme applied to the viscous system (1a)-(1c). The effective +mesh resolution is provided for each test case in the corresponding figure caption. In all cases we can note +an excellent agreement between the numerical solution obtained with the new HTC schemes forwarded +in this paper and the available exact or numerical reference solutions. +Test problem RP1s was proposed by Toro in [57] and includes a sonic rarefaction. Simulations are +carried out on several meshes and the obtained quantities ρ, p, u = v1 and S are shown in Fig. 3. We +observe that the thermodynamically compatible schemes proposed in this paper do not exhibit any sonic +glitch, compared to other Godunov-type finite volume schemes, see [57]. +A quantitative study concerning the influence of the number of Gauss-Legendre quadrature nodes nGP +and the chosen time discretization on the total energy conservation error can be found for a smoothed +version of RP1 with initial data q(x, 0) = 1 +2(qL + qR) + 1 +2(qR − qL) erf(x/χ) with χ = 0.01 in Table 6.3. +As expected, the conservation error of the semi-discrete schemes is dominated by the time discretization +and the chosen time step size (CFL number), while the energy conservation error of the fully discrete +scheme is independent of the time step size and is dominated only by the numerical quadrature rule used +in (38). +Table 3: Initial states left (L) and right (R) for density ρ, velocity v = (u, v, 0) and pressure p for a set +of Riemann problems solved on the domain Ω = [− 1 +2, + 1 +2] using the new HTC schemes. The Riemann +problems include the pure Euler equations (RP1, RP2 and RP1s), as well as the fluid and solid limit of +the GPR model (RP3 and RP4). For the GPR model (RP3 and RP4) we initialize A and J as A = +3√ρ I +and J = 0 and set cs = ch = 1. The relaxation times have been chosen as τ1 = τ2 = 2 · 10−5 for RP3 and +τ1 = τ2 = 1020 for RP4. In all cases we set γ = 1.4. +RP +ρL +uL +vL +pL +ρR +uR +vR +pR +RP1 +1.0 +0.0 +0.0 +1.0 +0.125 +0.0 +0.0 +0.1 +RP1s +1.0 +0.75 +0.0 +1.0 +0.125 +0.0 +0.0 +0.1 +RP2 +5.99924 +19.5975 +0.0 +460.894 +5.99242 +-6.19633 +0.0 +46.095 +RP3 +1.0 +0.0 +-0.2 +1.0 +0.5 +0.0 ++0.2 +0.5 +RP4 +1.0 +0.0 +-0.2 +1.0 +0.5 +0.0 ++0.2 +0.5 +6.4 +Viscous shock wave +Consider a stationary viscous shock wave at a shock Mach number of Ms = 2. For Prandtl number +Pr= 0.75 there exists an exact solution of the compressible Navier-Stokes equations, see [4, 20]. The +computational domain Ω = [−0.5, +0.5] is covered with 1024 control volumes and the shock wave is +centered at x = 0. We assume that the fluid is moving into the shock wave from right to left. The data +in front of the shock are ρ0 = 1, v0 +1 = −2, v0 +2 = v3 = 0 and p0 = 1/γ so that the associated sound +speed is c0 = 1 and the corresponding Reynolds number based on a reference length L = 1 is given by +Res = ρ0 c0 Ms L µ−1. The parameters are set as γ = 1.4, cv = 2.5, ch = cs = 50, µ = 2 · 10−2 and +λ = 9 1 +3 ·10−2, hence the shock Reynolds number is Res = 100. At t = 0 we set A = +3√ρ I and J = 0. The +comparison between the numerical solution obtained with the semi-discrete HTC scheme applied to (1) +and the exact solution of the compressible Navier-Stokes equations is shown in Fig. 6. For all quantities +19 + +Table 4: Total energy conservation error depending on the time discretization and the number of Gauss- +Legendre quadrature points nGP for the calculation of the thermodynamically compatible flux (38). +CFL +0.5 +0.4 +0.3 +0.2 +0.1 +semi-discrete HTC scheme + TVD Runge-Kutta O3 +nGP = 3 +2.90 · 10−5 +1.55 · 10−5 +6.30 · 10−6 +2.00 · 10−6 +3.00 · 10−7 +nGP = 5 +2.90 · 10−5 +1.54 · 10−5 +6.31 · 10−5 +1.99 · 10−5 +2.45 · 10−7 +semi-discrete HTC scheme + classical Runge-Kutta O4 +nGP = 3 +2.23 · 10−6 +9.51 · 10−7 +3.07 · 10−7 +5.25 · 10−8 +8.33 · 10−9 +nGP = 5 +2.24 · 10−6 +9.64 · 10−7 +3.19 · 10−7 +6.53 · 10−8 +4.43 · 10−9 +Fully-discrete HTC scheme +nGP = 3 +1.80 · 10−9 +1.81 · 10−9 +1.82 · 10−9 +1.83 · 10−9 +1.84 · 10−9 +nGP = 5 +2.70 · 10−13 +2.70 · 10−13 +2.70 · 10−13 +2.70 · 10−13 +2.70 · 10−13 +x +rho +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Exact solution +Semi-discrete HTC scheme +Fully-discrete HTC scheme +x +rho +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Exact solution +Semi-discrete HTC scheme +Fully-discrete HTC scheme +Figure 2: Numerical results for Riemann problems RP1 (xc = 0) and RP2 (xc = −0.2) at times t = 0.2 +and t = 0.035, respectively, obtained with the semi-discrete (red solid line) and the fully-discrete (dashed +blue line) HTC schemes on 1024 elements applied to the compressible Euler equations. The exact solution +of the compressible Euler equations is represented by the black solid line. +20 + +x +rho +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +x +p +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Exact solution +Semi-discrete HTC (2048) +Semi-discrete HTC (1024) +Semi-discrete HTC (512) +Semi-discrete HTC (256) +Semi-discrete HTC (128) +Fully-discrete HTC (2048) +Fully-discrete HTC (1024) +Fully-discrete HTC (512) +Fully-discrete HTC (256) +Fully-discrete HTC (128) +x +u +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +x +S +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +Figure 3: Numerical results for Riemann problem RP1s (xc = −0.2) at time t = 0.2 obtained with the +semi-discrete (solid lines) and the fully-discrete (dashed lines) HTC schemes on 2048, 1024, 512, 256 and +128 elements applied to the compressible Euler equations. The exact solution of the compressible Euler +equations is represented by the black solid line. +21 + +x +rho +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +Exact solution +Vanishing viscosity limit +HTC scheme +x +v +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +Exact solution +Vanishing viscosity limit +HTC scheme +Figure 4: Numerical results at time t = 0.2 for Riemann problem RP3 (xc = 0) obtained with the HTC +scheme (red solid line) on 1024 elements, the fourth order ADER-DG scheme applied to the vanishing +viscosity limit of the viscous equations (1a)-(1c) using ϵ = 2 · 10−5 on 14400 elements (dashed blue line) +and the exact solution of the compressible Euler equations (black solid line). +an excellent agreement is achieved. +6.5 +Solid rotor problem +In this section we solve the solid rotor problem proposed in [7]. By setting τ1 = τ2 = 1020 the model +(1) describes a nonlinear hyperelastic solid. The computational domain is Ω = [−1, +1]2 with periodic +boundary conditions everywhere. The initial data for density, pressure, A and J is set to ρ = 1, p = 1, +A = I and J = 0, while the initial condition for the velocity field is v1 = −y/R, v2 = +x/R and v3 = 0 +within the circular region r ≤ R, where r = ∥x∥ and R = 0.2, while v = 0 for r > R. The parameters of +the GPR model are set to γ = 1.4, cs = 1.0 and ch = 1.0. We run the test problem until a final time of +t = 0.3 using the two-dimensional semi-discrete HTC scheme for the GPR model on a uniform Cartesian +grid composed of 512 × 512 elements. The artificial viscosity in the HTC scheme is set to a constant +value of ϵ = 5 · 10−4. To obtain a reference solution, on the same mesh of 512 × 512 elements we solve +the same problem again but using a classical second order MUSCL-Hancock scheme, see [57] for details. +We emphasize that in the MUSCL scheme, which is not thermodynamically compatible, we solve the +total energy conservation law (1f) rather than the entropy inequality (1c), as already suggested in [20]. +The obtained results are compared with each other in Fig. 7, where the contour colors of the velocity +component v1 are shown. The agreement between the numerical solution obtained with the new HTC +scheme and the reference solution is very good. Since the applied HTC scheme for this test problem is only +compatible with the semi-discrete total energy conservation law, we have explicitly monitored the total +energy conservation error during the entire simulation, finding a maximum relative energy conservation +error of 4.02 · 10−7. +6.6 +Double shear layer +In this section we present numerical results for the double shear layer test, see [5, 20, 6, 12]. +The +computational domain is Ω = [0, 1]2 with periodic boundary conditions everywhere. The initial condition +is given by v1 = tanh (˜ρ(y − 0.25)) for y ≤ 0.5 and v1 = tanh (˜ρ(0.75 − y)) if y > 0.5, v2 = δ sin(2πx), v3 = +0, ρ = ρ0 = 1, p = 102/γ, A = I, J = 0, with δ = 0.05 and ˜ρ = 30. The remaining parameters of the +GPR model are set to ν = µ/ρ0 = 2 · 10−3, γ = 1.4, ρ0 = 1, cv = 1, cs = 8, ch = 2 and τ2 = 4 · 10−3. +The characteristic Mach number of the flow resulting from this setup is M = 0.1. +Calculations are +22 + +x +rho +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +Reference solution +Vanishing viscosity limit +HTC scheme +x +v +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Reference solution +Vanishing viscosity limit +HTC scheme +x +A11 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +Reference solution +Vanishing viscosity limit +HTC scheme +x +J1 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +Reference solution +Vanishing viscosity limit +HTC scheme +Figure 5: Numerical results at time t = 0.2 for Riemann problem RP4 (xc = 0) obtained with the +HTC scheme (red solid line) on 10000 control volumes, a fourth order ADER-DG scheme applied to the +vanishing viscosity limit of the viscous equations (1a)-(1c) using ϵ = 2·10−5 (dashed blue line) on 144000 +elements and the reference solution obtained with a MUSCL-Hancock scheme applied to the model with +the energy conservation law (1f) instead of the entropy inequality (1c) (black solid line) using 128000 +elements. +x +rho +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Reference solution +HTC scheme +x +sigma11 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Reference solution +HTC scheme +x +h1 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.9 +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +Reference solution +HTC scheme +Figure 6: Exact solution of the compressible Navier-Stokes equations and numerical solution obtained +with the HTC scheme applied to the GPR model for a viscous shock at Ms = 2, Res = 100 and Pr = 0.75. +Density (left), stress σ11 (center) and heat flux h1 (right) at time t = 0.25. +23 + +Figure 7: Velocity component v1 for the solid rotor test problem at time t = 0.3 obtained by solving (1) +with the new HTC scheme (left) and by using a classical MUSCL scheme (right). +performed with the new HTC scheme up to a final time of t = 1.8. The computational grid is composed +of 4000 × 4000 control volumes and the numerical viscosity is chosen as ϵ = 1 · 10−6, hence three orders +of magnitude lower than the physical one. In Fig. 8 the results obtained with the new HTC scheme +are compared with a numerical reference solution that is based on the solution of the incompressible +Navier-Stokes equations using a hybrid FV/FE method on a triangular grid made of 2097152 elements +(Nx = 1000 divisions along each boundary), see [11, 6, 12] for details. The flow dynamics has already +been described in [5, 6, 12, 20, 7] and can be summarized by the development of several vortices from +the initially perturbed shear layers. The agreement between the Navier-Stokes reference solution and the +numerical solution of the GPR model computed with the new HTC schemes is rather good. In Fig. 9 we +present the temporal evolution of the distortion field component A12, which is qualitatively similar to the +results shown in [20], but for a lower physical viscosity µ. The maximum relative conservation error of +the total energy monitored during the simulation for the semi-discrete HTC scheme was 7.12 · 10−7. Due +to the low numerical viscosity of ϵ = 10−6 and fine mesh, one can observe small structures developing in +the distortion field A, which we would like to demonstrate in Fig. 9. +Figure 8: Vorticity contours for the double shear layer with a viscosity of µ = 2 · 10−3 at time t = 1.8. +Left: numerical solution of the GPR model obtained with the new thermodynamically compatible finite +volume scheme. Right: reference solution obtained by solving the incompressible Navier-Stokes equations +with the staggered semi-implicit hybrid FV/FE scheme [11, 6, 12]. +24 + +u +0.24 +0.22 +0.2 +0.18 +0.16 +0.14 +0.12 +0.1 +0.08 +0.06 +0.04 +0.02 +0 +-0.02 +-0.04 +-0.06 +-0.08 +-0.1 +-0.12 +-0.14 +-0.16 +-0.18 +-0.2 +-0.22 +-0.24u +0.24 +0.22 +0.2 +0.18 +0.16 +0.14 +0.12 +0.1 +0.08 +0.06 +0.04 +0.02 +0 +-0.02 +-0.04 +-0.06 +-0.08 +-0.1 +-0.12 +-0.14 +-0.16 +-0.18 +-0.2 +-0.22 +-0.2411 +10 +0.8 +9 +8 +7 +6 +5 +0.6 +4 +3 +2 +1 +0 +-1 +0.4 +-2 +-3 +456 +0.2 +-7 +-8 +-10 +-11 +0 +0 +0.2 +0.4 +0.6 +0.8 +x11 +10 +0.8 +9 +8 +7 +6 +5 +0.6 +4 +3 +2 +1 +0 +-1 +0.4 +-2 +-3 +-4 +-5 +-6 +0.2 +-7 +-8 +-9 +-10 +-11 +0 +0 +0.2 +0.4 +0.6 +0.8Figure 9: Distortion field component A12 for the double shear layer problem at times t = 1.2 and t = 1.8 +obtained by solving the GPR model (µ = 2 · 10−3) with the HTC scheme. +6.7 +Lid-driven cavity +As last numerical test case for the fluid limit of the model (1) we present the lid-driven cavity problem, +see [27], which can be used to validate compressible flow solvers in the low Mach number regime, see e.g. +[55, 6, 12] and which was already successfully solved with the GPR model in [20, 7], but the schemes +used in [20, 7] were not thermodynamically compatible. The computational domain is Ω = [0, 1] × [0, 1] +and the initial condition is set to ρ = 1, v = 0, p = 102, A = I and J = 0. We furthermore set γ = 1.4, +cv = 1, cs = 8, ρ0 = 1 and ch = 2, τ2 = 10−2 and µ = 10−2 so that the Reynolds number of the test +problem is Re = 100. The lid velocity on the upper boundary is set to v = (1, 0, 0), while on all other +boundaries v = 0 is imposed. The Mach number of this test is about M = 0.08. The new semi-discrete +HTC scheme is run until t = 10 using 256 × 256 elements and a constant artificial viscosity of ϵ = 10−3. +The numerical results are shown in Fig. 10, where also a comparison with the Navier-Stokes reference +solution of Ghia et al. [27] is provided. We note an excellent agreement between the numerical solution +of the GPR model and the incompressible Navier-Stokes reference solution. +x,y +u,v +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +GPR model (HTC scheme) - u(0,y) +GPR model (HTC scheme) - v(x,0) +Reference solution - u(0,y) +Reference solution - v(x,0) +Figure 10: Lid-driven cavity at Reynolds number Re = 100. Results obtained at time t = 10 with the +new HTC scheme applied to the GPR model. Color contours of the velocity component v1 (left) and +comparison of the velocity components v1 and v2 on 1D cuts along the x and y axis with the reference +solution of Ghia et al. [27] (right). +25 + +A12 +0.8 +0.9 +0.8 +0.7 +0.6 +0.5 +0.6 +0.4 +0.3 +0.2 +> +0.1 +0 +-0.1 +0.4 +-0.2 +-0.3 +-0.4 +-0.5 +-0.6 +-0.7 +0.2 +-0.8 +-0.9 +-1 +0 +0 +0.2 +0.4 +0.6 +0.8 +xA12 +0.8 +0.9 +0.8 +0.7 +0.6 +0.5 +0.6 +0.4 +0.3 +0.2 +> +0.1 +0 +-0.1 +0.4 +-0.2 +-0.3 +-0.4 +-0.5 +-0.6 +-0.7 +0.2 +-0.8 +-0.9 +-1 +0 +0 +0.2 +0.4 +0.6 +0.8 +xu +0.95 +0.9 +0.85 +0.8 +0.8 +0.75 +0.7 +0.65 +0.6 +0.6 +0.55 +0.5 +0.45 +y +0.4 +0.35 +0.3 +0.4 +0.25 +0.2 +0.15 +0.1 +0.05 +0.2 +0 +-0.05 +-0.1 +-0.15 +-0.2 +0 +0 +0.2 +0.4 +0.6 +0.8 +x7 +Conclusions +In this paper, we have presented two novel thermodynamically compatible finite volume schemes for first +order hyperbolic PDE systems (HTC schemes). The first method is a semi-discrete finite volume scheme +for the unified first order hyperbolic model of solid and fluid mechanics that goes back to the work of +Godunov, Peshkov and Romenski on symmetric hyperbolic and thermodynamically compatible (SHTC) +systems, see [29, 31, 51, 33, 46]. We have furthermore introduced a new fully-discrete HTC scheme for +the compressible Euler equations, establishing a fully discrete analogy of the continuous framework in- +troduced by Godunov in [29]. All schemes under consideration in this paper have in common that they +directly discretize the entropy inequality rather than the usual total energy conservation law. Instead, +total energy conservation is obtained at the discrete level as a mere consequence of a suitable and ther- +modynamically compatible discretization of all the other equations. As such, the new schemes can be +proven to be nonlinearly marginally stable in the energy norm and they furthermore satisfy a discrete +entropy inequality by construction. The new HTC schemes have been applied to several test problems +for fluid and solid mechanics, obtaining an excellent agreement with available reference solutions. In +future work, we will investigate the possible use of symplectic time integrators in order to preserve exact +total energy conservation of our new semi-discrete thermodynamically compatible scheme also on the +fully discrete level. We also plan an extension to higher order in space at the aid of thermodynami- +cally compatible discontinuous Galerkin (DG) finite element schemes, similar to entropy compatible DG +schemes introduced in [19, 38, 26] for the shallow water equations and magnetohydrodynamics (MHD), +as well as an extension to general unstructured meshes. Another major challenge left to future work is +the development of HTC schemes that are not only thermodynamically compatible, but which are also +able to preserve the curl involution constraints of the governing PDE system exactly at the semi-discrete +level and that are also consistent with the low Mach number limit of the equations. In this context we +will consider staggered semi-implicit finite volume schemes [7], as well as staggered semi-implicit hybrid +finite volume / finite element methods [11, 6, 12] and staggered DG schemes [55, 13], which are not yet +thermodynamically compatible in the sense of the HTC schemes presented in this paper. +Acknowledgments +S.B., M.D. and I.P. are members of the INdAM GNCS group and acknowledge the financial support +received from the Italian Ministry of Education, University and Research (MIUR) in the frame of the +Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (grant +L. 232/2016) and in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary +partial differential equations and applications. S.B. was also funded by INdAM via a GNCS grant for +young researchers and by an UniTN starting grant of the University of Trento. E.R., M.D. and I.P. +were supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 +with the Ministry of Science and Higher Education of the Russian Federation. The authors would like to +acknowledge support from the Leibniz Rechenzentrum (LRZ) in Garching, Germany, for granting access +to the SuperMUC-NG supercomputer under project number pr63qo. The authors are very grateful to +the two anonymous referees for their constructive and insightful comments, which helped to improve the +clarity and quality of this paper. +References +[1] R. Abgrall. A general framework to construct schemes satisfying additional conservation relations. +Application to entropy conservative and entropy dissipative schemes. J. Comput. Phys., 372:640–666, +2018. +[2] R. Abgrall, P. Bacigaluppi, and S. Tokareva. A high-order nonconservative approach for hyperbolic +equations in fluid dynamics. Computers and Fluids, 169:10–22, 2018. +26 + +[3] A.L. Bauera, D.E.Burton, E.J. Caramana, R.Loub`ere, M.J. Shashkov, and P.P. Whalen. The in- +ternal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian +hydrodynamics. Journal of Computational Physics, 218:572–593, 2006. +[4] R. Becker. Stosswelle und Detonation. Physik, 8:321, 1923. +[5] J. B. Bell, P. Coletta, and H. M. Glaz. A second-order projection method for the incompressible +Navier-Stokes equations. J. Comput. Phys., 85:257–283, 1989. +[6] A. Berm´udez, S. Busto, M. Dumbser, J.L. Ferr´ın, L. Saavedra, and M.E. V´azquez-Cend´on. +A +staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows. J. Comput. +Phys., 421:109743, 2020. +[7] W. Boscheri, M. Dumbser, M. Ioriatti, I. Peshkov, and E. Romenski. A structure-preserving staggered +semi-implicit finite volume scheme for continuum mechanics. J. Comput. Phys., 424:109866, 2021. +[8] C. Buet and B. Despr´es. Asymptotic preserving and positive schemes for radiation hydrodynamics. +J. Comput. Phys., 215(2):717–740, 2006. +[9] S. Busto, S. Chiocchetti, M. Dumbser, E. Gaburro, and I. Peshkov. High order ADER schemes for +continuum mechanics. Frontiers in Physics, 8:32, 2020. +[10] S. Busto, M. Dumbser, S. Gavrilyuk, and K. Ivanova. +On thermodynamically compatible finite +volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow +water flows. Journal of Scientific Computing, 88:28, 2021. +[11] S. Busto, J.L. Ferr´ın, E.F. Toro, and M.E. V´azquez-Cend´on. A projection hybrid high order finite +volume/finite element method for incompressible turbulent flows. J. Comput. Phys., 353:169–192, +2018. +[12] S. Busto, L. Del Rio, M.E. V´azquez-Cend´on, and M. Dumbser. A semi-implicit hybrid finite volume +/ finite element scheme for all Mach number flows on staggered unstructured meshes. Appl. Math. +Comput., 402:126117, 2021. +[13] S. Busto, M. Tavelli, W. Boscheri, and M. Dumbser. Efficient high order accurate staggered semi- +implicit discontinuous Galerkin methods for natural convection problems. +Computers & Fluids, +198:104399, 2020. +[14] R.E. Caflish, S. Jin, and G. Russo. Uniformly accurate schemes for hyperbolic systems with relax- +ation. SIAM J. Numer. Anal., 34:246–281, 1997. +[15] E.J. Caramana and R.Loub`ere. The force/work differencing of exceptional points in the discrete, +compatible formulation of Lagrangian hydrodynamics. Journal of Computational Physics, 216:1–18, +2006. +[16] M.J. Castro, J.M. Gallardo, and C. Par´es. High-order finite volume schemes based on reconstruction +of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water +systems. Math. Comput., 75:1103–1134, 2006. +[17] N. Chatterjee and U.S. Fjordholm. Convergence of second-order, entropy stable methods for multi- +dimensional conservation laws. ESAIM Math. Model. Numer. Anal., 54(4):1415–1428, 2020. +[18] T. Cheng and C.W. Shu. Entropy stable high order discontinuous Galerkin methods with suitable +quadrature rules for hyperbolic conservation laws. J. Comput. Phys., 345:427–461, 2017. +[19] D. Derigs, A. R. Winters, G. Gassner, S. Walch, and M. Bohm. Ideal GLM-MHD: About the entropy +consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations. +J. Comput. Phys., 364:420–467, 2018. +27 + +[20] M. Dumbser, I. Peshkov, E. Romenski, and O. Zanotti. High order ADER schemes for a unified first +order hyperbolic formulation of continuum mechanics: Viscous heat–conducting fluids and elastic +solids. J. Comput. Phys., 314:824–862, 2016. +[21] M. Dumbser, I. Peshkov, E. Romenski, and O. Zanotti. High order ADER schemes for a unified first +order hyperbolic formulation of Newtonian continuum mechanics coupled with electro–dynamics. J. +Comput. Phys., 348:298–342, 2017. +[22] M. Dumbser and E. F. Toro. A simple extension of the Osher Riemann solver to non-conservative +hyperbolic systems. J. Sci. Comput., 48:70–88, 2011. +[23] U.S. Fjordholm and S. Mishra. Accurate numerical discretizations of non-conservative hyperbolic +systems. ESAIM Math. Model. Numer. Anal., 46(1):187–206, 2012. +[24] K.O. Friedrichs. Symmetric positive linear differential equations. Comm. Pure Appl. Math., 11:333– +418, 1958. +[25] K.O. Friedrichs and P.D. Lax. Systems of conservation equations with a convex extension. Proc. +Nat. Acad. Sci. USA, 68:1686–1688, 1971. +[26] G. Gassner, A.R. Winters, and D.A. Kopriva. A well balanced and entropy conservative discontinuous +Galerkin spectral element method for the shallow water equations. Appl. Math. Comput., 272:291– +308, 2016. +[27] U. Ghia, K. N. Ghia, and C. T. Shin. High-Re solutions for incompressible flow using Navier-Stokes +equations and multigrid method. J. Comput. Phys., 48:387–411, 1982. +[28] S. K. Godunov. Thermodynamic formalization of the fluid dynamics equations for a charged dielectric +in an electromagnetic field. Comput. Math. Math. Phys., 52:787–799, 2012. +[29] S.K. Godunov. An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR, 139(3):521–523, +1961. +[30] S.K. Godunov. Symmetric form of the magnetohydrodynamic equation. Numerical Methods for +Mechanics of Continuum Medium, 3(1):26–34, 1972. +[31] S.K. Godunov and E.I. Romenski. Nonstationary equations of the nonlinear theory of elasticity in +Euler coordinates. J. Appl. Mech. Tech. Phys., 13:868–885, 1972. +[32] S.K. Godunov and E.I. Romenski. Thermodynamics, conservation laws, and symmetric forms of +differential equations in mechanics of continuous media. In Computational Fluid Dynamics Review +95, pages 19–31. John Wiley, NY, 1995. +[33] S.K. Godunov and E.I. Romenski. Elements of continuum mechanics and conservation laws. Kluwer +Academic/Plenum Publishers, 2003. +[34] S. Gottlieb and C.W. Shu. +Total variation diminishing Runge-Kutta schemes. +Math. Comput., +67:73–85, 1998. +[35] S. Hennemann, A.M. Rueda-Ram´ırez, F.J. Hindenlang, and G.J. Gassner. A provably entropy stable +subcell shock capturing approach for high order split form DG for the compressible Euler equations. +J. Comput. Phys., 426, 2021. +[36] C. Hu and C.W. Shu. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. +Phys., 150:97–127, 1999. +[37] S. Jin, L. Pareschi, and G. Toscani. Uniformly accurate diffusive relaxation scheme for multiscale +transport equations. SIAM J. Numer. Anal., 38(3):913–936, 2001. +28 + +[38] Y. Liu, C.W. Shu, and M. Zhang. Entropy stable high order discontinuous Galerkin methods for +ideal compressible MHD on structured meshes. J. Comput. Phys., 354:163–178, 2018. +[39] G. Naldi and L. Pareschi. Numerical schemes for hyperbolic systems of conservation laws with stiff +diffusive relaxation. SIAM J. Numer. Anal., 37(4):1246–1270, 2000. +[40] S. Osher and F. Solomon. +Upwind difference schemes for hyperbolic conservation laws. +Math. +Comput., 38:339–374, 1982. +[41] C. Par´es. +Numerical methods for nonconservative hyperbolic systems: a theoretical framework. +SIAM J. Numer. Anal., 44:300–321, 2006. +[42] L. Pareschi and G. Russo. Implicit-explicit Runge-Kutta schemes for stiff systems of differential +equations. Advances in the Theory of Computational Mathematics, 3:269–288, 2000. +[43] L. Pareschi and G. Russo. Implicit-explicit Runge-Kutta schemes and applications to hyperbolic +systems with relaxation. J. Sci. Comput., 25:129–155, 2005. +[44] I. Peshkov, M. Dumbser, W. Boscheri, E. Romenski, S. Chiocchetti, and M. Ioriatti. Simulation +of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure- +preserving semi-implicit scheme. Computers & Fluids, page 104963, 2021. +[45] I. Peshkov, M. Pavelka, E. Romenski, and M. Grmela. Continuum mechanics and thermodynamics in +the Hamilton and the Godunov-type formulations. Continuum Mech. Thermodyn., 30(6):1343–1378, +2018. +[46] I. Peshkov and E. Romenski. A hyperbolic model for viscous Newtonian flows. Continuum Mech. +Thermodyn., 28:85–104, 2016. +[47] I. Peshkov, E. Romenski, and M. Dumbser. Continuum mechanics with torsion. Continuum Mech. +Thermodyn., 31:1517–1541, 2019. +[48] H. Ranocha, L. Dalcin, and M. Parsani. Fully discrete explicit locally entropy-stable schemes for the +compressible Euler and Navier–Stokes equations. Comput. Math. Appl., 80(5):1343–1359, 2020. +[49] E. Romenski, D. Drikakis, and E.F. Toro. Conservative models and numerical methods for com- +pressible two-phase flow. J. Sci. Comput., 42:68–95, 2010. +[50] E. Romenski, I. Peshkov, M. Dumbser, and F. Fambri. A new continuum model for general relativistic +viscous heat-conducting media. Philos. Trans. R. Soc. A, 378:20190175, 2020. +[51] E.I. Romenski. Hyperbolic systems of thermodynamically compatible conservation laws in continuum +mechanics. Math. Comput. Modell., 28(10):115–130, 1998. +[52] C.W. Shu and S. Osher. +Efficient implementation of essentially non-oscillatory shock capturing +schemes. J. Comput. Phys., 77:439–471, 1988. +[53] P.K. Subbareddy and G.V. Candler. A fully discrete, kinetic energy consistent finite–volume scheme +for compressible flows. Journal of Computational Physics, 228:1347–1364, 2009. +[54] E. Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws I. +Math. Comput., 49:91–103, 1987. +[55] M. Tavelli and M. Dumbser. +A pressure-based semi-implicit space-time discontinuous Galerkin +method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equa- +tions at all Mach numbers. J. Comput. Phys., 341:341–376, 2017. +[56] M. Tavelli, E. Romenski, S. Chiocchetti, A. Gabriel, and M. Dumbser. Space-time adaptive ADER +discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure. J. Comput. Phys., +422:109758, 2020. +[57] E.F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 2009. +29 + diff --git a/kNE_T4oBgHgl3EQf5Rz9/content/tmp_files/load_file.txt b/kNE_T4oBgHgl3EQf5Rz9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f95e20840f1a090615d98e7e84f13d4441a1a77f --- /dev/null +++ b/kNE_T4oBgHgl3EQf5Rz9/content/tmp_files/load_file.txt @@ -0,0 +1,1870 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf,len=1869 +page_content='On thermodynamically compatible finite volume schemes for continuum mechanics S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser2, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov3, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski4 (1) Department of Applied Mathematics I, Universidade de Vigo, Campus As Lagoas, 36310 Vigo, Spain (2,3) Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy (4) Sobolev Institute of Mathematics, 4 Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Koptyug Avenue, 630090 Novosibirsk, Russia Abstract In this paper we present a new family of semi-discrete and fully-discrete finite volume schemes for overdetermined, hyperbolic and thermodynamically compatible PDE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In the following we will denote these methods as HTC schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In particular, we consider the Euler equations of compressible gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model of continuum me- chanics, which, at the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first order hyperbolic model of continuum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The main novelty of the schemes presented in this paper lies in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual total energy conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Go- dunov’s seminal paper An interesting class of quasilinear systems of 1961 exactly at the discrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' All other terms in the governing equations of the more general GPR model, including non-conservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Keywords: thermodynamically compatible finite volume schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' semi-discrete and fully-discrete Go- dunov formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' vanishing viscosity limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' entropy inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' hyperbolic thermodynamically compatible PDE systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' overdetermined hyperbolic PDE systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' unified GPR model for solid mechanics and fluid mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 1 Introduction In his groundbreaking work An interesting class of quasilinear systems [29] published 60 years ago in 1961 Godunov discovered the connection between symmetric hyperbolicity in the sense of Friedrichs [24] and thermodynamic compatibility, 10 years before the work of Friedrichs & Lax on the same subject [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In subsequent work by Godunov & Romenski and collaborators, the theory of symmetric hyperbolic and thermodynamic compatible (SHTC) systems was extended to a wide class of mathematical models in con- tinuum physics, ranging from the magnetohydrodynamics (MHD) equations over nonlinear hyperelasticity to compressible multi-phase flows and relativistic gasdynamics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [30, 31, 32, 33, 51, 49, 28, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' All SHTC systems can be rigorously derived from an underlying variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A connection between SHTC systems and Hamiltonian mechanics was established in [45], emphasizing a peculiar role of the 1saray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='busto@uvigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='es 2michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='dumbser@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='it 3ilya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='peshkov@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='it 4evrom@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='nsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ru 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='08358v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='NA] 19 Jan 2023 energy potential (Hamiltonian), while an extension to continuum mechanics with torsion was provided in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Notwithstanding the mathematical elegance of the SHTC framework, to the best knowledge of the authors it was up to now never directly carried over to the discrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Most existing papers on thermodynamically compatible schemes are based on the ideas of the seminal work of Tadmor [54], in which a discrete compatibility with the entropy equation is sought, rather than a discrete compatibility with the total energy conservation, as suggested by the SHTC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A fully discrete entropy-stable scheme has been recently forwarded in [48], while the convergence of entropy-stable schemes was proven in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For high order entropy-compatible schemes the reader is referred to [26, 18, 38, 19, 35] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In [23, 2] entropy compatible schemes were applied to non-conservative hyperbolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Last, but not least, we also would like to mention the general framework for the construction of numerical methods that satisfy additional extra conservation laws recently introduced by Abgrall in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A first attempt to achieve discrete energy conservation as a consequence of all other equations was made in [10] for a novel hyperbolic model of unsteady turbulent shallow water flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For compatible schemes in the context of Lagrangian hydrodynamics, where total energy conservation is obtained as a consequence of the discrete mass, momentum and internal energy equations, see the interesting papers [15, 3], while a fully-discrete compatible kinetic energy preserving scheme was forwarded in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' However, all aforementioned schemes address only the compressible Euler equations and not the full GPR model of continuum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The main contribution of this paper is thus a new thermodynamically compatible finite volume scheme for the GPR model of continuum mechanics [51, 46, 20] in which the discrete energy conservation law is obtained as a consequence of a compatible discretization of all the other equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' To the best knowledge of the authors, this is the first time that such a provably thermodynamically compatible scheme is proposed for the PDE system (1), which is able to describe solid mechanics and fluid mechanics at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We stress that the main objective of this paper is not to introduce a better or more efficient numerical scheme compared to existing methods, but to introduce a radically new concept: direct discretization of the entropy inequality in order to obtain the discrete total energy conservation law as a consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We also would like to clearly indicate the three main shortcomings of the new method introduced in this paper: i) the numerical fluxes are only known implicitly via path integrals of the physical flux in phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' however, also other numerical methods are based on path integrals, like the Osher-Solomon flux [40], the entropy-consistent scheme of Tadmor [54] and the family of path-conservative schemes of Castro and Par´es [16, 41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ii) currently, in our new framework, a numerical scheme that provably satisfies total energy conserva- tion at the fully-discrete level can only be achieved at the aid of a special implicit time integrator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' iii) in the case of the semi-discrete scheme, total energy conservation is in general lost at the fully- discrete level once a standard, nonsymplectic Runge-Kutta time discretization is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In Section 2 we present the unified first order hyperbolic model of continuum mechanics (GPR model) under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In Sections 3 and 5 the construction of thermodynamically compatible semi-discrete and fully-discrete finite volume schemes is explained for the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In Section 4 an extension to the general multi-dimensional case is presented, together with a proof of nonlinear stability in the energy norm and a proof of the entropy inequality satisfied by the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numerical results are shown in Section 6 for the fluid and the solid limits of the governing PDE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The paper closes with some concluding remarks and an outlook to future work in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 2 Mathematical model and its structure We consider the following first order hyperbolic model of continuum mechanics regularized with vanishing viscosity terms and which goes back to the work of Godunov [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov & Romenski [31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 33] and 2 Peshkov & Romenski,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' see [46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 20]: ∂ρ ∂t + ∂(ρvk) ∂xk − ∂ ∂xm � ϵ ∂ρ ∂xm � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1a) ∂ρvi ∂t + ∂ (ρvivk + p δik + σik + ωik) ∂xk − ∂ ∂xm � ϵ∂ρvi ∂xm � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1b) ∂ρS ∂t + ∂ (ρSvk + βk) ∂xk − ∂ ∂xm � ϵ ∂ρS ∂xm � = Π + αikαik θ1(τ1)T + βiβi θ2(τ2)T ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1c) ∂Aik ∂t + ∂(Aimvm) ∂xk + vm �∂Aik ∂xm − ∂Aim ∂xk � − ∂ ∂xm � ϵ∂Aik ∂xm � = − αik θ1(τ1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1d) ∂Jk ∂t + ∂ (Jmvm + T) ∂xk + vm � ∂Jk ∂xm − ∂Jm ∂xk � − ∂ ∂xm � ϵ ∂Jk ∂xm � = − βk θ2(τ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1e) ∂E ∂t + ∂ (vk (E1+E2+E3 + E4) + vi (p δik+σik + ωik)+hk) ∂xk − ∂ ∂xm � ϵ ∂E ∂xm � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1f) In the overdetermined system above q = {qi} = (ρ, ρvi, ρS, Aik, Jk)T denotes the state vector, the total energy potential is E = ρE = E1 + E2 + E3 + E4 with Ei = ρEi, ϵ > 0 is a vanishing viscosity and the nonnegative entropy production term due to the viscous terms is given by Π = ϵ T ∂xmqi ∂2 qiqjE ∂xmqj ≥ 0, (2) since ϵ > 0 and we assume that the temperature T > 0 and that the Hessian of the total energy potential is at least positive semi-definite, Hij := ∂2 qiqjE ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Throughout this paper, we use the notations ∂p = ∂/∂p and ∂2 pq = ∂2/(∂p∂q) for the first and second partial derivatives w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' generic coordinates or quantities p and q, which may also be vectors or components of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Furthermore, we make use of the Einstein summation convention over repeated indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Last but not least, in some occasions we also use bold face symbols in order to denote vectors and matrices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' q = {qi} and A = {Aik}, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In the above model the four contributions to the total energy density are E1 = ργ γ − 1eS/cv, E2 = 1 2ρvivi, E3 = 1 4ρc2 s ˚ Gij ˚ Gij, E4 = 1 2c2 hρJiJi, (3) with the metric tensor G components and its trace-free part ˚ G given by Gik = AjiAjk, and ˚ Gik = Gik − 1 3 Gmmδik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The vector of thermodynamic dual variables reads p = ∂qE = {pi} = (r, vi, T, αik, βk)T with r = ∂ρE, vi = ∂ρviE, T = ∂ρSE, αik = ∂AikE, βk = ∂JkE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (4) The pressure is defined as p = ρ ∂ρE + ρvi ∂ρviE + ρS ∂ρSE − E = ρ2∂ρE, the stress tensors due to shear stress and thermal stress are, respectively, σik = Aji∂AjkE = Ajiαjk = ρc2 sGij ˚ Gjk, ωik = Ji∂JkE = Jiβk = ρc2 hJiJk, (5) while the heat flux vector is given by hk = ∂ρSE ∂JkE = Tβk = ρc2 hTJk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (6) Note that for our convenience, we use the opposite sign in the definition of the stress tensor compared to the generally accepted notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Furthermore, θ1(τ1) > 0 and θ2(τ2) > 0 are two algebraic functions of the state vector q and the positive relaxation times τ1 > 0 and τ2 > 0: θ1 = 1 3ρz1τ1 c2 s |A|− 5 3 , θ2 = ρz2τ2 c2 h, z1 = ρ0 ρ , z2 = ρ0T0 ρ T , (7) 3 with ρ0 and T0 being some reference density and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' It is easy to check that (1f) is a consequence of (1a)-(1e), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (1f) = r · (1a) + vi · (1b) + T · (1c) + αik · (1d) + βk · (1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (8) In [20] a formal asymptotic analysis of the model (1a)-(1f) was carried out, revealing that in the stiff limit the stress tensor σik and the heat flux hk tend to σik = −1 6ρ0c2 sτ1 � ∂kvi + ∂ivk − 2 3 (∂mvm) δik � , hk = −ρ0T0c2 hτ2∂kT, (9) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' when the relaxation times τ1, τ2 → 0, the Navier-Stokes-Fourier equations are retrieved with effective shear viscosity µ = 1 6ρ0c2 sτ1 and heat conductivity κ = ρ0T0c2 hτ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 3 Thermodynamically compatible semi-discrete finite volume scheme for the complete model in one space dimension In this section, we derive the thermodynamically compatible semi-discrete finite volume scheme for model (1) in one space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' To this end, we start analysing the black terms on the system which enter into the original Godunov formalism [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Once compatibility of these terms is established for the Euler subsystem, we can include dissipative terms which require the consideration of the non-negative entropy production term, coloured in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The third step is the study of the red terms of (1b)-(1f) corresponding to the discretization of the distortion field and the thermal impulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Finally, also the relaxation terms, in green, are addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Throughout the discretization, we will employ lower case subscripts, i, j, k, for tensor indices while lower case superscripts, ℓ, refer to the spatial discretization index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Accordingly, we denote by Ωℓ = [xℓ− 1 2 , xℓ+ 1 2 ] a spatial control volume in one space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The Godunov form [29] of the Euler subsystem (black terms in (1)) reads (∂pL)t + ∂x (∂p(v1L)) = 0, (10) q = ∂pL, p = ∂qE, f = ∂p(v1L), F = p · f − v1L, (11) with the generating potential L = p·q−E, which is the Legendre transform of the total energy potential E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The semi-discrete finite volume discretization of (10) reads d dtqℓ = −f ℓ+ 1 2 − f ℓ− 1 2 ∆x = − � f ℓ+ 1 2 − f ℓ� − � f ℓ− 1 2 − f ℓ� ∆x (12) with f ℓ = f(qℓ) and f(q) = (ρv1, ρviv1 + pδi1, ρSv1, 0, 0)T , containing only the fluxes of the Euler subsystem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' the black terms in (1), and F being the corresponding energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Just like on the continuous level, our first objective is to get a discrete form of the energy conservation as consequence of the discrete form of equations (1a)-(1c), see (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Therefore we proceed alike we would do on the continuous level and we perform the dot product of the discrete dual variables, pℓ = ∂qE(qℓ), with the discrete equations, obtaining pℓ · d dtqℓ = d dtEℓ = −pℓ · (f ℓ+ 1 2 − f ℓ) + (f ℓ − f ℓ− 1 2 ) ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (13) We now introduce the fluctuations D ℓ+ 1 2 ,− E = pℓ · (f ℓ+ 1 2 − f ℓ), D ℓ− 1 2 ,+ E = pℓ · (f ℓ − f ℓ− 1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In order to achieve a flux conservative expression for the discrete formulation of (1f), we must be able to rewrite the fluctuations related to an interface as a flux difference D ℓ+ 1 2 ,− E + D ℓ+ 1 2 ,+ E = F ℓ+1 − F ℓ, (14) 4 with F ℓ a consistent approximation of the total energy flux F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' This condition (14) is mandatory in order to guarantee total energy conservation for vanishing energy flux at the boundary via the telescopic-sum property � ℓ ∆x d dtEℓ = − � ℓ � D ℓ+ 1 2 ,− E + D ℓ− 1 2 ,+ E � = − � ℓ � F ℓ+1 − F ℓ� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (15) Substitution of the fluctuations in the former definition gives pℓ · (f ℓ+ 1 2 − f ℓ) + pℓ+1 · (f ℓ+1 − f ℓ+ 1 2 ) = −f ℓ+ 1 2 · � pℓ+1 − pℓ� + pℓ+1 · f ℓ+1 − pℓ · f ℓ = F ℓ+1 − F ℓ (16) and, taking into account definition (11) for f and F, we conclude −∂p(v1L)ℓ+ 1 2 · � pℓ+1 − pℓ� + pℓ+1 · f ℓ+1 − pℓ · f ℓ = pℓ+1 · f ℓ+1 − (v1L)ℓ+1 − pℓ · f ℓ + (v1L)ℓ, (17) where F ℓ = pℓ ·f ℓ −(v1L)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Accordingly, the numerical flux f ℓ+ 1 2 = ∂p(v1L)ℓ+ 1 2 must verify the Roe-type property, f ℓ+ 1 2 · � pℓ+1 − pℓ� = ∂p(v1L)ℓ+ 1 2 · � pℓ+1 − pℓ� = (v1L)ℓ+1 − (v1L)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (18) Next, we make use of the key idea on which path conservative schemes are based, see [16, 41], and construct a path integral in phase-space by recalling the fundamental theorem of calculus (v1L)ℓ+1 − (v1L)ℓ = pℓ+1 � pℓ ∂p(v1L) · dp = 1 � 0 ∂p(v1L) · ∂ψ ∂s ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (19) Note that a similar methodology has already been successfully used in the construction of entropy- conservative fluxes [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Since the path, ψ(s), s ∈ [0, 1], can be freely chosen, we can select any parametrization convenient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As a path connecting pℓ and pℓ+1 we choose the sim- ple straight line segment path in p variables: ψ(s) = pℓ + s � pℓ+1 − pℓ� , ∂ψ ∂s = pℓ+1 − pℓ, 0 ≤ s ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (20) So (19) together with (20) leads to (v1L)ℓ+1 − (v1L)ℓ = � � 1 � 0 f(ψ(s))ds � � · � pℓ+1 − pℓ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (21) Therefore, the corresponding thermodynamically compatible numerical flux, f ℓ+ 1 2 p = 1 � 0 f(ψ(s))ds = � f ℓ+ 1 2 ρ , f ℓ+ 1 2 ρv , f ℓ+ 1 2 ρS , 0, 0 �T , (22) guarantees (18) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The subscript p refers to the segment path in p variables (p-scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For a different choice of path in terms of q variables (q-scheme) the reader is referred to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' From the numerical point of view all path integrals appearing in this paper are approximated using a sufficiently accurate numerical quadrature rule, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' If not stated otherwise, throughout this paper, we use a standard Gauss-Legendre quadrature rule with nGP = 3 points in order to compute the path integral appearing in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For a quantitative study of the influence of the quadrature rule on total energy conservation, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 Compatible scheme with dissipation terms So far we have presented a compatible discretization for the black terms in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' To derive a dissipative scheme, we still need to include a compatible numerical dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Let us enlarge (12) with an additional dissipative flux and corresponding production terms: d dtqℓ + f ℓ+ 1 2 − f ℓ− 1 2 ∆x = gℓ+ 1 2 − gℓ− 1 2 ∆x + Pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (23) We first focus on the numerical flux gℓ+ 1 2 = ϵℓ+ 1 2 ∆qℓ+ 1 2 ∆x , ∆qℓ+ 1 2 = qℓ+1 − qℓ, (24) whose scalar numerical dissipation is either chosen to be constant, ϵℓ+ 1 2 = ϵ, or taken of the form ϵℓ+ 1 2 = 1 2 � 1 − φℓ+ 1 2 � ∆x s ℓ+ 1 2 max ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (25) In the former expression, we have denoted by s ℓ+ 1 2 max the maximum signal speed at the cell interface and introduced φℓ+ 1 2 which allows the use of a flux limiter, hence a reduction of the numerical dissipation in smooth regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In particular, we consider the minbee flux limiter given by φℓ+ 1 2 = min � φ ℓ+ 1 2 − , φ ℓ+ 1 2 + � , with φ ℓ+ 1 2 ± = max � 0, min � 1, h ℓ+ 1 2 ± �� , (26) where h ℓ+ 1 2 − = Eℓ − Eℓ−1 Eℓ+1 − Eℓ , and h ℓ+ 1 2 + = Eℓ+2 − Eℓ+1 Eℓ+1 − Eℓ (27) are the ratios of the total energy potential slopes, see the SLIC scheme presented in [57] for further details on flux limiting strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Note that an alternative approach to the use of flux limiters is the definition of a fixed numerical dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The dot product of pℓ by (23) yields dEℓ dt + 1 ∆x � F ℓ+ 1 2 − F ℓ− 1 2 � = 1 ∆xpℓ · � gℓ+ 1 2 − gℓ− 1 2 � + pℓ · Pℓ, (28) where the left hand side has already been studied in the Godunov formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We therefore focus on the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='right hand side of the former equation obtaining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pℓ · Pℓ + pℓ · gℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 − gℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='= pℓ · Pℓ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ · gℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ+1 · gℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ · gℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ+1 · gℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ · gℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ−1 · gℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ · gℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2pℓ−1 · gℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='= pℓ · Pℓ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pℓ+1 + pℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆qℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pℓ + pℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆qℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pℓ+1 − pℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆qℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pℓ − pℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆qℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (29) Besides, applying path integration yields qℓ+1 � qℓ p · dq = qℓ+1 � qℓ ∂qE · dq = Eℓ+1 − Eℓ = ∆Eℓ+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (30) 6 So 1 2(pℓ+1 + pℓ) · ∆qℓ+ 1 2 can be seen as an approximation of ∆Eℓ+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As a consequence of (29) and (30), the energy flux including convective and diffusive terms is F ℓ+ 1 2 d = F ℓ+ 1 2 − 1 2(pℓ+1 + pℓ) · ϵℓ+ 1 2 ∆qℓ+ 1 2 ∆x ≈ F ℓ+ 1 2 − ϵℓ+ 1 2 ∆Eℓ+ 1 2 ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (31) To transform the jumps in p variables into jumps in q variables, we need to introduce a Roe-type matrix ∂2 qq ˜Eℓ+ 1 2 verifying the Roe property ∂2 qq ˜Eℓ+ 1 2 · (qℓ+1 − qℓ) = pℓ+1 − pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (32) For its calculation, we introduce another segment path ˜ψ, written in terms of q ˜ψ(s) = qℓ + s � qℓ+1 − qℓ� , 0 ≤ s ≤ 1, (33) allowing to compute the sought Roe matrix as ˜H ℓ+ 1 2 = ∂2 qq ˜Eℓ+ 1 2 = 1 � 0 ∂2 qqE � ˜ψ(s) � ds =: � ∂2 pp ˜Lℓ+ 1 2 �−1 , (34) which satisfies (32) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Substituting the obtained flux in (28) and taking into account (32) gives d dtEℓ + F ℓ+ 1 2 d − F ℓ− 1 2 d ∆x = pℓ · Pℓ (35) −1 2ϵℓ+ 1 2 qℓ+1 − qℓ ∆x ˜H ℓ+ 1 2 qℓ+1 − qℓ ∆x − 1 2ϵℓ− 1 2 qℓ − qℓ−1 ∆x ˜H ℓ− 1 2 qℓ − qℓ−1 ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Thus, defining the production term Pℓ = (0, 0, Πℓ, 0, 0)T pℓ · Pℓ = T ℓΠℓ = 1 2ϵℓ+ 1 2 ∆qℓ+ 1 2 ∆x ˜H ℓ+ 1 2 ∆qℓ+ 1 2 ∆x + 1 2ϵℓ− 1 2 ∆qℓ− 1 2 ∆x ˜H ℓ− 1 2 ∆qℓ− 1 2 ∆x , (36) we obtain the sought semi-discrete total energy conservation law d dtEℓ + F ℓ+ 1 2 d − F ℓ− 1 2 d ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (37) Note that, as expected, the above definition provides a zero production term for all equations but for (1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The final compatible flux including convective and diffusive terms reads f ℓ+ 1 2 p,d = 1 � 0 f(ψ(s))ds − ϵℓ+ 1 2 ∆x � qℓ+1 − qℓ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (38) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 Compatible discretization of the terms related to the distortion field The momentum flux in (1b) gathers four terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The first two, in black, belong to the Euler subsystem and have already been studied in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The third term, σik, is related to the distortion field and thus its compatibility must be analysed together with the distortion transport equations, (1d), and the terms E3 and σik in the energy equation (1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Let us consider the red terms in (1d) (except for the convective term v1∂xAik), ∂(Aimvm) ∂x − vm ∂Aim ∂x = Aim ∂vm ∂x , (39) 7 and the following chosen discretization ∆xAim∂x∂vm ≈ A ℓ+ 1 2 im � vℓ+1 m − vℓ m � with A ℓ+ 1 2 im = 1 2 � Aℓ+1 im + Aℓ im � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (40) Multiplication of equations (1b), (1d) by ∂ρviE = vi, ∂AikE = αik and assuming a compatible discretiza- tion with the term ∂x (viσik) in (1f) leads to vℓ i � σ ℓ+ 1 2 ik − σℓ ik � + vℓ+1 i � σℓ+1 ik − σ ℓ+ 1 2 ik � + αℓ ik 1 2A ℓ+ 1 2 im � vℓ+1 m − vℓ m � +αℓ+1 ik 1 2A ℓ+ 1 2 im � vℓ+1 m − vℓ m � = vℓ+1 i σℓ+1 ik − vℓ iσℓ ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (41) We therefore obtain the following discretization for σ ℓ+ 1 2 ik : σ ℓ+ 1 2 ik = 1 2 � αℓ+1 mk + αℓ mk � A ℓ+ 1 2 mi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (42) We now focus on the remaining flux term v1∂xAik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We multiply the continuity equation (1a) by the dual variable ∂ρE3 = E3 and (1d) by ∂AikE = αik and impose the compatibility condition with the energy conservation equation yielding Eℓ 3 � ρv ℓ+ 1 2 1 − ρvℓ 1 � + Eℓ+1 3 � ρvℓ+1 1 − ρv ℓ+ 1 2 1 � + αℓ ik 1 2 ˜v ℓ+ 1 2 A 1 � Aℓ+1 ik − Aℓ ik � +αℓ+1 ik 1 2 ˜v ℓ+ 1 2 A 1 � Aℓ+1 ik − Aℓ ik � = ρvℓ+1 1 Eℓ+1 3 − ρvℓ 1Eℓ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (43) where the approximation of the averaged velocity ˜v ℓ+ 1 2 A 1 still needs to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Collecting terms, we get −ρv ℓ+ 1 2 1 � Eℓ+1 3 − Eℓ 3 � + 1 2 ˜v ℓ+ 1 2 A 1 � αℓ+1 ik + αℓ ik � � Aℓ+1 ik − Aℓ ik � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (44) Hence, the average velocity must be discretised as ˜v ℓ+ 1 2 A 1 = ρv ℓ+ 1 2 1 � Eℓ+1 3 − Eℓ 3 � 1 2 � αℓ+1 ik + αℓ ik � � Aℓ+1 ik − Aℓ ik � (45) if the denominator in (45) is non-zero, otherwise we set ˜v ℓ+ 1 2 A 1 = 1 2 � vℓ+1 1 + vℓ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Finally, if Eℓ+1 3 − Eℓ 3 = 0, from (44), we get ˜v ℓ+ 1 2 A 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 Compatible discretization of the terms related to the thermal impulse Similarly to what has been done for the distortion field, in this section we derive the discretization of the red terms in (1b), (1e), (1f) related to the heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' First, we focus on terms ∂xωi1 in (1b), Jm∂xvm = ∂x(Jmvm) − vm∂xJm in (1e) and ∂x (viωi1) in (1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Multiplying the momentum equation by ∂ρviE = vi, the thermal impulse equation by ∂J1E = β1 and requiring compatibility with the energy equation we get vℓ i � ω ℓ+ 1 2 i1 − ωℓ i1 � + vℓ+1 i � ωℓ+1 i1 − ω ℓ+ 1 2 i1 � + βℓ 1 1 2J ℓ+ 1 2 m � vℓ+1 m − vℓ m � +βℓ+1 1 1 2J ℓ+ 1 2 m � vℓ+1 m − vℓ m � = vℓ+1 i ωℓ+1 i1 − vℓ iωℓ i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (46) Relabeling the repeated index m and defining J ℓ+ 1 2 i = 1 2 � Jℓ+1 i + Jℓ i � , yields −ω ℓ+ 1 2 i1 � vℓ+1 i − vℓ i � + 1 2 � βℓ+1 1 + βℓ 1 � J ℓ+ 1 2 i � vℓ+1 i − vℓ i � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (47) 8 Thus choosing ω ℓ+ 1 2 i1 = 1 2 � βℓ+1 1 + βℓ 1 � J ℓ+ 1 2 i gives the sought compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Next, we need to compute the discretization related to the term v1∂xJk in (1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Multiplication of (1a) by ∂ρE4 = E4 and addition of (1e) multiplied by ∂JkE = βk yields Eℓ 4 � ρv ℓ+ 1 2 1 − ρvℓ 1 � + Eℓ+1 4 � ρvℓ+1 1 − ρv ℓ+ 1 2 1 � + 1 2 ˜v ℓ+ 1 2 J 1 βℓ k � Jℓ+1 k − Jℓ k � +1 2 ˜v ℓ+ 1 2 J 1 βℓ+1 k � Jℓ+1 k − Jℓ k � = ρvℓ+1 1 Eℓ+1 4 − ρvℓ 1Eℓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (48) Hence, ˜v ℓ+ 1 2 J 1 = ρv ℓ+ 1 2 1 � Eℓ+1 4 − Eℓ 4 � 1 2 � βℓ+1 k + βℓ k � � Jℓ+1 k − Jℓ k � (49) is the compatible discretization for the advection speed related to the thermal impulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Analogous to the previous section, for null denominator and Eℓ+1 4 − Eℓ 4 ̸= 0 we define ˜v ℓ+ 1 2 J 1 as the arithmetic average of the velocity in the two related cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' It now just remains to establish the discrete compatibility between the term βk in equation (1c), T in equation (1e) and hk in equation (1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Let us assume we have the following given discretization for the gradient of T: ∆x∂xT ≈ 1 2 � T ℓ+1 − T ℓ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (50) Then, multiplication of the thermal impulse equation (1e) by ∂JkE = βk and the entropy relation by ∂ρSE = T gives T ℓ � β ℓ+ 1 2 k − βℓ k � + T ℓ+1 � βℓ+1 k − β ℓ+ 1 2 k � + βℓ k 1 2 � T ℓ+1 − T ℓ� +βℓ+1 k 1 2 � T ℓ+1 − T ℓ� = βℓ+1 k T ℓ+1 − βℓ kT ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (51) Hence, by simply defining β ℓ+ 1 2 k = 1 2 � βℓ+1 k + βℓ k � we get a compatible discretization of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 Compatible discretization of relaxation terms Finally, it is easy to see that the relaxation terms, in green in (1c)-(1e), cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Multiplication of ∂ρSE = T, ∂AikE = αik and ∂JkE = βk by the green terms in (1c)-(1e), respectively, and adding the result gives T αikαik θ1(τ1)T + T βiβi θ2(τ2)T − αik αik θ1(τ1) − βk βk θ2(τ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (52) Thus, the compatibility is proven by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 4 Thermodynamically compatible semi-discrete finite volume scheme for the complete model in two space dimensions The derivation of the thermodynamically compatible semi-discrete finite volume scheme for the complete model in two space dimensions can be done following the steps described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Here we summarize the final scheme and provide the mathematical proofs of the marginal nonlinear stability in the energy norm and of the semi-discrete cell entropy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Let us consider the spatial control volume Ωℓ with circumcenter xℓ, one of its neighbors Ωr and the common edge ∂Ωℓr, n = (n1, n2)T being the outward unit normal vector to the face ∂Ωℓr and Nℓ being the set of neighbors of cell Ωℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The final semi-discrete finite volume scheme reads ∂ρℓ ∂t = − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ + 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (53a) 9 ∂(ρvℓ i) ∂t = − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρvi − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� σℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' − ik nk − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� ωℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' − ik nk+ 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr ρvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (53b) ∂(ρSℓ) ∂t = − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρS − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� � βℓr k − βℓ k � nk + 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr ρS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n+ 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Πℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n + αℓ ikαℓ ik θℓ 1 (τ1) T ℓ + βℓ i βℓ i θℓ 2 (τ2) T ℓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (53c) ∂Aℓ ik ∂t =− 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� 1 2Aℓr im � vr m − vℓ m � nk − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� 1 2 ˜uℓr A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Ar ik − Aℓ ik � + 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr Aik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n− αℓ ik θℓ 1 (τ1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (53d) ∂Jℓ k ∂t =− 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� 1 2Jℓr i � vr m − vℓ m � nk− 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� 1 2T ℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='−nk − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� 1 2 ˜uℓr J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Jr k − Jℓ k � + 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr Jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n− βℓ i θℓ 2 (τ2) (53e) with Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− q = � f ℓr q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k − f ℓ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k � nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (54) gℓr q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n = ϵℓr qr − qℓ δℓr = ϵℓr ∆qℓr δℓr ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' δℓr = ��xr − xℓ�� = ∆xn1 + ∆yn2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (55) σℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− jk = σℓr jk − σℓ jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' σℓr jk = 1 2Aℓr ij � αℓ ik + αr ik � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (56) ωℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− jk = ωℓr jk − ωℓ jk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ωℓr jk = 1 2 � βℓ k + βr k � Jℓr i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (57) Πℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n = 1 2ϵℓr ∆qℓr T ℓ ∂2 qqEℓr ∆qℓr δℓr ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' T ℓ = � ρℓ�γ−1 (γ − 1) cv e Sℓ cv ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (58) Aℓr im = 1 2 � Aℓ im + Ar im � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ˜uℓr A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n = ˜vℓr A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' jnj = f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' jnj � Er 3 − Eℓ 3 � 1 2 � αℓ ik + αr ik � � Ar ik − Aℓ ik �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (59) Jℓr i = 1 2 � Jℓ i + Jr i � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ˜uℓr J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n =˜vℓr J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' jnj = f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' jnj � Er 4 − Eℓ 4 � 1 2 � βℓ k + βr k � � Jr k − Jℓ k �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' T ℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− =T r − T ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (60) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The thermodynamically compatible semi-discrete finite volume scheme (53) admits the semi-discrete energy conservation law ∂Eℓ ∂t = − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Dℓr,− E + 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr E, n (61) with Dℓr,− E + Dℓr,+ E = Dℓr,− E + Drℓ,− E = F r − F ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (62) Assuming that the jumps on the boundary vanish, the scheme is nonlinearly marginally stable in the energy norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' the scheme satisfies the identity � Ω ∂Eℓ ∂t dV = � ℓ |Ωℓ|∂Eℓ ∂t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (63) 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We start considering the contributions of the dot product of vector pℓ = ∂qEℓ = � ∂ρEℓ, ∂ρviEℓ, ∂ρSEℓ, ∂AikEℓ, ∂JkEℓ�T with the time derivative terms in (53): ∂ρEℓ ∂ρℓ ∂t + ∂ρviEℓ ∂ρvℓ i ∂t + ∂ρSEℓ ∂ρSℓ ∂t + ∂AikEℓ ∂Aℓ ik ∂t + ∂JkEℓ ∂Jℓ k ∂t = ∂qEℓ ∂qℓ ∂t = ∂Eℓ ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (64) We now define the fluctuations associated to the total energy equation as Dℓr,− E = ∂ρEℓ 1Dℓr,− ρ + ∂ρEℓ 2Dℓr,− ρ +∂ρEℓ 3Dℓr,− ρ +∂ρEℓ 4Dℓr,− ρ + ∂ρviEℓDℓr,− ρvi +∂ρviEℓ � σℓr,− ik nk + ωℓr,− ik nk � + ∂ρSEℓDℓr,− ρS +∂ρSEℓ � βℓr k − βℓ k � nk +∂AikEℓ 1 2δℓr Aℓr im � vr m − vℓ m � nk+∂AikEℓ 1 2 ˜uℓr A, n � Ar ik − Aℓ ik � +∂JkEℓ 1 2δℓr Jℓr i � vr m − vℓ m � nk+∂JkEℓ 1 2 ˜uℓr J, n � Jr k − Jℓ k � + ∂JkEℓ 1 2T ℓr,−nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (65) On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' from (55) and applying relations analogous to the ones introduced in (29) and (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' we have 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� � pℓ · Pℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n + pℓ · gℓr n � = � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � pℓ · Pℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n + pℓ · ϵℓr ∆qℓr δℓr � = � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � pℓ· Pℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n + 1 2pℓ· ϵℓr ∆qℓr δℓr + 1 2pr· ϵℓr ∆qℓr δℓr + 1 2pℓ· ϵℓr ∆qℓr δℓr − 1 2pr· ϵℓr ∆qℓr δℓr � = � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � pℓ · Pℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n + 1 2 � pℓ + pr� ϵℓr ∆qℓr δℓr − 1 2 � pr − pℓ� ϵℓr ∆qℓr δℓr � = � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � pℓ · Pℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− n + ϵℓr ∆Eℓr δℓr − 1 2ϵℓr ∆qℓr δℓr ∂2 qqEℓr∆qℓr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (66) Substitution of Pℓr,− n = � 0, 0, Πℓr,− n , 0, 0 � combined with (58) yields 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� � pℓ · gℓr n + pℓ · Pℓr,− n � = 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� ϵℓr ∆Eℓr δℓr = 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr E, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (67) Finally, taking into account the dot product of pℓ by the diffusion terms in (53) and applying (67), we get pℓ · 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr n + ∂ρSEℓ 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Πℓr n (68) = � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � pℓ · ϵℓr ∆qℓr δℓr + ∂ρSEℓ 1 4ϵℓr ∆qℓr T ℓ ∂2 qqEℓr ∆qℓr δℓr � = 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr E, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' From (64), (65), (68) and noting that the dot product of ∂qEℓ by the green terms in (53a)-(53e) is zero, we conclude ∂Eℓ ∂t = − 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Dℓr,− E + 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� gℓr E, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The second part of the proof concerns marginal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Integration of equation (61) over the computational domain Ω gives � Ω ∂Eℓ ∂t dV = � ℓ ��Ωℓ�� ∂Eℓ ∂t = − � ℓ � r∈Nℓ ��∂Ωℓr�� Dℓr,− E + � ℓ � r∈Nℓ ��∂Ωℓr�� gℓr E, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 11 Assuming that the solution on the boundaries of the domain tends to a constant value, the jumps on q become zero at ∂Ω and fluctuations and dissipative terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Besides, the remaining dissipative terms can be seen as a telescopic sum which cancels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Reordering of the first summation in the right hand side of the former equation to cluster the contributions at each face we obtain � Ω ∂Eℓ ∂t dV = � ℓ ��Ωℓ�� ∂Eℓ ∂t = − � ℓr ��∂Ωℓr�� � Dℓr,− E + Drℓ,− E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Consequently, marginal stability is proven given that the contributions of fluctuations in the interior cell boundaries cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Let us focus on a face ∂Ωℓr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We start analysing the terms corresponding with the Godunov formalism (black terms) in (65): ∂ρEℓ 1Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ + ∂ρEℓ 2Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ + ∂ρviEℓDℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρvi + ∂ρSEℓDℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρS +∂ρEr 1Drℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ + ∂ρEr 2Drℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ + ∂ρviErDrℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρvi + ∂ρSErDrℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρS = − � ∂ρEr 1 − ∂ρEℓ 1 � f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − � ∂ρEr 2 − ∂ρEℓ 2 � f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − � ∂ρviEr − ∂ρviEℓ� f ℓr ρvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − � ∂ρSEr − ∂ρSEℓ� f ℓr ρS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − ∂ρEℓ 1f ℓ ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − ∂ρEℓ 2f ℓ ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − ∂ρviEℓf ℓ ρvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk −∂ρSEℓf ℓ ρS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk + ∂ρEr 1f r ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk + ∂ρEr 2f r ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk + ∂ρviErf r ρvi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk + ∂ρSErf r ρS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk = − � pr − pℓ� f ℓr q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk + pr · f r q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk − pℓ · f ℓ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' knk = � pr · f r q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k − (vkL)r� nk − � pℓ · f ℓ q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k − (vkL)ℓ� nk = F r G − F ℓ G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (69) with FG standing for the black terms in the energy flux in (1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Regarding the red terms in (65),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' we have ∂ρEℓ 3Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ +∂ρEℓ 4Dℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ +∂ρviEℓ � σℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ik nk + ωℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ik nk � +∂ρSEℓ � βℓr k − βℓ k � nk +∂AikEℓ 1 2δℓr Aℓr im � vr m − vℓ m � nk+∂AikEℓ 1 2 ˜uℓr A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Ar ik − Aℓ ik � +∂JkEℓ 1 2δℓr Jℓr i � vr m − vℓ m � nk+∂JkEℓ 1 2 ˜uℓr J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Jr k − Jℓ k � +∂JkEℓ 1 2T ℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='−nk +∂ρEr 3Drℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ +∂ρEr 4Drℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ρ −∂ρviEr � σrℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ik nk + ωrℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ik nk � −∂ρSEr � βrℓ k − βr k � nk −∂AikEr 1 2Arℓ im � vℓ m − vr m � nk+∂AikEr 1 2 ˜urℓ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Aℓ ik − Ar ik � −∂JkEr 1 2Jrℓ i � vℓ m − vr m � nk+∂JkEr 1 2 ˜urℓ J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Jℓ k − Jr k � −∂JkEr 1 2T rℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='−nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Substitution of ∂qE by its expression in state variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' together with (54) yields Eℓ 3 � f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k −f ℓ ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k � nk−Er 3 � f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k −f r ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k � nk+αℓ ik 1 2 ˜uℓr A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Ar ik −Aℓ ik � +αr ik 1 2 ˜urℓ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Aℓ ik −Ar ik � +Eℓ 4 � f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k − f ℓ ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k � nk−Er 4 � f ℓr ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k − f r ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' k � nk+βℓ k 1 2 ˜uℓr J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Jr k − Jℓ k � +βr k 1 2 ˜urℓ J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' n � Jℓ k − Jr k � + � vℓ i � σℓr ik −σℓ ik � − vr i � σℓr ik −σr ik � + αℓ ik 1 2Aℓr im � vr m −vℓ m � − αr ik 1 2Arℓ im � vℓ m −vr m �� nk + � vℓ i � ωℓr ik − ωℓ ik � − vr i � ωℓr ik − ωr ik � + βℓ k 1 2Jℓr i � vr m − vℓ m � − βr k 1 2Jrℓ i � vℓ m − vr m �� nk + � T ℓ � βℓr k − βℓ k � − T r � βℓr k − βr k � + βℓ k 1 2T ℓr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− − βr k 1 2T rℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− � nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Taking into account (56)-(60) and collecting terms gives (ρvr kEr 3 + ρvr kEr 4 + vr iσr ik + vr iωr ik + βr kT r) nk − � ρvℓ kEℓ 3 + ρvℓ kEℓ 4 + vr iσr ik + vr iωr ik + βℓ kT ℓ� nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (70) 12 Gathering (69) and (70), we obtain Dℓr,− E + Drℓ,− E = F r G + (ρvr kEr 3 + ρvr kEr 4 + vr iσr ik + vr iωr ik + βr kT r) nk − F ℓ G − � ρvℓ kEℓ 3 + ρvℓ kEℓ 4 + vℓ iσℓ ik + vℓ iωℓ ik + βℓ kT ℓ� nk = F r − F ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (71) Thus the fluctuations can be seen as the difference between fluxes which will cancel out when adding the contributions of all cells, and hence the scheme is marginally stable in the energy norm, as claimed: � Ω ∂Eℓ ∂t dV = � ℓ |Ωℓ|∂Eℓ ∂t = − � ℓr � Dℓr,− E + Dℓr,+ E � = − � ℓr � F r − F ℓ� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (72) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Assuming T ℓ > 0 and Hℓr = ∂2 qqEℓr ≥ 0 the semi-discrete finite volume scheme (53) with production term (58) satisfies the semi-discrete cell entropy inequality ∂ρSℓ ∂t + � r∈Nℓ ��∂Ωℓr�� |Ωℓ| Dℓr,− ρS + � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � βℓr k − βℓ k � nk− � r∈Nℓ ��∂Ωℓr�� |Ωℓ| gℓr ρS, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (73) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The proof is an immediate consequence of the discretization (53c) with (58): ∂ρSℓ ∂t + � r∈Nℓ ��∂Ωℓr�� |Ωℓ| Dℓr,− ρS + � r∈Nℓ ��∂Ωℓr�� |Ωℓ| � βℓr k − βℓ k � nk− � r∈Nℓ ��∂Ωℓr�� |Ωℓ| gℓr ρS, n = 1 |Ωℓ| � r∈Nℓ ��∂Ωℓr�� Πℓr,− n + αℓ ikαℓ ik θℓ 1 (τ1) T ℓ + βℓ i βℓ i θℓ 2 (τ2) T ℓ ≥ 0, (74) since Πℓr,− n = 1 2ϵℓr ∆qℓr T ℓ ∂2 qqEℓr ∆qℓr δℓr ≥ 0 due to ∂2 qqE ≥ 0 and θℓ 1,2 > 0 as well as T ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 5 Thermodynamically compatible fully-discrete finite volume scheme for the Euler subsystem In this section we present a fully-discrete finite volume scheme for the Euler subsystem of (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' for the black and blue terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For simplicity, we restrict the considerations to one space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As before, the spatial control volumes are denoted by Ωℓ = [xℓ− 1 2 , xℓ+ 1 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The scheme reads qn+1,ℓ − qn,ℓ ∆t = − (f ℓ+ 1 2 ˜p − f ℓ) − (f ℓ− 1 2 ˜p − f ℓ) ∆x + g ℓ+ 1 2 ˜p − g ℓ+ 1 2 ˜p ∆x + ˜Pℓ, (75) Again, the subscript ˜p refers to the fact that the flux is evaluated using the segment path in p variables defined below, similar to (20)-(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In order to construct a thermodynamically compatible fully-discrete scheme, where the total energy conservation law (1f) is a consequence of the discrete equations (75), we introduce a new average quantity ˜pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Since by construction one has qn+1,ℓ � qn,ℓ ∂qE · dq = En+1,ℓ − En,ℓ, (76) for any path connecting qn,ℓ with qn+1,ℓ, we define the quantity ˜pℓ as ˜pℓ = 1 � 0 ∂qE(τ(s))ds, (77) 13 with the straight-line segment path τ = τ(s) = qn,ℓ + s � qn+1,ℓ − qn,ℓ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (78) Therefore, ˜pℓ satisfies the Roe-type property ˜pℓ · � qn+1,ℓ − qn,ℓ� = En+1,ℓ − En,ℓ, (79) which is fundamental for the construction of our thermodynamically compatible fully-discrete scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We now multiply (75) with ˜pℓ from the left and neglecting the viscous fluxes gℓ± 1 2 leads to ˜pℓ · qn+1,ℓ − qn,ℓ ∆t = En+1,ℓ − En,ℓ ∆t = −˜pℓ · (f ℓ+ 1 2 ˜p − f ℓ) + (f ℓ − f ℓ− 1 2 ˜p ) ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (80) To obtain a conservative form of the fully discrete energy conservation law, we require ˜pℓ · (f ℓ+ 1 2 ˜p − f ℓ) + ˜pℓ+1 · (f ℓ+1 − f ℓ+ 1 2 ˜p ) = ˜F ℓ+1 − ˜F ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (81) Using the parametrization (10) and the associated relations (11) we get −∂p(� v1L)ℓ+ 1 2 · �˜pℓ+1 − ˜pℓ� + ˜pℓ+1 · f ℓ+1 − ˜pℓ · f ℓ = ˜pℓ+1 · f ℓ+1 − � v1L ℓ+1 − ˜pℓ · f ℓ + � v1L ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (82) Hence, the numerical flux f ℓ+ 1 2 ˜p = ∂p(� v1L)ℓ+ 1 2 must satisfy the following jump condition: f ℓ+ 1 2 ˜p �˜pℓ+1 − ˜pℓ� = ∂p(� v1L)ℓ+ 1 2 · �˜pℓ+1 − ˜pℓ� = � v1L ℓ+1 − � v1L ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (83) We choose again a simple straight line segment path,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' this time in the ˜p variables: ψ(s) = ψ(˜pℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ˜pℓ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' s) = ˜pℓ + s �˜pℓ+1 − ˜pℓ� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 0 ≤ s ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (84) Using the same reasoning as for the semi-discrete scheme (19)-(38),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' we find the thermodynamically compatible numerical flux of the fully discrete p-scheme as f ℓ+ 1 2 ˜p = � f ℓ+ 1 2 ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' f ℓ+ 1 2 ρvi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' f ℓ+ 1 2 ρS �T = 1 � 0 f(ψ(˜pℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ˜pℓ+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' s))ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (85) with the jump ∆˜pℓ+ 1 2 = ˜pℓ+1 − ˜pℓ and the numerical viscosity flux defined as g ℓ+ 1 2 ˜p = � g ℓ+ 1 2 ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' g ℓ+ 1 2 ρvi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' g ℓ+ 1 2 ρS �T = ϵℓ+ 1 2 ∂2 pp ˜Lℓ+ 1 2 ˜pℓ+1 − ˜pℓ ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (86) The corresponding production term reads ˜Pℓ = (0, 0, ˜Πℓ)T with ˜pℓ · ˜Pℓ = ˜T ℓ ˜Πℓ = 1 2ϵℓ+ 1 2 ∆˜pℓ+ 1 2 ∆x ∂2 pp ˜Lℓ+ 1 2 ∆˜pℓ+ 1 2 ∆x + 1 2ϵℓ− 1 2 ∆˜pℓ− 1 2 ∆x ∂2 pp ˜Lℓ− 1 2 ∆˜pℓ− 1 2 ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (87) The disadvantage of the p-scheme is that it requires the expression of the physical flux f in terms of the p variables, or, equivalently, it requires the variable transformation q = q(p), which in general may be quite cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' However, for the Euler subsystem at least this conversion is simple and analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Recall that ∂2 pp ˜Lℓ− 1 2 = (∂2 qq ˜Eℓ− 1 2 )−1, see (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Note that the proposed fully-discrete scheme is implicit, since ˜pℓ is a function of qn,ℓ and qn+1,ℓ, see (77) and (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In order to obtain a simple and straightforward implementation of the fully-discrete scheme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' we propose the following predictor- corrector approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' based on a Picard-type iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' similar to the iterative procedure employed in the fully-discrete kinetic energy-preserving scheme proposed for the Euler equations in [53]: qn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ k+1 = qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ − ∆t ∆x � f ℓ+ 1 2 ˜pk − f ℓ− 1 2 ˜pk � + ∆t ∆x � g ℓ+ 1 2 ˜pk − g ℓ− 1 2 ˜pk � + ˜Pℓ k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (88) 14 with the quantity ˜pℓ k defined as ˜pℓ k = 1 � 0 ∂qE(τ)ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' with τ = qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ + s � qn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ k − qn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (89) As initial guess for the iterative scheme we set qn+1,ℓ 0 = qn,ℓ and the iterations are stopped when the following condition is satisfied: � ℓ � En+1,ℓ k − E � qn+1,ℓ k+1 ��2 < δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (90) with δ > 0 an arbitrarily small tolerance, typically of the order of the machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Recall that En+1,ℓ k = En,ℓ k +˜pℓ k· � qn+1,ℓ k − qn,ℓ� , see (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' This completes the description of the fully-discrete Godunov formalism for the inviscid Euler subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The thermodynamically compatible fully-discrete finite volume scheme qn+1,ℓ − qn,ℓ ∆t = − (f ℓ+ 1 2 ˜p − f ℓ) − (f ℓ− 1 2 ˜p − f ℓ) ∆x + g ℓ+ 1 2 ˜p − g ℓ− 1 2 ˜p ∆x + ˜Pℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (91) with production term ˜Pℓ according to (87) and fluxes (85) and (86) verifies the fully discrete energy conservation law En+1,ℓ − En,ℓ ∆t = − 1 ∆x � ˜D ℓ+ 1 2 ,− E + ˜D ℓ− 1 2 ,+ E � + g ℓ+ 1 2 E − g ℓ− 1 2 E ∆x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (92) The fluctuations above are defined as ˜D ℓ+ 1 2 ,− E = ˜pℓ · (f ℓ+ 1 2 ˜p − f ℓ), ˜D ℓ− 1 2 ,+ E = ˜pℓ · (f ℓ − f ℓ− 1 2 ˜p ) (93) and satisfy ˜D ℓ+ 1 2 ,− E + ˜D ℓ+ 1 2 ,+ E = ˜F ℓ+1 − ˜F ℓ = F(˜pℓ+1) − F(˜pℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (94) The numerical viscosity flux in (92) reads g ℓ+ 1 2 E = 1 2 ϵℓ+ 1 2 ∆x �˜pℓ + ˜pℓ+1� ∂pp ˜Lℓ+ 1 2 �˜pℓ+1 − ˜pℓ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (95) As a consequence, for vanishing jumps on the boundary, the scheme is nonlinearly marginally stable in the energy norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Multiplying (91) with ˜p defined according to (77) and using the Roe property (79) yields ˜pℓ · qn+1,ℓ − qn,ℓ ∆t = En+1,ℓ − En,ℓ ∆t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (96) Furthermore, using (93) one has immediately ˜pℓ · � f ℓ+ 1 2 ˜p,d − f ℓ� + ˜pℓ · � f ℓ − f ℓ+ 1 2 ˜p,d � = ˜D ℓ+ 1 2 ,− E + ˜D ℓ− 1 2 ,+ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (97) By construction, (81)-(85), the fluctuations satisfy (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For the numerical viscosity we get after some calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' see (29) and (66) for the semi-discrete case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='˜pℓ · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='˜p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='˜p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='+ ˜pℓ · ˜P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='= ϵℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ˜pℓ · ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pp ˜Lℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 �˜pℓ+1 − ˜pℓ� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− ϵℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ˜pℓ · ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pp ˜Lℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 �˜pℓ − ˜pℓ−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='+ ˜pℓ · ˜P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='˜pℓ+1 + ˜pℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pp ˜Lℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆˜pℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='˜pℓ + ˜pℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pp ˜Lℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆˜pℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆˜pℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pp ˜Lℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆˜pℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆˜pℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ϵℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='pp ˜Lℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ∆˜pℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='+ ˜pℓ · ˜P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='= g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='− g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='ℓ− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='∆x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='(98) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='due to the definition (95) and the production term that satisfies (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Multiplication of (92) by ∆t∆x and summation over Ωℓ yields � ℓ ∆x � En+1,ℓ − En,ℓ� = − � ℓ ∆t � ˜D ℓ+ 1 2 ,− E + ˜D ℓ− 1 2 ,+ E + g ℓ+ 1 2 E − g ℓ− 1 2 E � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' (99) The terms on the right hand side of (99) are a telescopic sum that vanishes because the fluctuations satisfy (94) and since the jumps vanish at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The fully-discrete finite volume scheme (91) with production term ˜Pℓ according to (87) and fluxes (85) and (86) satisfies the fully discrete cell entropy inequality (ρS)n+1,ℓ ≥ (ρS)n,ℓ − ∆t ∆x � f ℓ+ 1 2 ρS − f ℓ− 1 2 ρS � + ∆t ∆x � g ℓ+ 1 2 ρS − g ℓ− 1 2 ρS � , (100) assuming that ˜T ℓ > 0 and ∂2 pp ˜Lℓ± 1 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The fully-discrete form of the entropy equation (1c) according to the scheme (91) reads (ρS)n+1,ℓ = (ρS)n,ℓ − ∆t ∆x � f ℓ+ 1 2 ρS − f ℓ− 1 2 ρS � + ∆t ∆x � g ℓ+ 1 2 ρS − g ℓ− 1 2 ρS � + ∆t ˜Πℓ, (101) with ˜Πℓ = 1 ˜T ℓ � 1 2ϵℓ+ 1 2 ∆˜pℓ+ 1 2 ∆x ∂2 pp ˜Lℓ+ 1 2 ∆˜pℓ+ 1 2 ∆x + 1 2ϵℓ− 1 2 ∆˜pℓ− 1 2 ∆x ∂2 pp ˜Lℓ− 1 2 ∆˜pℓ− 1 2 ∆x � ≥ 0, (102) since we assume ˜T ℓ > 0 and ∂2 pp ˜Lℓ± 1 2 > 0, hence one directly obtains the inequality (100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6 Numerical results The new schemes for hyperbolic and thermodynamically compatible PDE systems (HTC schemes) pro- posed in this paper do not discretize the energy conservation law (1f) explicitly, but consider the entropy inequality (1c) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Semi-discrete / fully-discrete energy conservation is obtained as a mere conse- quence of the thermodynamically compatible discretization of the PDEs (1a)-(1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As such, the proposed approach is different from most existing finite volume discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The main aim of the following numerical test problems is therefore to show that the scheme is able to compute correct solutions to problems with shock waves, as predicted by Theorems 1 and 3 on the semi-discrete and fully-discrete energy conservation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Furthermore, we check numerically whether the relaxation limit of the model (Navier-Stokes limit) is properly captured, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' when for sufficiently small relaxation times τ1 and τ2 the behaviour of the medium becomes the one of a viscous heat-conducting Newtonian fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For more advanced applications of the GPR model, the reader is referred to [20, 21, 9, 56, 44, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In the following tests, when a viscosity coefficient µ is specified together with a shear sound speed cs, the corresponding relaxation time τ1 is calculated as τ1 = 6µ/(ρ0c2 s), according to (9) and the results of the asymptotic analysis carried out in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In all tests of this section, the semi-discrete HTC schemes are integrated in time using an explicit third order TVD Runge-Kutta scheme, see [52, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For more efficient 16 Table 1: Numerical convergence results for the isentropic vortex problem, obtained with the semi-discrete HTC scheme proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The reported L2 error norms refer to a final time of t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Nx = Ny ∥ρ∥2 ∥ρv1∥2 ∥ρS∥2 O(ρ) O(ρv1) O(ρS) 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1094E-03 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1324E-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='7896E-04 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5602E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3633E-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3256E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9 128 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9230E-04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9585E-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3972E-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 256 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8232E-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4928E-04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5455E-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 512 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4626E-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='7369E-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1397E-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 Table 2: Numerical convergence results for the isentropic vortex problem, obtained with the fully-discrete HTC scheme proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The reported L2 error norms refer to a final time of t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Nx = Ny ∥ρ∥2 ∥ρv1∥2 ∥ρS∥2 O(ρ) O(ρv1) O(ρS) 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2046E-03 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1891E-03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8312E-04 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5749E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3742E-03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3384E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9 128 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9424E-04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9737E-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4170E-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 256 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8481E-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4948E-04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5718E-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 512 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4658E-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='7394E-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1430E-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 IMEX Runge-Kutta schemes in the case of stiff relaxation source terms, see the work of Pareschi & Russo [42, 43] as well as [39, 14, 8, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In all numerical tests carried out with the semi-discrete scheme, we assume the time step ∆t to be small enough so that time discretization errors can be neglected concening the conservation of total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In the following, if not stated otherwise, the numerical viscosity ϵℓ+ 1 2 is chosen according to (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' When explicit values of ϵ are provided, the numerical dissipation is chosen as a constant, ϵℓ+ 1 2 = ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 Numerical convergence study In this section, we solve the isentropic vortex problem forwarded in [36] in order to verify the accuracy of the proposed HTC schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We apply the schemes to the pure inviscid Euler equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' to the black terms in (1a)-(1c), setting γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4, cs = 0, ch = 0 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The computational domain is Ω = [0, 10]2 with periodic boundaries everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The initial conditions for the perturbations are given in [36, 20] and are not repeated here to save space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The background velocity is chosen as v0 = 0 so that a stationary vortex is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In this situation, the exact solution is given by the initial condition for all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Simulations are run with the semi-discrete HTC scheme until a final time of t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25 using an equidistant Cartesian grid composed of Nx × Ny control volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The L2 errors obtained with the semi-discrete HTC schemes at the final time for the density ρ, the momentum density ρv1 and the entropy density ρS are shown in Table 1 together with the corresponding convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The results for the fully discrete HTC scheme are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' One can observe that all proposed HTC schemes are of second order of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 Simple shear motion in solids and fluids We first apply the new HTC schemes to simple shear motion in solids and fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The one-dimensional computational domain is Ω = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5] and the initial condition of the problem, which is also prescribed at the boundaries of Ω, is given by ρ = 1, v1 = v3 = 0, p = 1, A = I, J = 0, while the velocity component v2 is v2 = −v0 for x < 0 and v2 = +v0 for x ≥ 0, with v0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The remaining parameters of this test are γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4, cv = 1, ρ0 = 1, cs = 1 and ch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The calculations are carried out with the new HTC 17 schemes on a grid composed of 1024 control volumes up to a final time of t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In the Navier-Stokes limit of the GPR model, a reference solution can be obtained by the exact solution of the incompressible Navier-Stokes equations for the first problem of Stokes, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [20, 7, 6, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For the solid limit of the GPR model (τ1 → ∞), this initial condition leads to two shear waves traveling to the left and right, respectively, with speed cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A reference solution can be obtained using a classical second order MUSCL- Hancock scheme [57] on a fine mesh of 32000 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We stress that for all cases with µ > 0 the HTC scheme has been run without any numerical viscosity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' setting ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The comparison between the numerical solutions obtained with the new HTC schemes and the aforementioned reference solutions is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 1, where one can observe an excellent agreement for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' x v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 Reference solution HTC scheme x v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 Reference solution HTC scheme x v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 Reference solution HTC scheme x v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 Reference solution HTC scheme Figure 1: Numerical solution at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 obtained with the new thermodynamically compatible HTC schemes for the GPR model applied to a simple shear flow in fluids and in an elastic solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Results for the solid (top left) and for fluids with different viscosities: µ = 10−2 (top right), µ = 10−3 (bottom left) and µ = 10−4 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For fluids, this test corresponds to the first problem of Stokes, which has an exact analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 Riemann problems In this section, we solve a set of Riemann problems with initial data according to Table 3, for both the Euler equations of compressible gasdynamics, which are a subset of the GPR model (black terms in (1)), and for the full GPR model in both its fluid and solid limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The initial discontinuity is located in xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For the Euler equations, we consider semi-discrete as well as fully-discrete schemes and the exact solution of the Riemann problem has been provided in [57], while for the GPR model we consider two types of completely independent numerical reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The first reference solution is obtained by using a classical MUSCL-Hancock finite volume scheme on a fine mesh of 128000 elements, discretizing the total energy conservation law (1f) instead of the entropy inequality (1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' An alternative reference solution is 18 obtained by solving the GPR model (1a)-(1c) with the entropy inequality in its vanishing viscosity limit, using a fourth order ADER-DG scheme on a fine mesh composed of 14400 order elements, including also the quadratic entropy production term in (1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In this case, thermodynamic compatibility is achieved simply at the aid of a fully resolved simulation employing sufficiently fine meshes in combination with high order of accuracy in space and time, see [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The numerical results obtained with the semi-discrete and fully-discrete HTC schemes for the compressible Euler equations are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 2 and 3, while the numerical results obtained with the semi-discrete HTC scheme applied to the fluid and solid limits of the GPR model are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 4 and 5, respectively, together with the reference solution obtained with the MUSCL-Hancock scheme solving the energy conservation law (1f), as well as the reference solution obtained with the high order ADER-DG scheme applied to the viscous system (1a)-(1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The effective mesh resolution is provided for each test case in the corresponding figure caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In all cases we can note an excellent agreement between the numerical solution obtained with the new HTC schemes forwarded in this paper and the available exact or numerical reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Test problem RP1s was proposed by Toro in [57] and includes a sonic rarefaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Simulations are carried out on several meshes and the obtained quantities ρ, p, u = v1 and S are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We observe that the thermodynamically compatible schemes proposed in this paper do not exhibit any sonic glitch, compared to other Godunov-type finite volume schemes, see [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A quantitative study concerning the influence of the number of Gauss-Legendre quadrature nodes nGP and the chosen time discretization on the total energy conservation error can be found for a smoothed version of RP1 with initial data q(x, 0) = 1 2(qL + qR) + 1 2(qR − qL) erf(x/χ) with χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='01 in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As expected, the conservation error of the semi-discrete schemes is dominated by the time discretization and the chosen time step size (CFL number), while the energy conservation error of the fully discrete scheme is independent of the time step size and is dominated only by the numerical quadrature rule used in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Table 3: Initial states left (L) and right (R) for density ρ, velocity v = (u, v, 0) and pressure p for a set of Riemann problems solved on the domain Ω = [− 1 2, + 1 2] using the new HTC schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The Riemann problems include the pure Euler equations (RP1, RP2 and RP1s), as well as the fluid and solid limit of the GPR model (RP3 and RP4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For the GPR model (RP3 and RP4) we initialize A and J as A = 3√ρ I and J = 0 and set cs = ch = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The relaxation times have been chosen as τ1 = τ2 = 2 · 10−5 for RP3 and τ1 = τ2 = 1020 for RP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In all cases we set γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' RP ρL uL vL pL ρR uR vR pR RP1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 RP1s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 RP2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='99924 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='894 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='99242 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='19633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='095 RP3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 RP4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 Viscous shock wave Consider a stationary viscous shock wave at a shock Mach number of Ms = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For Prandtl number Pr= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='75 there exists an exact solution of the compressible Navier-Stokes equations, see [4, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The computational domain Ω = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5] is covered with 1024 control volumes and the shock wave is centered at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We assume that the fluid is moving into the shock wave from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The data in front of the shock are ρ0 = 1, v0 1 = −2, v0 2 = v3 = 0 and p0 = 1/γ so that the associated sound speed is c0 = 1 and the corresponding Reynolds number based on a reference length L = 1 is given by Res = ρ0 c0 Ms L µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The parameters are set as γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4, cv = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5, ch = cs = 50, µ = 2 · 10−2 and λ = 9 1 3 ·10−2, hence the shock Reynolds number is Res = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' At t = 0 we set A = 3√ρ I and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The comparison between the numerical solution obtained with the semi-discrete HTC scheme applied to (1) and the exact solution of the compressible Navier-Stokes equations is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' For all quantities 19 Table 4: Total energy conservation error depending on the time discretization and the number of Gauss- Legendre quadrature points nGP for the calculation of the thermodynamically compatible flux (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' CFL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 semi-discrete HTC scheme + TVD Runge-Kutta O3 nGP = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='90 · 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='55 · 10−5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='30 · 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='00 · 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='00 · 10−7 nGP = 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='90 · 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='54 · 10−5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='31 · 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='99 · 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='45 · 10−7 semi-discrete HTC scheme + classical Runge-Kutta O4 nGP = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='23 · 10−6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='51 · 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='07 · 10−7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25 · 10−8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='33 · 10−9 nGP = 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='24 · 10−6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='64 · 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='19 · 10−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='53 · 10−8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='43 · 10−9 Fully-discrete HTC scheme nGP = 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='80 · 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='81 · 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='82 · 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='83 · 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='84 · 10−9 nGP = 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='70 · 10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='70 · 10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='70 · 10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='70 · 10−13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='70 · 10−13 x rho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 Exact solution Semi-discrete HTC scheme Fully-discrete HTC scheme x rho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0 5 10 15 20 25 30 35 40 Exact solution Semi-discrete HTC scheme Fully-discrete HTC scheme Figure 2: Numerical results for Riemann problems RP1 (xc = 0) and RP2 (xc = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2) at times t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='035, respectively, obtained with the semi-discrete (red solid line) and the fully-discrete (dashed blue line) HTC schemes on 1024 elements applied to the compressible Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The exact solution of the compressible Euler equations is represented by the black solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 20 x rho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 x p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 Exact solution Semi-discrete HTC (2048) Semi-discrete HTC (1024) Semi-discrete HTC (512) Semi-discrete HTC (256) Semi-discrete HTC (128) Fully-discrete HTC (2048) Fully-discrete HTC (1024) Fully-discrete HTC (512) Fully-discrete HTC (256) Fully-discrete HTC (128) x u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 x S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 1 Figure 3: Numerical results for Riemann problem RP1s (xc = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2) at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 obtained with the semi-discrete (solid lines) and the fully-discrete (dashed lines) HTC schemes on 2048, 1024, 512, 256 and 128 elements applied to the compressible Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The exact solution of the compressible Euler equations is represented by the black solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 21 x rho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 Exact solution Vanishing viscosity limit HTC scheme x v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 Exact solution Vanishing viscosity limit HTC scheme Figure 4: Numerical results at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 for Riemann problem RP3 (xc = 0) obtained with the HTC scheme (red solid line) on 1024 elements, the fourth order ADER-DG scheme applied to the vanishing viscosity limit of the viscous equations (1a)-(1c) using ϵ = 2 · 10−5 on 14400 elements (dashed blue line) and the exact solution of the compressible Euler equations (black solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' an excellent agreement is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 Solid rotor problem In this section we solve the solid rotor problem proposed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' By setting τ1 = τ2 = 1020 the model (1) describes a nonlinear hyperelastic solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The computational domain is Ω = [−1, +1]2 with periodic boundary conditions everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The initial data for density, pressure, A and J is set to ρ = 1, p = 1, A = I and J = 0, while the initial condition for the velocity field is v1 = −y/R, v2 = +x/R and v3 = 0 within the circular region r ≤ R, where r = ∥x∥ and R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2, while v = 0 for r > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The parameters of the GPR model are set to γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4, cs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0 and ch = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We run the test problem until a final time of t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 using the two-dimensional semi-discrete HTC scheme for the GPR model on a uniform Cartesian grid composed of 512 × 512 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The artificial viscosity in the HTC scheme is set to a constant value of ϵ = 5 · 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' To obtain a reference solution, on the same mesh of 512 × 512 elements we solve the same problem again but using a classical second order MUSCL-Hancock scheme, see [57] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We emphasize that in the MUSCL scheme, which is not thermodynamically compatible, we solve the total energy conservation law (1f) rather than the entropy inequality (1c), as already suggested in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The obtained results are compared with each other in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 7, where the contour colors of the velocity component v1 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The agreement between the numerical solution obtained with the new HTC scheme and the reference solution is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Since the applied HTC scheme for this test problem is only compatible with the semi-discrete total energy conservation law, we have explicitly monitored the total energy conservation error during the entire simulation, finding a maximum relative energy conservation error of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='02 · 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 Double shear layer In this section we present numerical results for the double shear layer test, see [5, 20, 6, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The computational domain is Ω = [0, 1]2 with periodic boundary conditions everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The initial condition is given by v1 = tanh (˜ρ(y − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25)) for y ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 and v1 = tanh (˜ρ(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='75 − y)) if y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5, v2 = δ sin(2πx), v3 = 0, ρ = ρ0 = 1, p = 102/γ, A = I, J = 0, with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 and ˜ρ = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The remaining parameters of the GPR model are set to ν = µ/ρ0 = 2 · 10−3, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4, ρ0 = 1, cv = 1, cs = 8, ch = 2 and τ2 = 4 · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The characteristic Mach number of the flow resulting from this setup is M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Calculations are 22 x rho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} 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x J1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25 Reference solution Vanishing viscosity limit HTC scheme Figure 5: Numerical results at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 for Riemann problem RP4 (xc = 0) obtained with the HTC scheme (red solid line) on 10000 control volumes, a fourth order ADER-DG scheme applied to the vanishing viscosity limit of the viscous equations (1a)-(1c) using ϵ = 2·10−5 (dashed blue line) on 144000 elements and the reference solution obtained with a MUSCL-Hancock scheme applied to the model with the energy conservation law (1f) instead of the entropy inequality (1c) (black solid line) using 128000 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' x rho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 Reference solution HTC scheme Figure 6: Exact solution of the compressible Navier-Stokes equations and numerical solution obtained with the HTC scheme applied to the GPR model for a viscous shock at Ms = 2, Res = 100 and Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Density (left), stress σ11 (center) and heat flux h1 (right) at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 23 Figure 7: Velocity component v1 for the solid rotor test problem at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='3 obtained by solving (1) with the new HTC scheme (left) and by using a classical MUSCL scheme (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' performed with the new HTC scheme up to a final time of t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The computational grid is composed of 4000 × 4000 control volumes and the numerical viscosity is chosen as ϵ = 1 · 10−6, hence three orders of magnitude lower than the physical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 8 the results obtained with the new HTC scheme are compared with a numerical reference solution that is based on the solution of the incompressible Navier-Stokes equations using a hybrid FV/FE method on a triangular grid made of 2097152 elements (Nx = 1000 divisions along each boundary), see [11, 6, 12] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The flow dynamics has already been described in [5, 6, 12, 20, 7] and can be summarized by the development of several vortices from the initially perturbed shear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The agreement between the Navier-Stokes reference solution and the numerical solution of the GPR model computed with the new HTC schemes is rather good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 9 we present the temporal evolution of the distortion field component A12, which is qualitatively similar to the results shown in [20], but for a lower physical viscosity µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The maximum relative conservation error of the total energy monitored during the simulation for the semi-discrete HTC scheme was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='12 · 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Due to the low numerical viscosity of ϵ = 10−6 and fine mesh, one can observe small structures developing in the distortion field A, which we would like to demonstrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Figure 8: Vorticity contours for the double shear layer with a viscosity of µ = 2 · 10−3 at time t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Left: numerical solution of the GPR model obtained with the new thermodynamically compatible finite volume scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Right: reference solution obtained by solving the incompressible Navier-Stokes equations with the staggered semi-implicit hybrid FV/FE scheme [11, 6, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 24 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='22 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 7 8 10 11 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 x11 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 9 8 7 6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 4 3 2 1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 7 8 9 10 11 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8Figure 9: Distortion field component A12 for the double shear layer problem at times t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 obtained by solving the GPR model (µ = 2 · 10−3) with the HTC scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='7 Lid-driven cavity As last numerical test case for the fluid limit of the model (1) we present the lid-driven cavity problem, see [27], which can be used to validate compressible flow solvers in the low Mach number regime, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [55, 6, 12] and which was already successfully solved with the GPR model in [20, 7], but the schemes used in [20, 7] were not thermodynamically compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The computational domain is Ω = [0, 1] × [0, 1] and the initial condition is set to ρ = 1, v = 0, p = 102, A = I and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We furthermore set γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4, cv = 1, cs = 8, ρ0 = 1 and ch = 2, τ2 = 10−2 and µ = 10−2 so that the Reynolds number of the test problem is Re = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The lid velocity on the upper boundary is set to v = (1, 0, 0), while on all other boundaries v = 0 is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The Mach number of this test is about M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The new semi-discrete HTC scheme is run until t = 10 using 256 × 256 elements and a constant artificial viscosity of ϵ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The numerical results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 10, where also a comparison with the Navier-Stokes reference solution of Ghia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [27] is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We note an excellent agreement between the numerical solution of the GPR model and the incompressible Navier-Stokes reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' x,y u,v 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 1 GPR model (HTC scheme) - u(0,y) GPR model (HTC scheme) - v(x,0) Reference solution - u(0,y) Reference solution - v(x,0) Figure 10: Lid-driven cavity at Reynolds number Re = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Results obtained at time t = 10 with the new HTC scheme applied to the GPR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Color contours of the velocity component v1 (left) and comparison of the velocity components v1 and v2 on 1D cuts along the x and y axis with the reference solution of Ghia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [27] (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 25 A12 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='8 x7 Conclusions In this paper, we have presented two novel thermodynamically compatible finite volume schemes for first order hyperbolic PDE systems (HTC schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The first method is a semi-discrete finite volume scheme for the unified first order hyperbolic model of solid and fluid mechanics that goes back to the work of Godunov, Peshkov and Romenski on symmetric hyperbolic and thermodynamically compatible (SHTC) systems, see [29, 31, 51, 33, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We have furthermore introduced a new fully-discrete HTC scheme for the compressible Euler equations, establishing a fully discrete analogy of the continuous framework in- troduced by Godunov in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' All schemes under consideration in this paper have in common that they directly discretize the entropy inequality rather than the usual total energy conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Instead, total energy conservation is obtained at the discrete level as a mere consequence of a suitable and ther- modynamically compatible discretization of all the other equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' As such, the new schemes can be proven to be nonlinearly marginally stable in the energy norm and they furthermore satisfy a discrete entropy inequality by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The new HTC schemes have been applied to several test problems for fluid and solid mechanics, obtaining an excellent agreement with available reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In future work, we will investigate the possible use of symplectic time integrators in order to preserve exact total energy conservation of our new semi-discrete thermodynamically compatible scheme also on the fully discrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' We also plan an extension to higher order in space at the aid of thermodynami- cally compatible discontinuous Galerkin (DG) finite element schemes, similar to entropy compatible DG schemes introduced in [19, 38, 26] for the shallow water equations and magnetohydrodynamics (MHD), as well as an extension to general unstructured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Another major challenge left to future work is the development of HTC schemes that are not only thermodynamically compatible, but which are also able to preserve the curl involution constraints of the governing PDE system exactly at the semi-discrete level and that are also consistent with the low Mach number limit of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In this context we will consider staggered semi-implicit finite volume schemes [7], as well as staggered semi-implicit hybrid finite volume / finite element methods [11, 6, 12] and staggered DG schemes [55, 13], which are not yet thermodynamically compatible in the sense of the HTC schemes presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Acknowledgments S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' are members of the INdAM GNCS group and acknowledge the financial support received from the Italian Ministry of Education, University and Research (MIUR) in the frame of the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (grant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 232/2016) and in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary partial differential equations and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' was also funded by INdAM via a GNCS grant for young researchers and by an UniTN starting grant of the University of Trento.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' were supported by the Mathematical Center in Akademgorodok under agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The authors would like to acknowledge support from the Leibniz Rechenzentrum (LRZ) in Garching, Germany, for granting access to the SuperMUC-NG supercomputer under project number pr63qo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The authors are very grateful to the two anonymous referees for their constructive and insightful comments, which helped to improve the clarity and quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Abgrall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A general framework to construct schemes satisfying additional conservation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Application to entropy conservative and entropy dissipative schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 372:640–666, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Abgrall, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Bacigaluppi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Tokareva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A high-order nonconservative approach for hyperbolic equations in fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Computers and Fluids, 169:10–22, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 26 [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Bauera, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='Burton, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Caramana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='Loub`ere, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shashkov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Whalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The in- ternal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Journal of Computational Physics, 218:572–593, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Becker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Stosswelle und Detonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Physik, 8:321, 1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Bell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Coletta, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Glaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A second-order projection method for the incompressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 85:257–283, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Berm´udez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ferr´ın, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Saavedra, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' V´azquez-Cend´on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 421:109743, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [7] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Boscheri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ioriatti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A structure-preserving staggered semi-implicit finite volume scheme for continuum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 424:109866, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Buet and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Despr´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Asymptotic preserving and positive schemes for radiation hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 215(2):717–740, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Chiocchetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gaburro, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' High order ADER schemes for continuum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Frontiers in Physics, 8:32, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gavrilyuk, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ivanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' On thermodynamically compatible finite volume methods and path-conservative ADER discontinuous Galerkin schemes for turbulent shallow water flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Journal of Scientific Computing, 88:28, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ferr´ın, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Toro, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' V´azquez-Cend´on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A projection hybrid high order finite volume/finite element method for incompressible turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 353:169–192, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Del Rio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' V´azquez-Cend´on, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A semi-implicit hybrid finite volume / finite element scheme for all Mach number flows on staggered unstructured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 402:126117, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [13] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Busto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Tavelli, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Boscheri, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Efficient high order accurate staggered semi- implicit discontinuous Galerkin methods for natural convection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Computers & Fluids, 198:104399, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Caflish, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Jin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Russo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Uniformly accurate schemes for hyperbolic systems with relax- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 34:246–281, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Caramana and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='Loub`ere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Journal of Computational Physics, 216:1–18, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Castro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gallardo, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Par´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Applications to shallow-water systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 75:1103–1134, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Chatterjee and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Fjordholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Convergence of second-order, entropy stable methods for multi- dimensional conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ESAIM Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 54(4):1415–1428, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Cheng and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 345:427–461, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Derigs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Winters, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gassner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Walch, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Bohm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 364:420–467, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 27 [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Zanotti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat–conducting fluids and elastic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 314:824–862, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Zanotti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro–dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 348:298–342, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Toro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 48:70–88, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [23] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Fjordholm and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Mishra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Accurate numerical discretizations of non-conservative hyperbolic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' ESAIM Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 46(1):187–206, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [24] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Friedrichs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Symmetric positive linear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 11:333– 418, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Friedrichs and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Lax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Systems of conservation equations with a convex extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' USA, 68:1686–1688, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [26] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gassner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Winters, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Kopriva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 272:291– 308, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [27] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ghia, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ghia, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' High-Re solutions for incompressible flow using Navier-Stokes equations and multigrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 48:387–411, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 52:787–799, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' An interesting class of quasilinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Nauk SSSR, 139(3):521–523, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Symmetric form of the magnetohydrodynamic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numerical Methods for Mechanics of Continuum Medium, 3(1):26–34, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 13:868–885, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Thermodynamics, conservation laws, and symmetric forms of differential equations in mechanics of continuous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' In Computational Fluid Dynamics Review 95, pages 19–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' John Wiley, NY, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Godunov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Elements of continuum mechanics and conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Kluwer Academic/Plenum Publishers, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gottlieb and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Total variation diminishing Runge-Kutta schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 67:73–85, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Hennemann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Rueda-Ram´ırez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Hindenlang, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gassner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 426, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [36] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Hu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Weighted essentially non-oscillatory schemes on triangular meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 150:97–127, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [37] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Pareschi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Toscani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Uniformly accurate diffusive relaxation scheme for multiscale transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 38(3):913–936, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 28 [38] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 354:163–178, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [39] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Naldi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Pareschi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 37(4):1246–1270, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Osher and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Solomon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Upwind difference schemes for hyperbolic conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 38:339–374, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Par´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numerical methods for nonconservative hyperbolic systems: a theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 44:300–321, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [42] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Pareschi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Russo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Advances in the Theory of Computational Mathematics, 3:269–288, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [43] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Pareschi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Russo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 25:129–155, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [44] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Boscheri, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Chiocchetti, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ioriatti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure- preserving semi-implicit scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Computers & Fluids, page 104963, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [45] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Pavelka, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Grmela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Continuum Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Thermodyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 30(6):1343–1378, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [46] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A hyperbolic model for viscous Newtonian flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Continuum Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Thermodyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 28:85–104, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [47] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Continuum mechanics with torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Continuum Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Thermodyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 31:1517–1541, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [48] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Ranocha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dalcin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Parsani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Fully discrete explicit locally entropy-stable schemes for the compressible Euler and Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 80(5):1343–1359, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [49] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Drikakis, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Toro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Conservative models and numerical methods for com- pressible two-phase flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 42:68–95, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [50] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Peshkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Fambri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A new continuum model for general relativistic viscous heat-conducting media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A, 378:20190175, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [51] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Modell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 28(10):115–130, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [52] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Shu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Osher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Efficient implementation of essentially non-oscillatory shock capturing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 77:439–471, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [53] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Subbareddy and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Candler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A fully discrete, kinetic energy consistent finite–volume scheme for compressible flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Journal of Computational Physics, 228:1347–1364, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [54] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Tadmor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' The numerical viscosity of entropy stable schemes for systems of conservation laws I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 49:91–103, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Tavelli and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equa- tions at all Mach numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 341:341–376, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Tavelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Romenski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Chiocchetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Gabriel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Dumbser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=', 422:109758, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' [57] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Toro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Riemann Solvers and Numerical Methods for Fluid Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE_T4oBgHgl3EQf5Rz9/content/2301.08358v1.pdf'} diff --git a/lNE3T4oBgHgl3EQfhwqh/content/tmp_files/2301.04574v1.pdf.txt b/lNE3T4oBgHgl3EQfhwqh/content/tmp_files/2301.04574v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d791bc6f87a3369f8f89fbf3125002c4448997e0 --- /dev/null +++ b/lNE3T4oBgHgl3EQfhwqh/content/tmp_files/2301.04574v1.pdf.txt @@ -0,0 +1,452 @@ +Prototyping Vehicle Control Applications +Using the CAT Vehicle Simulator +Rahul Bhadani∗ +Jonathan Sprinkle† +Abstract +This paper demonstrates the integration model-based design approaches for vehicle control, with +validation in a freely available open-source simulator. Continued interest in autonomous vehicles and their +deployment is driven by the potential benefits of their use. However, it can be challenging to transition +new theoretical approaches into unknown simulation environments. Thus, it is critical for experts from +other fields, whose insights may be necessary to continue to advance autonomy, to be able to create +control applications with the potential to transition to practice. In this article, I will explain how to use +the CAT Vehicle simulator and ROS packages to create and test vehicle controllers. The methodology of +developing the control system in this article takes the approach of model-based design using Simulink, +and the ROS Toolbox, followed by code generation to create a standalone C++ ROS node. Such ROS +nodes can be integrated through roslaunch in the CAT Vehicle ROS package. +1 +Introduction +Recent advances in computing, control, and sensor technology have brought autonomous systems – especially +autonomous ground vehicles (AV) into the limelight of not only academic research but also media [16]. The +goal of autonomous vehicle control and related research is to improve passenger comfort and safety [6, 10] +and reduce road accidents[9]. At the same time, some other researchers have been investigating the use of +autonomous vehicle control in reducing traffic congestion [4, 14, 1, 3] and fuel consumption reduction [8, 11]. +Such objectives and research endeavors encompass a mix of simulation study as well as experimental research +using physical platforms. While there are several simulation software and packages have been developed to +prototype autonomous vehicle control – both general-purpose simulators such as AirSim [13], CARLA [5]; +and application-specific simulators such as CAT Vehicle [2], and [15], not all simulators are created equally. +Some provide the ability to prototype a wide variety of use cases but at the same arduous and difficult to get +familiar with, while others are limited in use cases but easier to understand. +In this article, we present ways in which a previously proposed autonomous vehicle simulator CAT Vehicle, +written as a ROS package [12] can be used to prototype longitudinal vehicle control. The CAT Vehicle +simulator is a multi-vehicle simulator that uses rigid body dynamics from the Gazebo physics engine [7]. The +methodology presented in this article takes the approach of model-based design using Mathworks’ Simulink. +Simulink provides ROS Toolbox that can be used to prototype ROS components along with custom control +law. Further, Simulink allows code-generation of C++ standalone ROS node provides open-source C++ code, +capable of executing on any Ubuntu machine. +The rest of the article is divided into the following parts. Section 2 provides a step-by-step guide to installing +the CAT Vehicle ROS package. Section 3 provides a brief overview of CAT Vehicle APIs available for +controller prototype. Section 4 provides an example of vehicle control. We end the article with a conclusion. +∗Vanderbilt University, rahul.bhadani@vanderbilt.edu, rahulbhadani@email.arizona.edu +†Vanderbilt University, jonathan.sprinkle@vanderbilt.edu +1 +arXiv:2301.04574v1 [cs.RO] 11 Jan 2023 + +2 +Installing CAT Vehicle Package +The CAT Vehicle is a ROS-based simulator written as a ROS package to facilitate the development of +autonomous vehicle control applications. The simulator utilizes Gazebo 3D world and ROS tools for deploying +and testing a control application in a 3D environment with realistic vehicle dynamics. In this section, we +provide a hands-on on how to install the CAT Vehicle ROS package that will help you in getting started +with writing an autonomous control application. The examples presented in this article use ROS Noetic on +Ubuntu 20.04. +2.1 +Installing ROS Noetic +Open the terminal, and execute the following commands +sudo sh -c 'echo "deb http://packages.ros.org/ros/ubuntu $(lsb_release -sc) main" \ +> /etc/apt/sources.list.d/ros-latest.list' +sudo apt install curl # if you haven't already installed curl +curl -s https://raw.githubusercontent.com/ros/rosdistro/master/ros.asc | \ +sudo apt-key add - +sudo apt-get update +sudo apt install ros-noetic-desktop-full +echo "source /opt/ros/noetic/setup.bash" >> ~/.bashrc +sudo apt install python3-rosdep python3-rosinstall \ +python3-rosinstall-generator python3-wstool build-essential python3-rosdep +Once successfully executed above commands, close the terminal, reopen it, and execute the following command +sudo rosdep init +rosdep update +In addition, we require a few additional packages that can be installed using the following command: +sudo apt-get install python-yaml +sudo apt-get install ros-noetic-controller-manager \ +ros-noetic-ros-control ros-noetic-ros-controllers \ +ros-noetic-gazebo-ros-control libpcap-dev ros-noetic-velodyne +2.2 +Creating Catkin Workspace +The first step in using the CAT Vehicle ROS package is to create a catkin workspace. Open a Terminal in +your Ubuntu machine and type the following: +cd ~ +mkdir -p catvehicle_ws/src +cd catvehicle_ws/src +catkin_init_workspace +cd .. +catkin_make +Next, we will clone a few essential repositories that are dependencies for the CAT Vehicle package +git clone https://github.com/jmscslgroup/catvehicle +git clone https://github.com/jmscslgroup/obstaclestopper +git clone https://github.com/jmscslgroup/control_toolbox +2 + +git clone https://github.com/jmscslgroup/sicktoolbox +git clone https://github.com/jmscslgroup/sicktoolbox_wrapper +git clone https://github.com/jmscslgroup/stepvel +git clone https://github.com/jmscslgroup/cmdvel2gazebo +cd catvehicle +git checkout noetic_gazebo-11 +cd ~/catvehicle_ws/ +catkin_make +catkin_make compiles all packages and generates two folders in ~/catvehicle_ws with the name devel and +build. They contain executables and other artifacts to run the program written in ROS packages. +2.3 +Sourcing Workspace to the Environment Path +We also need to tell the terminal where to find the desired program we want to run. For that, we need to +“source” the catvehicle_ws catkin workspace. We do this by typing the following in the terminal: +echo "source ~/catvehicle_ws/devel/setup.bash" >> ~/.bashrc +source ~/.bashrc +Once done, close your terminal and reopen it. To test your installation, type the following in one terminal +roslaunch catvehicle catvehicle_neighborhood.launch +and the following in the second terminal: +gzclient +gzclient should open a Gazebo window that should look like the one shown in Figure 1. +Figure 1: A Gazebo window with an example simulated environment. +3 +CAT Vehicle APIs for Vehicle Control Applications +While this section is not a tutorial on how to use ROS, it is necessary to understand a few basic things about +ROS that can help create some simple control applications. we explain the basics using what is provided +through the CAT Vehicle package. +3 + +Gazebo +口 +X +File +Edit +Camera +View +Window +Help +World +Insert +Layers +EO +GUI +Scene +Spherical Coordinates +Physics +Atmosphere +wind + Models +Lights +Property +Value +.Real Time Factor: 1.0o +simTime:0:21:00.840Real Time:3.1 +The Launch File +ROS provides a methodology to execute a specialized program called ROS nodes through launch files. ROS +nodes do some meaningful tasks (such as executing a control law) and publish messages on a named topic or +subscribe to some other messages through a named topic. At the same time, some other ROS nodes can +subscribe to messages being published through topics. Topics are like slots and nodes put messages on those +slots – some other node can read those slots to get messages. +Launch files have an extension .launch and they are generally in the launch directory of a ROS package. In +the case of the CAT Vehicle package, consider the launch file catvehicle_empty.launch. It can be used to +create a simulation by typing the following in a terminal +roslaunch catvehicle catvehicle_empty.launch +To see the visual, we type the following in another terminal: +gzclient +The above command launches a window in the Gazebo program showing a virtual world with a ground plane +and the center coordinates as shown in Figure 2. +Figure 2: An empty Gazebo window. +3.2 +Spawning a Vehicle +Spawning a vehicle in the virtual world is done using the catvehicle_spawn.launch file in another terminal. +roslaunch catvehicle catvehicle_spawn.launch +By default, it creates a vehicle at the origin with the name catvehicle. The launch file provides several +command line arguments that can be revealed by pressing the tab a couple of times after typing roslaunch +catvehicle catvehicle_spawn.launch in the terminal. We have the following command line arguments: +• camera_left: to enable the left camera mounted on the car. +• camera_right: to enable the right camera mounted on the car. +• laser_sensor: to enable front 2-D Lidar sensor. +• obstaclestopper: to enable a custom control node that prevents collision. +• pitch: specify the pitch in radian. +• robot: the name of the car. you must specify a unique name when spawning multiple cars in the +simulation. +• roll: specify the roll in radian. +• triclops: enable front-mounted camera on the car. +• updateRate: specify the update rate of speed of the car published. +4 + +Gazebo +口 +X +File +Edit +Camera +View +Window +HeLp +World +Insert +Layers +GUI +Scene +Spherical Coordinates +Physics +Atmosphere +wind + Models +Lights +Property +Value +Real Time Factor:1.00 +simTime:D:01:31.550Real Time::• velodyne_points: enable 3D Velodyne Lidar sensor. +• X: specify the X coordinate of the car. +• Y: specify the Y coordinate of the car. +• yaw: specify the yaw of the car in radian. +• Z: specify the Z coordinate of the car. +With some of the most essential options, we can spawn a car with the following command-line arguments: +roslaunch catvehicle spawn.launch robot:=ego X:=0.0 laser_sensor:=true +The above command spawns a car at the center with the name ego and a front 2D Lidar sensor enabled. +Figure 3 displays the outcome. +Figure 3: A car spawned at the center with a 2-D laser sensor +3.3 +Important ROStopics in the CAT Vehicle package +To develop a control application, we will need to know about some important ROS topics. A full list of topics +can be obtained by typing rostopic list and anything that starts with /ego are topics associated with +the above car we spawned. /ego/vel is where we get the current driving speed of the car on its linear.x +component. Note that each topic has a message type that is equivalent to a C++ structure. You can see the +message types of each topic in the output of the rostopic list. Interested readers can learn more about +ROS messages at http://wiki.ros.org/msg. The relative speed of any car being followed by a car directly in +its front can be found on the linear.z component of /ego/rel_vel. It will be zero if there is no car in the +front. Headway distance of the leader car in front of the ego car is obtained on the topic /ego/lead_dist on +the data component. A control command to the car can be sent on the topic /ego/cmd_vel where you can +specify speed on the linear.x component and steering angle on the angular.z component. +4 +Controller Modeling Example +For controller modeling, we take the approach of model-based design using Simulink software which is a part +of Mathworks’ MATLAB. Simulink provides a library of blocks for specific purposes. One such blocks are +ROS toolbox that can be used for creating controller models. +4.1 +Modeling in Simulink +Open MATLAB, in the MATLAB command prompt, type simulink. Select “Create Model” in the Blank +Model option. In the empty model workspace, you can see Library Browser where you can choose, +drag-and-drop blocks to perform certain tasks. We are interested in blocks from ROS Toolbox. Note that +my example is built in MATLAB 2022b. I am interested in a very stupid velocity control shown in Equation 1 +5 + +Gazebo +口 +X +File +Edit +Camera +View +Window +HeLp +World +Insert +Layers +EO +GUI +Scene +Spherical Coordinates +Physics +Atmosphere +wind + Models + Lights +Property +Value +.Real Time Factor: 0.93 +Sim Time:0:00:22.890 Real Time:0:ewhich I arbitrarily came up with. This control law is merely for following a vehicle in its front if there is one. +vcmd = +� +� +� +� +� +r + 0.5vlead +if h > 30 +r +if h = 30 +r − 0.5vlead +if h < 30 +(1) +In Equation (1), r is the desired velocity for the ego vehicle (The ego vehicle is the one we are interested +in controlling). vlead is the speed of the vehicle or an object directly in the range of the ego vehicle’s front +LiDAR sensor. vlead is reconstructed from LiDAR data by differentiating headway h (that is available on the +/ego/lead_dist topic) and adding to the ego’s current velocity v (obtained from the topic /ego/vel). Differ- +entiated relative velocity is published on /ego/rel_veltopic. vcmd is published on the topic /ego/cmd_vel. +Note that /ego/cmd_vel and /ego/vel are different because a vehicle has dynamics so it won’t exactly be +driving with what it is commanded to do so. We have a hidden transfer function to represent the vehicle +dynamics but we don’t model it separately. It is done by rigid body dynamics implemented in the CAT +Vehicle package. +A full model of Equation (1) in Simulink is shown in Figure 4 where the Equation is contained in MATLAB +function block. +Figure 4: The Simulink model of the velocity controller +4.2 +Settings for the model +In the Simulink Simulation tab, we set the stop time of the simulation to be inf. Now, we specify ROS- +related parameters in the Modeling tab -> Model Settings to generate a ROS node. In Model Settings, +we use the following settings: +1. Solver -> Type: Fixed Step, Fixed-Step Size: 0.05 (which is in seconds) +2. Hardware implementation-> Hardware Board: +Robot Operating System, +Target Hardware +Resources-> Build Options: +Build and Load, +Catkin Workspace: +~/catvehicle_ws/ (or +/home//catvehicle_ws/) +Then we press OK. We save the model as velocity_control.slx. The Simulink file used in this example +can be downloaded from https://github.com/rahulbhadani/medium.com/blob/master/10-30-2022/velocit +y_control.slx. +6 + +sNew +Msg +vel + +geometry_msgs/Twist +lead_vel +sNew +headway +cmd_vel +:= Linear.X +Bus +ROS +Msg +rel_vel +velocity_control + +cmd_vel += Angular.Z +VelocityController +sNew +lead_dist +Msg + +ROS +Value +ErrorCode4.3 +Generating ROS node and corresponding launchfile from the Simulink +model +To generate the ROS node, we type roscore in a terminal window, and then in the Simulink ROS tab, press +Build & Load. It compiles the model and generates a C++ standalone ROS node in ~/catvehicle_ws/src. +The first step in running the simulation is to create a launch file. We first create a new text file in an editor +and copy the following code: + + + + + + + + + +We save the text file as velocity_control.launch in the launch folder of the catvehicle package (which +may be in the ~/catvehicle_ws/src/catvehicle/launch directory). +4.4 +Simulation Setup +We consider a two-vehicle simulation where the first vehicle or the leader vehicle drives with an open loop +trajectory specified from a data file. The data file can be downloaded from https://github.com/rahulbh +adani/medium.com/releases/download/data/test_data.csv The leader vehicle control is executed using +velinjector.launch. For the purpose of this tutorial, we save data in the home directory. A whole setup is +illustrated in Figure 4. +Figure 5: Two-car simulation setup for the velocity control example +4.5 +Running the Simulation +To run the simulation with our velocity controller developed in the Simulink, we need to execute several +roslaunch files in different terminal windows. To make things easier, we can use the bash script below which +executes all roslaunch one by one. +#!/bin/bash +gnome-terminal -- roslaunch catvehicle catvehicle_empty.launch +sleep 5 +gnome-terminal -- roslaunch catvehicle catvehicle_spawn.launch robot:=leader X:=30.0 +sleep 5 +gnome-terminal -- gzclient +sleep 5 +gnome-terminal -- roslaunch catvehicle spawn.launch robot:=ego X:=0.0 laser_sensor:=true +sleep 5 +velinjectfile="roslaunch catvehicle velinjector.launch +7 + +Leader, driven in +Ego, to be controlled +open-loop mannercsvfile:=/home/ubuntu/test_data.csv input_type:=CSV +time_col:=Time vel_col:=speed robot:=leader str_angle:=0.0" +gnome-terminal -- $velinjectfile +sleep 5 +gnome-terminal -- roslaunch catvehicle velocity_control.launch robot:=ego r:=2.5 +sleep 5 +gnome-terminal -- rosparam set /execute true +We save the above bash script as run_controller.sh and execute the following to run the simulation +chmod +x run_controller.sh +./run_controller.sh +The above command opens a series of terminal windows and executes all commands one by one. +To log the data in a .bag format, type rosbag record -a. The .bag file can be analyzed using the bagpy +python package. How to use the bagpy package can be found at https://jmscslgroup.github.io/bagpy. To +terminate the simulation, we press Ctrl-C in every terminal window that was opened through the bash script. +To stop the rosbag recording, we also need to press Ctrl-C. +5 +Conclusion and Discussion +In this article, we have discussed how to use the CAT Vehicle ROS package and Simulink’s model-based +design approach to prototype a vehicle control law and test it in the Simulation. The example presented +in this article uses local data corresponding to the ego vehicle, however, a more complex control law that +uses non-local data is also possible. Further other sensor information such as front and side-camera can +be used for improving the decision-making ability of the velocity control. However, based on the current +implementation of the simulator in the CAT Vehicle package, only a velocity control command is possible. If +acceleration-based control law needs to be prototyped, one needs to take an indirect approach of adding an +integrator block in Simulink to integrate the commanded acceleration to produce a velocity command. +References +[1] Rahul Bhadani, Matthew Bunting, Benjamin Seibold, Raphael Stern, Shumo Cui, Jonathan Sprinkle, +Benedetto Piccoli, and Daniel B Work. Real-time distance estimation and filtering of vehicle headways +for smoothing of traffic waves. In Proceedings of the 10th ACM/IEEE International Conference on +Cyber-Physical Systems, pages 280–290, 2019. +[2] Rahul Bhadani, Jonathan Sprinkle, and Matthew Bunting. The CAT Vehicle Testbed: A Simulator with +Hardware in the Loop for Autonomous Vehicle Applications. In Proceedings 2nd International Workshop +on Safe Control of Autonomous Vehicles (SCAV 2018), Porto, Portugal, 10th April 2018, Electronic +Proceedings in Theoretical Computer Science 269, volume 269, pages 32–47, 2018. +[3] Rahul Kumar Bhadani, Benedetto Piccoli, Benjamin Seibold, Jonathan Sprinkle, and Daniel Work. +Dissipation of emergent traffic waves in stop-and-go traffic using a supervisory controller. In 2018 IEEE +Conference on Decision and Control (CDC), pages 3628–3633. IEEE, 2018. +[4] Maria Laura Delle Monache, Thibault Liard, Anais Rat, Raphael Stern, Rahul Bhadani, Benjamin +Seibold, Jonathan Sprinkle, Daniel B Work, and Benedetto Piccoli. Feedback control algorithms for the +dissipation of traffic waves with autonomous vehicles. In Computational Intelligence and Optimization +Methods for Control Engineering, pages 275–299. Springer, 2019. +[5] Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, and Vladlen Koltun. Carla: An open +urban driving simulator. In Conference on robot learning, pages 1–16. PMLR, 2017. +8 + +[6] Yuchuan Du, Chenglong Liu, and Yishun Li. Velocity control strategies to improve automated vehicle +driving comfort. IEEE Intelligent transportation systems magazine, 10(1):8–18, 2018. +[7] Nathan Koenig and Andrew Howard. Design and use paradigms for gazebo, an open-source multi-robot +simulator. In 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)(IEEE +Cat. No. 04CH37566), volume 3, pages 2149–2154. IEEE, 2004. +[8] Nathan Lichtle, Eugene Vinitsky, George Gunter, Akash Velu, and Alexandre M Bayen. Fuel consumption +reduction of multi-lane road networks using decentralized mixed-autonomy control. In 2021 IEEE +International Intelligent Transportation Systems Conference (ITSC), pages 2068–2073. IEEE, 2021. +[9] Maria Michalowska and Mariusz Oglozinski. Autonomous vehicles and road safety. In International +Conference on Transport Systems Telematics, pages 191–202. Springer, 2017. +[10] Navid Mohajer, Saeid Nahavandi, Hamid Abdi, and Zoran Najdovski. Enhancing passenger comfort in +autonomous vehicles through vehicle handling analysis and optimization. IEEE Intelligent Transportation +Systems Magazine, 13(3):156–173, 2020. +[11] Yanyan Qin, Hao Wang, and Bin Ran. Stability analysis of connected and automated vehicles to +reduce fuel consumption and emissions. Journal of Transportation Engineering, Part A: Systems, +144(11):04018068, 2018. +[12] Morgan Quigley, Ken Conley, Brian Gerkey, Josh Faust, Tully Foote, Jeremy Leibs, Rob Wheeler, +Andrew Y Ng, et al. Ros: an open-source robot operating system. In ICRA workshop on open source +software, volume 3, page 5. Kobe, Japan, 2009. +[13] Shital Shah, Debadeepta Dey, Chris Lovett, and Ashish Kapoor. Airsim: High-fidelity visual and physical +simulation for autonomous vehicles. In Field and service robotics, pages 621–635. Springer, 2018. +[14] Raphael E Stern, Shumo Cui, Maria Laura Delle Monache, Rahul Bhadani, Matt Bunting, Miles Churchill, +Nathaniel Hamilton, Hannah Pohlmann, Fangyu Wu, Benedetto Piccoli, et al. Dissipation of stop-and-go +waves via control of autonomous vehicles: Field experiments. Transportation Research Part C: Emerging +Technologies, 89:205–221, 2018. +[15] Cathy Wu, Aboudy Kreidieh, Kanaad Parvate, Eugene Vinitsky, and Alexandre M Bayen. Flow: Archi- +tecture and benchmarking for reinforcement learning in traffic control. arXiv preprint arXiv:1710.05465, +10, 2017. +[16] Ge Zhu, Yuche Chen, and Jiali Zheng. Modelling the acceptance of fully autonomous vehicles: a +media-based perception and adoption model. Transportation research part F: traffic psychology and +behaviour, 73:80–91, 2020. +9 + diff --git a/lNE3T4oBgHgl3EQfhwqh/content/tmp_files/load_file.txt b/lNE3T4oBgHgl3EQfhwqh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc79f93617784ea6bfc5f7ce8f3d8f34621fbb42 --- /dev/null +++ b/lNE3T4oBgHgl3EQfhwqh/content/tmp_files/load_file.txt @@ -0,0 +1,289 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf,len=288 +page_content='Prototyping Vehicle Control Applications Using the CAT Vehicle Simulator Rahul Bhadani∗ Jonathan Sprinkle† Abstract This paper demonstrates the integration model-based design approaches for vehicle control, with validation in a freely available open-source simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Continued interest in autonomous vehicles and their deployment is driven by the potential benefits of their use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' However, it can be challenging to transition new theoretical approaches into unknown simulation environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Thus, it is critical for experts from other fields, whose insights may be necessary to continue to advance autonomy, to be able to create control applications with the potential to transition to practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In this article, I will explain how to use the CAT Vehicle simulator and ROS packages to create and test vehicle controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The methodology of developing the control system in this article takes the approach of model-based design using Simulink, and the ROS Toolbox, followed by code generation to create a standalone C++ ROS node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Such ROS nodes can be integrated through roslaunch in the CAT Vehicle ROS package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 1 Introduction Recent advances in computing, control, and sensor technology have brought autonomous systems – especially autonomous ground vehicles (AV) into the limelight of not only academic research but also media [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The goal of autonomous vehicle control and related research is to improve passenger comfort and safety [6, 10] and reduce road accidents[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' At the same time, some other researchers have been investigating the use of autonomous vehicle control in reducing traffic congestion [4, 14, 1, 3] and fuel consumption reduction [8, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Such objectives and research endeavors encompass a mix of simulation study as well as experimental research using physical platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' While there are several simulation software and packages have been developed to prototype autonomous vehicle control – both general-purpose simulators such as AirSim [13], CARLA [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' and application-specific simulators such as CAT Vehicle [2], and [15], not all simulators are created equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Some provide the ability to prototype a wide variety of use cases but at the same arduous and difficult to get familiar with, while others are limited in use cases but easier to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In this article, we present ways in which a previously proposed autonomous vehicle simulator CAT Vehicle, written as a ROS package [12] can be used to prototype longitudinal vehicle control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The CAT Vehicle simulator is a multi-vehicle simulator that uses rigid body dynamics from the Gazebo physics engine [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The methodology presented in this article takes the approach of model-based design using Mathworks’ Simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Simulink provides ROS Toolbox that can be used to prototype ROS components along with custom control law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Further, Simulink allows code-generation of C++ standalone ROS node provides open-source C++ code, capable of executing on any Ubuntu machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The rest of the article is divided into the following parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Section 2 provides a step-by-step guide to installing the CAT Vehicle ROS package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Section 3 provides a brief overview of CAT Vehicle APIs available for controller prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Section 4 provides an example of vehicle control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We end the article with a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' ∗Vanderbilt University, rahul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bhadani@vanderbilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='edu, rahulbhadani@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='edu †Vanderbilt University, jonathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='sprinkle@vanderbilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='04574v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='RO] 11 Jan 2023 2 Installing CAT Vehicle Package The CAT Vehicle is a ROS-based simulator written as a ROS package to facilitate the development of autonomous vehicle control applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The simulator utilizes Gazebo 3D world and ROS tools for deploying and testing a control application in a 3D environment with realistic vehicle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In this section, we provide a hands-on on how to install the CAT Vehicle ROS package that will help you in getting started with writing an autonomous control application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The examples presented in this article use ROS Noetic on Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='1 Installing ROS Noetic Open the terminal, and execute the following commands sudo sh -c \'echo "deb http://packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='org/ros/ubuntu $(lsb_release -sc) main" \\ > /etc/apt/sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='d/ros-latest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content="list' sudo apt install curl # if you haven't already installed curl curl -s https://raw." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='githubusercontent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/ros/rosdistro/master/ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='asc | \\ sudo apt-key add - sudo apt-get update sudo apt install ros-noetic-desktop-full echo "source /opt/ros/noetic/setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bash" >> ~/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bashrc sudo apt install python3-rosdep python3-rosinstall \\ python3-rosinstall-generator python3-wstool build-essential python3-rosdep Once successfully executed above commands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' close the terminal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' reopen it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' and execute the following command sudo rosdep init rosdep update In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' we require a few additional packages that can be installed using the following command: sudo apt-get install python-yaml sudo apt-get install ros-noetic-controller-manager \\ ros-noetic-ros-control ros-noetic-ros-controllers \\ ros-noetic-gazebo-ros-control libpcap-dev ros-noetic-velodyne 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='2 Creating Catkin Workspace The first step in using the CAT Vehicle ROS package is to create a catkin workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Open a Terminal in your Ubuntu machine and type the following: cd ~ mkdir -p catvehicle_ws/src cd catvehicle_ws/src catkin_init_workspace cd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='. catkin_make Next, we will clone a few essential repositories that are dependencies for the CAT Vehicle package git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/catvehicle git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/obstaclestopper git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/control_toolbox 2 git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/sicktoolbox git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/sicktoolbox_wrapper git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/stepvel git clone https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/jmscslgroup/cmdvel2gazebo cd catvehicle git checkout noetic_gazebo-11 cd ~/catvehicle_ws/ catkin_make catkin_make compiles all packages and generates two folders in ~/catvehicle_ws with the name devel and build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' They contain executables and other artifacts to run the program written in ROS packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='3 Sourcing Workspace to the Environment Path We also need to tell the terminal where to find the desired program we want to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' For that, we need to “source” the catvehicle_ws catkin workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We do this by typing the following in the terminal: echo "source ~/catvehicle_ws/devel/setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bash" >> ~/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bashrc source ~/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bashrc Once done, close your terminal and reopen it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' To test your installation, type the following in one terminal roslaunch catvehicle catvehicle_neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch and the following in the second terminal: gzclient gzclient should open a Gazebo window that should look like the one shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Figure 1: A Gazebo window with an example simulated environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 3 CAT Vehicle APIs for Vehicle Control Applications While this section is not a tutorial on how to use ROS, it is necessary to understand a few basic things about ROS that can help create some simple control applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' we explain the basics using what is provided through the CAT Vehicle package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 3 Gazebo 口 X File Edit Camera View Window Help World Insert Layers EO GUI Scene Spherical Coordinates Physics Atmosphere wind Models Lights Property Value .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='Real Time Factor: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='0o simTime:0:21:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='840Real Time:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='1 The Launch File ROS provides a methodology to execute a specialized program called ROS nodes through launch files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' ROS nodes do some meaningful tasks (such as executing a control law) and publish messages on a named topic or subscribe to some other messages through a named topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' At the same time, some other ROS nodes can subscribe to messages being published through topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Topics are like slots and nodes put messages on those slots – some other node can read those slots to get messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Launch files have an extension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch and they are generally in the launch directory of a ROS package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In the case of the CAT Vehicle package, consider the launch file catvehicle_empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' It can be used to create a simulation by typing the following in a terminal roslaunch catvehicle catvehicle_empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch To see the visual, we type the following in another terminal: gzclient The above command launches a window in the Gazebo program showing a virtual world with a ground plane and the center coordinates as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Figure 2: An empty Gazebo window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='2 Spawning a Vehicle Spawning a vehicle in the virtual world is done using the catvehicle_spawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch file in another terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' roslaunch catvehicle catvehicle_spawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch By default, it creates a vehicle at the origin with the name catvehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The launch file provides several command line arguments that can be revealed by pressing the tab a couple of times after typing roslaunch catvehicle catvehicle_spawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch in the terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We have the following command line arguments: camera_left: to enable the left camera mounted on the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' camera_right: to enable the right camera mounted on the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' laser_sensor: to enable front 2-D Lidar sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' obstaclestopper: to enable a custom control node that prevents collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' pitch: specify the pitch in radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' robot: the name of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' you must specify a unique name when spawning multiple cars in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' roll: specify the roll in radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' triclops: enable front-mounted camera on the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' updateRate: specify the update rate of speed of the car published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 4 Gazebo 口 X File Edit Camera View Window HeLp World Insert Layers GUI Scene Spherical Coordinates Physics Atmosphere wind Models Lights Property Value Real Time Factor:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='00 simTime:D:01:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='550Real Time::• velodyne_points: enable 3D Velodyne Lidar sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' X: specify the X coordinate of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Y: specify the Y coordinate of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' yaw: specify the yaw of the car in radian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Z: specify the Z coordinate of the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' With some of the most essential options, we can spawn a car with the following command-line arguments: roslaunch catvehicle spawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch robot:=ego X:=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='0 laser_sensor:=true The above command spawns a car at the center with the name ego and a front 2D Lidar sensor enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Figure 3 displays the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Figure 3: A car spawned at the center with a 2-D laser sensor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='3 Important ROStopics in the CAT Vehicle package To develop a control application, we will need to know about some important ROS topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' A full list of topics can be obtained by typing rostopic list and anything that starts with /ego are topics associated with the above car we spawned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' /ego/vel is where we get the current driving speed of the car on its linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='x component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Note that each topic has a message type that is equivalent to a C++ structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' You can see the message types of each topic in the output of the rostopic list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Interested readers can learn more about ROS messages at http://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='org/msg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The relative speed of any car being followed by a car directly in its front can be found on the linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='z component of /ego/rel_vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' It will be zero if there is no car in the front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Headway distance of the leader car in front of the ego car is obtained on the topic /ego/lead_dist on the data component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' A control command to the car can be sent on the topic /ego/cmd_vel where you can specify speed on the linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='x component and steering angle on the angular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='z component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 4 Controller Modeling Example For controller modeling, we take the approach of model-based design using Simulink software which is a part of Mathworks’ MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Simulink provides a library of blocks for specific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' One such blocks are ROS toolbox that can be used for creating controller models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='1 Modeling in Simulink Open MATLAB, in the MATLAB command prompt, type simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Select “Create Model” in the Blank Model option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In the empty model workspace, you can see Library Browser where you can choose, drag-and-drop blocks to perform certain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We are interested in blocks from ROS Toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Note that my example is built in MATLAB 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' I am interested in a very stupid velocity control shown in Equation 1 5 Gazebo 口 X File Edit Camera View Window HeLp World Insert Layers EO GUI Scene Spherical Coordinates Physics Atmosphere wind Models Lights Property Value .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='Real Time Factor: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='93 Sim Time:0:00:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='890 Real Time:0:ewhich I arbitrarily came up with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' This control law is merely for following a vehicle in its front if there is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' vcmd = � � � � � r + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='5vlead if h > 30 r if h = 30 r − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='5vlead if h < 30 (1) In Equation (1), r is the desired velocity for the ego vehicle (The ego vehicle is the one we are interested in controlling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' vlead is the speed of the vehicle or an object directly in the range of the ego vehicle’s front LiDAR sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' vlead is reconstructed from LiDAR data by differentiating headway h (that is available on the /ego/lead_dist topic) and adding to the ego’s current velocity v (obtained from the topic /ego/vel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Differ- entiated relative velocity is published on /ego/rel_veltopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' vcmd is published on the topic /ego/cmd_vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Note that /ego/cmd_vel and /ego/vel are different because a vehicle has dynamics so it won’t exactly be driving with what it is commanded to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We have a hidden transfer function to represent the vehicle dynamics but we don’t model it separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' It is done by rigid body dynamics implemented in the CAT Vehicle package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' A full model of Equation (1) in Simulink is shown in Figure 4 where the Equation is contained in MATLAB function block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Figure 4: The Simulink model of the velocity controller 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='2 Settings for the model In the Simulink Simulation tab, we set the stop time of the simulation to be inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Now, we specify ROS- related parameters in the Modeling tab -> Model Settings to generate a ROS node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In Model Settings, we use the following settings: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Solver -> Type: Fixed Step, Fixed-Step Size: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='05 (which is in seconds) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Hardware implementation-> Hardware Board: Robot Operating System, Target Hardware Resources-> Build Options: Build and Load, Catkin Workspace: ~/catvehicle_ws/ (or /home//catvehicle_ws/) Then we press OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We save the model as velocity_control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='slx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The Simulink file used in this example can be downloaded from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/rahulbhadani/medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/blob/master/10-30-2022/velocit y_control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='slx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 6 sNew Msg vel geometry_msgs/Twist lead_vel sNew headway cmd_vel := Linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='X Bus ROS Msg rel_vel velocity_control cmd_vel = Angular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='Z VelocityController sNew lead_dist Msg ROS Value ErrorCode4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='3 Generating ROS node and corresponding launchfile from the Simulink model To generate the ROS node, we type roscore in a terminal window, and then in the Simulink ROS tab, press Build & Load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' It compiles the model and generates a C++ standalone ROS node in ~/catvehicle_ws/src.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The first step in running the simulation is to create a launch file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' We first create a new text file in an editor and copy the following code: We save the text file as velocity_control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch in the launch folder of the catvehicle package (which may be in the ~/catvehicle_ws/src/catvehicle/launch directory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='4 Simulation Setup We consider a two-vehicle simulation where the first vehicle or the leader vehicle drives with an open loop trajectory specified from a data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The data file can be downloaded from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/rahulbh adani/medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='com/releases/download/data/test_data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='csv The leader vehicle control is executed using velinjector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' For the purpose of this tutorial, we save data in the home directory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' A whole setup is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Figure 5: Two-car simulation setup for the velocity control example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='5 Running the Simulation To run the simulation with our velocity controller developed in the Simulink, we need to execute several roslaunch files in different terminal windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' To make things easier, we can use the bash script below which executes all roslaunch one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' #!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='/bin/bash gnome-terminal -- roslaunch catvehicle catvehicle_empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch sleep 5 gnome-terminal -- roslaunch catvehicle catvehicle_spawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch robot:=leader X:=30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='0 sleep 5 gnome-terminal -- gzclient sleep 5 gnome-terminal -- roslaunch catvehicle spawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch robot:=ego X:=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='0 laser_sensor:=true sleep 5 velinjectfile="roslaunch catvehicle velinjector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch 7 Leader, driven in Ego, to be controlled open-loop mannercsvfile:=/home/ubuntu/test_data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='csv input_type:=CSV time_col:=Time vel_col:=speed robot:=leader str_angle:=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='0" gnome-terminal -- $velinjectfile sleep 5 gnome-terminal -- roslaunch catvehicle velocity_control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='launch robot:=ego r:=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='5 sleep 5 gnome-terminal -- rosparam set /execute true We save the above bash script as run_controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='sh and execute the following to run the simulation chmod +x run_controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='sh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='/run_controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='sh The above command opens a series of terminal windows and executes all commands one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' To log the data in a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bag format, type rosbag record -a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='bag file can be analyzed using the bagpy python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' How to use the bagpy package can be found at https://jmscslgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='io/bagpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' To terminate the simulation, we press Ctrl-C in every terminal window that was opened through the bash script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' To stop the rosbag recording, we also need to press Ctrl-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 5 Conclusion and Discussion In this article, we have discussed how to use the CAT Vehicle ROS package and Simulink’s model-based design approach to prototype a vehicle control law and test it in the Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The example presented in this article uses local data corresponding to the ego vehicle, however, a more complex control law that uses non-local data is also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Further other sensor information such as front and side-camera can be used for improving the decision-making ability of the velocity control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' However, based on the current implementation of the simulator in the CAT Vehicle package, only a velocity control command is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' If acceleration-based control law needs to be prototyped, one needs to take an indirect approach of adding an integrator block in Simulink to integrate the commanded acceleration to produce a velocity command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' References [1] Rahul Bhadani, Matthew Bunting, Benjamin Seibold, Raphael Stern, Shumo Cui, Jonathan Sprinkle, Benedetto Piccoli, and Daniel B Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Real-time distance estimation and filtering of vehicle headways for smoothing of traffic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In Proceedings of the 10th ACM/IEEE International Conference on Cyber-Physical Systems, pages 280–290, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [2] Rahul Bhadani, Jonathan Sprinkle, and Matthew Bunting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' The CAT Vehicle Testbed: A Simulator with Hardware in the Loop for Autonomous Vehicle Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In Proceedings 2nd International Workshop on Safe Control of Autonomous Vehicles (SCAV 2018), Porto, Portugal, 10th April 2018, Electronic Proceedings in Theoretical Computer Science 269, volume 269, pages 32–47, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [3] Rahul Kumar Bhadani, Benedetto Piccoli, Benjamin Seibold, Jonathan Sprinkle, and Daniel Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Dissipation of emergent traffic waves in stop-and-go traffic using a supervisory controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In 2018 IEEE Conference on Decision and Control (CDC), pages 3628–3633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [4] Maria Laura Delle Monache, Thibault Liard, Anais Rat, Raphael Stern, Rahul Bhadani, Benjamin Seibold, Jonathan Sprinkle, Daniel B Work, and Benedetto Piccoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In Computational Intelligence and Optimization Methods for Control Engineering, pages 275–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [5] Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, and Vladlen Koltun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Carla: An open urban driving simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In Conference on robot learning, pages 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 8 [6] Yuchuan Du, Chenglong Liu, and Yishun Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Velocity control strategies to improve automated vehicle driving comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' IEEE Intelligent transportation systems magazine, 10(1):8–18, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [7] Nathan Koenig and Andrew Howard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Design and use paradigms for gazebo, an open-source multi-robot simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)(IEEE Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 04CH37566), volume 3, pages 2149–2154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' IEEE, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [8] Nathan Lichtle, Eugene Vinitsky, George Gunter, Akash Velu, and Alexandre M Bayen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Fuel consumption reduction of multi-lane road networks using decentralized mixed-autonomy control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In 2021 IEEE International Intelligent Transportation Systems Conference (ITSC), pages 2068–2073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [9] Maria Michalowska and Mariusz Oglozinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Autonomous vehicles and road safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In International Conference on Transport Systems Telematics, pages 191–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [10] Navid Mohajer, Saeid Nahavandi, Hamid Abdi, and Zoran Najdovski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Enhancing passenger comfort in autonomous vehicles through vehicle handling analysis and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' IEEE Intelligent Transportation Systems Magazine, 13(3):156–173, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [11] Yanyan Qin, Hao Wang, and Bin Ran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Stability analysis of connected and automated vehicles to reduce fuel consumption and emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Journal of Transportation Engineering, Part A: Systems, 144(11):04018068, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [12] Morgan Quigley, Ken Conley, Brian Gerkey, Josh Faust, Tully Foote, Jeremy Leibs, Rob Wheeler, Andrew Y Ng, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Ros: an open-source robot operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In ICRA workshop on open source software, volume 3, page 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Kobe, Japan, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [13] Shital Shah, Debadeepta Dey, Chris Lovett, and Ashish Kapoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Airsim: High-fidelity visual and physical simulation for autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' In Field and service robotics, pages 621–635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [14] Raphael E Stern, Shumo Cui, Maria Laura Delle Monache, Rahul Bhadani, Matt Bunting, Miles Churchill, Nathaniel Hamilton, Hannah Pohlmann, Fangyu Wu, Benedetto Piccoli, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies, 89:205–221, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [15] Cathy Wu, Aboudy Kreidieh, Kanaad Parvate, Eugene Vinitsky, and Alexandre M Bayen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Flow: Archi- tecture and benchmarking for reinforcement learning in traffic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content='05465, 10, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' [16] Ge Zhu, Yuche Chen, and Jiali Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Modelling the acceptance of fully autonomous vehicles: a media-based perception and adoption model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' Transportation research part F: traffic psychology and behaviour, 73:80–91, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE3T4oBgHgl3EQfhwqh/content/2301.04574v1.pdf'} diff --git a/ltE5T4oBgHgl3EQfig-m/content/tmp_files/2301.05649v1.pdf.txt b/ltE5T4oBgHgl3EQfig-m/content/tmp_files/2301.05649v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15856f3f16277ca52c73e0187b92b0c0897e119e --- /dev/null +++ b/ltE5T4oBgHgl3EQfig-m/content/tmp_files/2301.05649v1.pdf.txt @@ -0,0 +1,1289 @@ +Filtering Down to Size: A Theory of Consideration∗ +Tonna Emenuga† +January 16, 2023 +Abstract +The standard rational choice model describes individuals as making choices by +selecting the best option from a menu. A wealth of evidence instead suggests that +individuals often filter menus into smaller sets — consideration sets — from which choices +are then made. I provide a theoretical foundation for this phenomenon, developing a +formal language of axioms to characterize how consideration sets are formed from menus. +I posit that consideration filters — mappings that translate a menu into one of its subsets +— capture this process, and I introduce several properties that consideration filters can +have. I then extend this core model to provide linkages with the sequential choice and +rational attention literatures. Finally, I explore whether utility representation is feasible +under this consideration model, conjecturing necessary and sufficient conditions for +consideration-mediated choices to be rationalizable. +∗I am grateful to Tomasz Strzalecki for his guidance and mentorship. I thank Jerry Green, Shengwu Li, +David Laibson, Yannai Gonczarowski, Nathaniel Hendren, Jeff Miron, Matthew Rabin, Angie Acquatella, +Sejal Aggarwal, Shani Cohen, Roberto Colarieti, Benny Goldman, Zo¨e Hitzig, Martin Koenen, Pierfrancesco +Mei, Akash Nandi, Cassidy Shubatt, Chris Walker and numerous seminar and workshop participants in +the Harvard Economics Department for helpful discussions. This work was supported by a grant from the +Harvard College Research Program. All errors are my own. +†Harvard University. Email: tonnaemenuga@college.harvard.edu +1 +arXiv:2301.05649v1 [econ.TH] 13 Jan 2023 + +1 +Introduction +Economists’ ability to deduce individual preferences from observed choices informs +much of microeconomic theory. In particular, the foundational concept of revealed +preference asserts that an individual’s choice of an item (an alternative) from a set of +options (a menu) is reflective of their underlying preference, allowing economists to +determine preferences simply from a summary of choices made from various menus. For +example, if when presented with two cars of equal cost1 — one red and one gray — a +consumer purchases the red car, revealed preference tells us that the they must prefer +the red car over the gray car. +In this standard model, individuals observe menus, analyze all available options, +and make choices that are most consistent with their tastes. This model, which forms +the foundation of rational choice theory as well as applied analysis across a variety of +subfields, relies on an assumption referred to as “full consideration.” This means that, +when analyzing a menu, a decision-making individual considers every item available +before making a choice. In the car choice example, this entails assuming the individual +considered the gray car, or was at least was aware of its presence. +Contrary to this assumption, a wealth of empirical evidence demonstrates that, +often, individuals will only consider a subset of a menu before making a choice. In many +cases, the entire menu is not fully examined: only the few alternatives which come to +mind are fully considered by the individual. This is known as limited consideration. +Rather than making a choice directly from a menu, individuals may filter menus into +consideration sets, which are smaller groupings of alternatives from which choices are +eventually made. +This filtered decision process creates some challenges for the classic revealed pref- +erence approach. For example, if an individual is presented with a menu consisting +of alternatives {x, y, z} and chooses y, can one state, as usual, that the individual +necessarily prefers y over x and z? In the case of a filtered process, in which limited +consideration holds, one cannot make this claim. Alternatives x and z may not have +been in the individual’s consideration set — x and z were not examined — and thus +the preference relation of y to x and z remains unknown. +Limited consideration also jeopardizes the ability attain utility representation of +individual preferences. As is well known,2 the ability to construct utility functions +corresponding to observed choices relies on preferences being both complete3 and +1Throughout, I will assume all alternatives within a menu are affordable; this is a standard assumption in +the literature. +2This is the utility representation theorem. +3Preferences are complete if there is a well-defined relation between any two alternatives in a menu. Given +two options, an individual with complete preferences will weakly prefer one of them, otherwise they are +indifferent. +2 + +transitive.4 +In the case of limited consideration, completeness is most clearly in +question, and transitivity can fail as well.5 To better understand these issues, I provide +in this paper a general model that can form the theoretical basis of work aimed at +reconciling the standard model with the challenges imposed by limited consideration. +In doing so, I develop a formal language of axioms to characterize how individuals +may not make choices from menus, but rather from consideration sets. I imagine that +individuals observe menus, filter them into smaller menus, and then make rational +choices from these smaller menus which are known as consideration sets. I posit that +the process by which individuals go from menus to consideration sets is mediated by +consideration filters,6 which are mappings that translate a given menu into one of its +subsets. +Such a model has been developed in the literature. The idea that only a subset +of available options are considered dates back as far as the Simon (1955) model of +satisficing and optimal stopping, whereby an individual browses options only up until +an acceptable one is found; at that point, search ceases. Masatlioglu et al. (2012) and +Masatlioglu and Nakajima (2015) develop models to capture the process of limited +consideration, focusing on the ability to infer consideration sets from observed choices +under limited consideration. +On a normative level Cherepanov et al. (2013) present a model in which individuals +only consider alternatives that can be rationalized. Ridout (2021) axiomatizes the +decision-making process of an individual “choosing for the right reasons,” modeling a +decision maker who only makes choices that can be justified to others. +Such axiomatic formulations forms a solid theoretical grounding to make sense of +much of the applied work on consideration. In particular, Erdem and Keane (1996), +Hauser and Wernerfelt (1990), and Roberts and Lattin (1991) test structural models +to describe the formation of consumers’ consideration sets over goods. More recently, +Abaluck and Gruber (2016) apply limited consideration to choices over healthcare plans, +and Abaluck and Adams-Prassl (2021) develop a structural demand model based on +limited consideration. +I contribute to the literature by uniting much of the above work into a general +framework for understanding limited consideration. I begin by formally outlining the +main feature of the model: individuals’ choice processes exhibit two mappings, one from +menus to consideration sets and another from consideration sets to eventual choices. +The timing of limited consideration features an individual observing a menu, considering +some subset of the available alternatives (the consideration set), and then making a +choice from said consideration set. +4Preferences are transitive if x ≿ y and y ≿ z implies x ≿ z. +5Masatlioglu et al. (2012) provide several examples. +6Also referred to as consideration set mappings in the literature. +3 + +I model consideration sets as generated by consideration filters, functions that map +the set of menus into subsets of themselves, thereby capturing the process of “filtering +out” certain alternatives according to some heuristic. A consideration heuristic is simply +a rule that determines which alternatives in a given menu are in the consideration set. +For example, an individual intending to select a banana from a grocery store is unlikely +to examine every banana in the produce section; rather, they may simply consider those +bananas which are in the front row. In this case, the menu is the set of all bananas, +the consideration set is the front row, and, importantly, the consideration filter is the +guiding heuristic “only look at bananas in the front row.” Consideration heuristics +may be intentional, in the case of the individual looking to purchase a banana, or +subconscious, in the case of an individual aiming to select a fruit of any kind, and +failing to see that there are apples, which they might very well prefer, in the next aisle. +The list of potential consideration heuristics is clearly enormous, at least as large +as the number of decision-making rules and behavioral biases that one could model +an individual as having. I outline several generic qualities that such heuristics, or +consideration filter properties, may have as well a the relationships that these properties +have with one another. +One of these properties, which I call Independence of Others (IO), is the focus +of many of the exercises contained in the proceeding sections and in the paper as a +whole. A consideration filter is IO if and only if alternatives in a menu are either always +considered when available, or never considered at all. The sense in which this makes +alternatives “independent” of each other is clear: the presence, or lack thereof, of other +alternatives in a menu has no bearing on whether a particular alternative is in the +final consideration set. IO closely approximates the rational model, insofar as IO filters +generate choices structures that cannot cycle7 and hence can be represented by the +utility function I derive in Section 6. I also present other potential filter properties +corresponding to different consideration heuristics. +I then extend the basic model of consideration to account for the use of filters +in a sequential fashion. The literature on sequential consideration begins with the +Tversky (1972) model of elimination by aspect, whereby one sequentially removes +alternatives from the consideration set based on particular qualities — aspects — that +these alternatives may or may not have. Manzini and Mariotti (2007) model a decision- +making process whereby an agent eliminates inferior alternatives sequentially, applying +a complete and transitive preference profile to the choice set until only one alternative +remains — the final choice. Apesteguia and Ballester (2013) take a game-theoretic +approach, using game trees to characterize which sequential choice processes of the +above sort are rationalizable. I propose a more general model, nesting the above models +7Choices cycle if x is chosen over y and x is chosen over y, yet z is chosen over x, violating transitivity. +4 + +into a general framework for understanding multi-step consideration. +I then develop a second model extension, constructing a “rational attention”-style +analog of the consideration model. This allows me to model preferences that decision- +makers may have over consideration filters themselves, rather than simply over goods. +In principle, there are may be a number of different consideration heuristics that an +individual may employ, and the ultimate choice in large part rests upon which of these +rules is applied to the menu they are presented with. I model an individuals who chooses +which filter to apply to a given menu, weighing two competing forces: the benefit of +a larger consideration set (a larger choice set may raise the chance of a particularly +good alternative being in the menu) and the cost of examining many items (it takes +time and effort to sift through many options). I derive two boundary conditions that +give a flavor for how subsequent work can unite the process of consideration with the +behavioral reality of limited attention. +Finally, I address the ability to rationalize choices that are made under limited +consideration. The key threat to rationality, and utility representation, is completeness +of preferences, which is clearly not the case when only a subset of available alternatives +are considered. Caplin and Dean (2011) discuss this challenge in a search-model setting. +I provide a strong condition, IO, under which consideration-mediated choices can +be represented by a utility function. Relatedly, I also provide a modification to the +Weak Axiom of Revealed Preference (WARP) that matches the consideration setting, +following in the vein of Lleras et al. (2017). Overall, this paper outlines a general +framework for understanding the phenomenon of limited consideration, extends the +basic model in novel ways, and discusses the implications of limited consideration for +utility representation and rational choice theory more broadly. +The paper proceeds as follows. Section 2 outlines the basic model of consideration +filters and resultant consideration sets, proving a language through which the ideas of the +papers can be discussed. Section 3 provides a number of qualities which consideration +filters may have, in particular IO. Sections 4 and 5 represent extensions to the base +model: Sections 4 allows for the use of more than one filter, and Section 5 models +individuals as having preferences over filters. Section 6 discusses the potential for utility +representation of consideration-mediated choices. Section 7 discusses the limitations of +my results and future directions that the literature can take. Section 8 concludes. An +appendix8, containing proofs of results, is also included. +5 + +2 +General Theory of Consideration +This section outlines the overall theory of limited consideration. I introduce the key +players — consideration sets and consideration filters — while providing the surrounding +definitions and axioms to close the model. +2.1 +Overview of Limited Consideration +The basic theory of limited consideration is that an individual will not pay attention to +every option they are presented with before making a choice. The simple translation +of menus to choices is inadequate to capture a key nuance of human decision-making, +namely that attention tends to be “costly” and thus individuals consider only a small +set number of alternatives in the choice process. Between menus and choices, there +exists an intermediate step known as consideration, which defines the smaller set of +alternatives that an individual will examine and ultimately choose from. +Rather than making choices from menus, individuals make choices from consideration +sets, which are nested within the original menus. Figure 2.1 makes clear the relationships +between menus, consideration sets, and choices. +Figure 2.1: Menu, Consideration Set, and Choice +I now provide the formal definitions and axioms to set up the environment. +2.2 +Definitions and Axioms +Equipped with an intuitive notion of the idea of consideration sets, I now define the +mathematical environment in more concrete terms. This section will use concepts that +are quite familiar to the decision theorist and discrete mathematician alike. +6 + +Menu +Consideration +Set +Choice2.2.1 +Set Notation +I use basic set-theoretic concepts to define the terms used in the process consideration- +mediated choice. +Definition 1. The set of alternatives is X. +The set of alternatives represents the total set of alternatives which may be presented +to an individual. For an individual browsing cars at a dealership, X is the total set +of cars in the lot. The set of cars consists of individual cars, which are its constituent +alternatives: +Definition 2. The constituent alternatives within X are known as xi, where xi ∈ X +for all i. +The individual’s final choice will be some alternative xi from the universe X. I +henceforth abuse notation by omitting the subscript i to refer to alternatives by x. +As discussed, individuals must first observe a menu — some subset of the available +alternatives. +Definition 3. The set of menus is M(X). +The set of menus constitutes every combination of alternatives {x1, x2, ..., xn} that +the individual may be presented with. +Axiom 1. There exist 2|X| menus. +Following from the definition of the power set, the set of menus M(X) will necessarily +contain 2|X| menus, where |X| is the cardinality of X, the number of alternatives +contained in it. This includes both the full set, containing every alternative, as well as +the null set ∅. +Definition 4. Generic menus are denoted A and B; A ⊆ B ∈ M(X). +I define two generic menus A and B, which will be used in proceeding sections when +discussing properties of the model. A is a subset of B — every alternative in A also +finds itself in B. I later sections, I will at times make use of other generic menus, such +as Q; in each instance, take Q to be some generic menu with the same properties as A. +2.2.2 +Mappings and Consideration Sets +I now define consideration filters and consideration sets. The consideration model +assumes that individuals observe menus, filter them into consideration sets, and then +make choices. The first step in the process is the mapping from menus to consideration +sets. Formally, a consideration filter is a mapping Γ : M(X) �→ M(X) that takes the +7 + +set of menus and returns corresponding subsets. That is, a consideration filter takes a +menu and returns one of its subsets. For the menu A, the filter returns Γ(A), where +Γ(A) ⊆ A for all menus A ∈ M(X). Recall the generic menus A and B, keeping in +mind that A is a subset of B. A consideration filter, in the most general sense, takes B +and returns some A: +Definition 5. Γ : B �→ A is a consideration filter. +Any function that takes a set and returns one its subsets qualifies as a consideration +filter, capturing the phenomenon of the limited consideration that motivates the +literature. Naturally, such filters can take various forms. For example, Masatlioglu et al. +(2012) restrict their analysis to those filters classified as “attention filters,” while Lleras +et al. (2017) study the case of “choice overload.” +Definition 6. Γ(B) is the consideration set of menu B. +The result of the consideration filter, applied to a menu, is the consideration set, +represented by the red region of Figure 2.1. +Choices, ultimately, are made from +consideration sets rather than menus. +Axiom 2. c(B) ∈ Γ(B) +The above axiom simply establishes that the choice from a menu must also be in +that menu’s consideration set. In essence, choices are made from consideration sets, not +menus. This provides a starting point for any attempts to identify consideration sets from +observed choice data. If an alternative is chosen, one can state with certainty that said +alternative must have been considered, and thus finds itself within the consideration set.8 +In order to determine the structure consideration sets beyond the chosen alternative, +assumptions must be made on the process of consideration, necessitating some language +for describing axioms of consideration filters. The following section provides such a +framework. +3 +Properties of Consideration Filters +This section introduces various properties that consideration filters may have, providing +a language through which consideration filters and consideration sets can be described. +Section 3.1 re-states the mathematical preliminaries, outlining a rigorous vocabulary for +describing the environment of interest. Section 3.2 lists properties that consideration +filters may have, providing examples, descriptions, and formal results regarding their +interrelations when necessary. +8Note that, if an alternative is not chosen, one cannot make any inference on whether it was considered, +without making additional assumptions on the consideration process. +8 + +3.1 +Environment +I reformulate the mathematical environment, using basic notions of sets and set rela- +tionships. I begin with an individual, who goes through a two-step process to make a +choice: the individual is presented with a menu of items, filters it down to a weakly +smaller consideration set, and then makes a final choice. In this section, I focus on the +first portion of the process: that is, the transformation of menus into consideration sets. +There is a set of alternatives, X, which captures the universe of goods that the +individual may potentially observe. The space X is non-empty, instead having con- +stituent elements x ∈ X, each of which is a particular alternative which could be +chosen. However, individuals do not ordinarily observe the entire set of alternatives X: +while this is possible, individuals more commonly are presented with menus, smaller +collections of alternatives which are drawn from the larger menu. +Formally, M(X) denotes the set of menus which can be derived from X, each of +which is a subset of X: for a menu A, A ∈ M(X) ⊆ X. There are 2X potential menus, +including the full set X and the null set ∅. +I refer to A as being a generic menu in M(X) and also make use of a generic menu +B representing a menu that constitutes a superset of menu A: A ⊆ B. +Individuals observe a menu and consider a subset of its alternatives. To describe this, +I introduce a mapping, a consideration filter, which takes an observed menu A ∈ M(X) +and produces another menu Γ(A). Because an individual can only consider alternatives +which are in the original menu, the consideration set is a subset of the original menu: +Γ(A) ⊆ A. The consideration filter is, then: +Γ : M(X) �→ M(X) +The following section, a listing of properties of consideration filters, puts more +structure on the nature of consideration filters. +3.2 +List of Properties +The following list of properties describes various features that a consideration filter +may have. A given consideration filter may satisfy some number of these properties, all +of them, or none at all.. Note that, while these properties are distinct, they are not +necessarily mutually exclusive, nor are they, as a rule, concomitant. +The first two properties represent adaptations of well-known axioms in rational +choice theory, translated here to axiomatize consideration filters rather than choice +functions. In the standard choice literature, axioms are frequently presented in the +following style: +9 + +Example 1. A Generic Choice Axiom If x is chosen from menu A, it must be +chosen from menu B. +The weak axiom, for example, takes this basic form. The idea behind this approach +is that, by making assumptions on the details of the decision process, one can make +claims about which alternatives from menu are chosen from another menu, simply from +taking note of the latter menu’s structure. +In this section, I adapt this familiar axiomatic style to accommodate consideration +rather than the final choice. That is, I present a group of axioms that, under certain +conditions, give clarity on which alternatives from a given menu are in the consideration +set. In doing so, I extend the rigour of axiomatic choice theory into the realm of +consideration, providing the literature with a set of sample axioms to form a basis for +further exploration and applied work. +Each axiom I present represents a property that a consideration filter may have, +outlining a condition under which the consideration set of a menu can, at least in part, +be identified. I pair each axiom with an accompanying example. Throughout these +examples, I make the following assumptions: +• The set of alternatives is X : {1, 2, 3, 4, 5, 6, 7, 8, 9}. +• The set of menus M(X) contains 29 = 512 menus. +• A is a generic menu; A : {1, 2, 3} +• B is a generic menu; B : {1, 2, 3, 4, 5} +These assumptions preserve generality and are made for the purpose of concretizing +each axiom. I also introduce generic menus other than A and B is specific cases. I +begin with Sen’s α: +Definition 7. Sen’s α. Γ satisfies Sen’s α if x ∈ Γ(B) and x ∈ A ⊆ B implies x ∈ +Γ(A). +This is the well-known Sen’s α9 axiom, adapted to the consideration setting. In +short, Sen’s α asserts that, whenever x is considered from a set B, x will also be +considered from all of B’s subsets in which it is present. Accordingly, I call this a +“large-to-small” condition, guaranteeing that any alternative considered from a larger +set will also be considered in its subsets. +Example 2. Sen’s α example +Suppose Γ satisfies Sen’s α. Further suppose there is a menu B = {1, 2, 3, 4, 5} with +consideration set Γ(B) = {2, 3}. If A : {1, 2, 3}, then 2, 3 ∈ Γ(A). +9Also known as Chernoff’s condition. +10 + +This example shows how Sen’s α, when assumed to hold with respect to a given +consideration filter Γ, allows an observer to determine which alternatives are in the +consideration set of another menu. One salient point to note is that the consideration +set of A, in this case, is not limited to 2 and 3. It is possible that 1 is also in the +consideration set of A. However, given Sen’s α, we can only say for sure that 2 and 3 are +in the consideration set of A. These axioms provide a lower bound on the alternatives +within a menu’s consideration set. +Sen’s β is another familiar result, here reformulated to match the consideration filter +environment. +Definition 8. Sen’s β. Γ satisfies Sen’s β if x1, x2 ∈ A ⊆ B and x2 ∈ Γ(B) implies +x1 ∈ Γ(B). +Sen’s β is another classical axiom of decision theory that I analogize to the consider- +ation setting. The axiom asserts what one might call the “take-with-me” property: if +two alternatives are considered from a menu, and then available in a larger menu, then +one cannot be considered without the other10. +Example 3. Sen’s β example +Suppose Γ satisfies Sen’s β. Further suppose there is a menu A = {1, 2, 3} with +consideration set Γ(A) = {2, 3}. If 2 ∈ Γ(B), where B ⊇ A, then 3 ∈ Γ(B). +The next property is a novel axiom I present as a counterpart to Sen’s α. +Definition 9. Condition τ. Γ satisfies Condition τ if x ∈ Γ(A) and A ⊆ B implies +x ∈ Γ(B). +Condition τ “reverses” Sen’s α, considering a “small-to-large case” rather than a +large-to-small. Condition τ states that if an alternative x is in the consideration set +of a menu A, then it is in the consideration set of every superset of A in which it is +present. This gives a characterization of the preservation of consideration sets under +“expansion”. When Condition τ holds, no alternatives are dropped from consideration +during this expansion. +Note that filters satisfying this Condition τ preclude the possibility of choice overload, +the “more is less” phenomenon described by Lleras et al. (2017). The concept of choice +overload posits that adding items to a menu may result in previously-considered +alternatives ceasing to be a part of the new consideration set, mirroring the well-known +literature of “overwhelmed” consumers losing track of alternatives and thus dropping +goods from consideration when confronted with larger menus. Condition τ is a direct +remedy for this, although it does not not apply realistically to every setting of interest. +10Of course, this abstracts away from externally-imposed limits on the size of consideration sets. I address +this in Section 5. +11 + +Example 4. Condition τ example +Suppose Γ satisfies Condition τ. Further suppose there is a menu A = {1, 2, 3} with +consideration set Γ(A) = {2, 3}. For menu B ⊇ A, where B = {1, 2, 3, 4, 5}, 2, 3 ∈ Γ(B). +Definition 10. Independence of Others Γ satisfies Independence of Others (IO) +if the following condition holds: +1. x ∈ Γ(A) ∀ A ∈ M(X) s.t. x ∈ A or +2. x /∈ Γ(A) ∀ A ∈ M(X) +Independence of Others, when true, states that every alternative x ∈ X is either +always considered whenever it is available (i.e. in the menu), otherwise it is never +considered. This gives an intuition for the name: an alternative’s inclusion in the +consideration set is purely independent of the the presence of all other alternatives. +There is no comparative process at this stage of the decision-making process. Naturally, +IO corresponds to cases in which there exist certain properties of goods that are not +subject to change, and individuals form consideration sets based on these unchangeable +qualities. For example, an individual deciding which car to purchase at a dealership, but +restricting their consideration to cars that travel at least 30 miles per gallon, chooses +to consider cars in a way that is IO. Any car in the available set with at least 30 +MPG is in the consideration set, independent of whether any other car is in the set. +Most consideration heuristics are not IO, particularly those which make consideration +dependent on comparison or position in some order. For example, the heuristic “when +shopping for fruits, only consider bananas in the front row” is not IO because moving a +banana’s position affects whether or not it is considered. IO consideration heuristics +can, more so than other rules, generate null consideration sets in the event that no +alternative in the menu satisfies the criterion of interest. +IO matches the standard rational choice model in the sense that revealed preference +analysis is always possible over alternatives that are ever chosen. Recall that the main +threat to revealed preference is incomplete preferences caused by lack of consideration. +IO largely closes this hole by guaranteeing that any alternative that chosen from any +menu is considered across all menus in which it is available. This gives greater scope to +infer preferences from observed choices. +Of course, IO is not realistic in every setting, and this exposes a weakness in +the standard rational choice model. +Any preference ordering taken directly from +observed choices implicitly assumes full consideration, which IO approximates. Thus, +this presents a challenge for economists who maintain the usefulness of the standard +revealed preference framework while naturally disputing the accuracy of IO in describing +real-world decision-making. +12 + +Example 5. IO example +Suppose Γ satisfies IO. Further suppose there is a menu A = {1, 2, 3} with con- +sideration set Γ(A) = {2, 3}. If we have menus B = {1, 2, 3, 4, 5}, Q = {1, 4, 7}, and +S{1, 6, 8}, 1 ∈ Γ(B), then 1 ∈ Γ(Q), and 1 ∈ Γ(S). +I now introduce a result regarding the relationship between IO and two preceding +properties. +Theorem 1. (Sen’s α and Condition τ ⇔ IO) Γ satisfies Sen’s α and Condition τ +if and only if it satisfies IO. +Proof. See appendix. +Intuitively, Sen’s α and Condition τ jointly provide both a “large to small” as well +as a “small to large” condition on filter Γ. The proof strategy relies on the singleton +set {x} being a subset of any menu A that includes x. If x is in the consideration set +of A, Γ(A), one can guarantee consideration of x in any menu that includes by going +“down and up,” a technique I demonstrate in the proof. +The next property, Dynamic Independence of Others (DIO) 11, extends IO, and the +process of consideration, into a dynamic setting. DIO states that, when alternatives in +a menu are filtered in some order, the order does not affect the final consideration set. +To characterize DIO formally, fix the elements of menu A ∈ M(X) so as to consider +every possible ordering of its elements. For example, the A = {1, 2} can equivalently be +represented as A = {2, 1}. For a menu A with |A| elements, there exist |A|! possible +orderings. Call the first ordering A1 and the second A2, giving us a general notation +where An denotes the nth ordering of menu A. For a menu A, imagine an individual is +endowed with |A| units of time, during each of which a singular element is considered. +That is, in period 1, the individual considers the first element, then in period 2 the +second element... until the final element is considered in period |A|. +Dynamic Independence of Others asserts that the final consideration set will be +equal among each such ordering. +Definition 11. Dynamic Independence of Others Γ satisfies Dynamic Indepen- +dence of Others (DIO) if Γ(A1) = Γ(A2) = · · · = Γ(An) where n = |A|!. +DIO rules out the possibility behavioral biases that make choices a function of the +manner in which alternatives are presented, rather than simply on the combination of +alternatives which constitute the observed menu. For this reason, DIO represents a +very strong condition on filtering in dynamic environments. +11Sejal Aggarwal coined the name of this property, helping me replace a previous term that was not as +clear. +13 + +Consideration heuristics satisfying DIO are incompatible with satisficing behavior +of the sort described by Simon (1955). In Simon’s model, consumers view alternatives +x ∈ X in a set order, and make a choice once some alternative meeting quality threshold +is observed. Clearly, order matters here. For example, in a setting with 10 alternatives +where 5 are “satisfying” but only 3 are to be considered, the 3 out of 5 satisfying +alternatives which come first will be considered, while 2 “miss out.” DIO precludes this +from occurring. Therefore, it is not applicable to satisficing scenarios and perhaps a +wide array of models in which one assumes a similar “first-mover” effect. However, DIO +gives the standard consideration filter model a benchmark against which such dynamic +behavioral frameworks can be tested. +Example 6. DIO example +Suppose Γ satisfies DIO and consideration is made in a ordered fashion. Further +suppose there are menus A1 = {1, 2, 3}, A2 = {1, 3, 2}, and A3 = {3, 1, 2}. Γ(A1) = +Γ(A2) = Γ(A3). +The next property, Constant Number, makes use of the notion of cardinality, the +number of elements in a set. As stated earlier, call |M(X)| the number of elements +(cardinality) of a menu M(X), by virtue of which |Γ(A)| is the number of elements +considered from a menu A. For example, if A = {1, 2}, then |A| = 2. +Definition 12. Constant Number (CN) Γ satisfies Constant Number if |Γ(A)| = n +for all A ∈ such that |A| ≥ n. +A constant number consideration heuristic always considers the same number of +alternatives from a menu, provided that the menu has that number of alternatives +available. This requires one to fix some n ∈ W.12 +Example 7. Constant Number example +Suppose Γ satisfies CN, with n = 2. Further suppose there is a menu A = {1, 2, 3}. +Gamma(A) must be {1, 2}, {1, 3}, or {2, 3}. +3.3 +Summary +This section restated the formal language of consideration sets and consideration filters, +allowing me to define various properties which consideration filters can potentially satisfy. +These properties are neither mutually exclusive as a rule nor necessarily concomitant. +The first two properties, Sen’s α and Sen’s β, are adaptations of well-known choice +axioms, herein translated to match the setting of interest: mappings from menus to +12W is the set of natural numbers and 0. +14 + +consideration sets, rather than the standard mapping from menus to choices. I then +introduce several novel properties that consideration filters may have: Condition τ +Independence of Others (IO), Dynamic Independence of Others (DIO), and Constant +Number. +Condition τ reverses Sen’s α to preclude the possibility of the choice overload +phenomenon. IO is a very strong condition, mandating that an alternative is either +always considered when available, or never considered. DIO extends IO to a dynamic +setting. As will be shown is later sections, IO matches the rational model and, while +perhaps unrealistic, is necessary for utility representation under the standard model. +Constant Number makes all consideration sets equal in cardinality. +In the next section, I extend the basic consideration model to allow for the use of +more than one consideration filter on a given menu. +4 +Extension 1: Sequential Consideration +Imagine an individual walks into a store looking for a new pair of shoes. In narrowing +down options, they focuses their attention on shoes in the first aisle. They then only +consider shoes of size 10, further narrowing down their options. How might one model +this process? As it stands, the current consideration model can only account for one +consideration filter, however this example requires two: one from all shoes to those in +the first aisle, and then another from those in the first aisle to those which are also size +10. A extension on the basic framework is needed to account for this. +In this section, I propose and develop a model of sequential consideration. While +the idea of consideration sets has been explored in the literature, two-step, or even +n-step (with more than 2 rounds of narrowing down), has not been formally explored. +Multiple rounds of consideration may better match real-world settings, by covering +scenarios in which individuals narrow down large sets based on multiple criteria. +Recall that, in the original case, we have the following mapping from menus to +consideration sets to eventual choices: +A �→ Γ(A) �→ c(Γ(A)) +Whereas there could exist more than one consideration filter: +M(X) �→ Γ1(A) �→ Γ2(A) �→ c(Γ12(A)) +One might imagine Figure 2.1, in this case, having series of circles within the original +one, each of which represents a new downsizing of the consideration set. I provide +a formal model for this, first by constructing a space of filters that contains any of +15 + +the many consideration filters an individual may use. These consideration filters may +satisfy different properties. For example, one filter may be IO and another may be CN. +Individuals can apply any number of filters to a given menu, narrowing it down to a +final consideration set in multiple steps. +This framework has precedent in the literature. Tversky (1972) introduced the +well-known process of elimination by aspect. In his model, menus contain alternatives, +each which has or does not have some aspect — a desirable features of goods. Individuals +make choices in a multi-step process, where each step involves eliminating all alternatives +which do not have a particular aspect. The process continues until only one alternative is +left, which becomes the final choice. Manzini and Mariotti (2007) imagine an individual +using multiple “rationales” — complete and transitive preference relations — to a +given menu, applying such rationale in a fixed order to arrive at a choice. The authors +then evaluate which sorts of choice functions are consistent with the unique alternative +selected by such a process. Apesteguia and Ballester (2013) provide a taxonomy for the +sort of model specified by Manzini and Mariotti (2007), using game trees to formalize +the idea of sequential rationalizability. +I propose a more general model, nesting the above models into a general framework +for understanding consideration in multiple steps. For example, elimination by aspects +can be modeled as applying a set of IO filters to a given menu, given that each “aspect” +represents an immutable characteristic of an alternative. My model differs from that of +Manzini and Mariotti (2007) in that I do not make use of rational preferences until after +the consideration set has been formed. I take a different approach from Apesteguia and +Ballester (2013), grounding my analysis in the standard choice framework rather than +the style of game-theoretic decision trees. +In addition, I introduce a property, commutativity, borrowing the term from algebra +to characterize relations between two or more filters. I say that two or more filters are +commutative if their successive application to a menu produces the same consideration +set, regardless of the order in which the filters are applied. Commutativity, as a concept, +has applications to any decision setting in which choices may be contingent on the +manner in which information is presented. I show that any number of IO filters are +always commutative, proving two results. +4.1 +Definitions +I begin by defining the space of filters, which contains the mass of consideration filters +that can be applied to a menu. +Definition 13. Space of Filters There exists a space of filters γ, comprising con- +stituent filters Γi ∈ γ. Γi represents the ith filter. +16 + +This defines the space of filters that are available to a decision maker, each of +which narrows a menu according to whatever properties it has and the heuristic that is +implicitly associated with it. Any filter that is applied to a menu comes from γ. The +example above involves Γ1 and Γ2, two are consideration filters which come from γ +and are applied to menu A ∈ M(X). There are multiple ways to represent two filters +applied to a menu. +Definition 14. Representation of 2-Step Consideration If Γ1, and then Γ2, are +applied to menu A, the resultant consideration set is Γ2(Γ1(A) or Γ12. +Notice that the nested notation, Γ2(Γ1(A), reads from right to left, whereas the +reduced form Γ12 reads from left to right. +The timing of 2-step consideration proceeds as follows: an individual observes +a menu, applied a consideration filter to that menu to get a consideration set, and +then applies another filter the consideration set to end with a final consideration set. +As such, consideration filters can be analogized to contracting mappings in dynamic +programming: each one, applied in succession, takes the current menu and returns a +smaller menu, which is still in the space of menus M(X). Similar notation and intuitions +hold for n-step consideration, in which any finite number of consideration filters within +γ are applied to a menu in succession: +Definition 15. Representation of N -Step Consideration If Γ1, Γ2, ..., Γn are +applied to menu A, the resultant consideration set is Γn(...(Γ2(Γ1(A)))) or Γ12...n. +Here, more than one filter can be applied to a menu. The timing of this process +is principle the same as that of 2-step consideration, albeit with a potentially large +number of new, smaller considerations sets being formed with each application of a +filter Γi from γ. +Equipped with notation to describe sequential consideration, I now introduce the +notion of commutativity, extending the framework to detail the importance of the order +in which a set of filters is applied. +4.2 +Commutative Filters +When different filters are applied sequentially to a menu, does order matter? That is, +can one apply them in any order and expect to get the same final consideration set? As +a concrete example, imagine an individual asking a librarian for help selecting a new +book to read. Alarmed by the incalculably-high number of books to choose from, they +use two rules of thumb: they want to read fiction, and also want to read one of the first +books that comes to the librarian’s mind when asked. The individual will make both +of these requests, but could either ask for fiction and then for the first few books that +17 + +come to mind, or ask for the first few books that come to mind, and then ask which +of these books are fiction. This section addresses whether the consideration set from +which they end up choosing the will be the same in both cases. +In order to answer these questions, I introduce a new concept: commutativity. I +adapt the basic axiom of algebra here to describe cases in which consideration filters +can be applied in any order to a menu, without the consideration set changing. Simply +put, order does not matter. As one may expect, this is a rather strong condition. In +particular, its viability will often depend on whether Independence of Others (IO) is +satisfied by the filters in question. Before introducing results, I define commutativity in +both the 2-step and n-step cases. Without loss of generality, Γ1 and Γ2 refer to two, +distinct, arbitrary filters in γ. +Definition 16. 2 Commutative Filters Γ1 and Γ2 are commutative if Γ12(A) = +Γ21(A) for all A. That is, x ∈ Γ12 iff x ∈ Γ21. +Two filters are commutative if their application to a menu A generates the same +final consideration, regardless of the order in which they are applied. This provides +language to describe consideration sets that are invariant to the order of 2 filters that +generated them. I extend this definition to the n-step case, which is a more demanding +condition. +Definition 17. N Commutative Filters N filters Γ1, Γ2, ..., Γn are commutative +if Γ12...n(A) is invariant to permutations in the order of {1, 2, ...n}. +Commutativity for n filters works in the same way that it does for 2 filters. However +this DIO requires checking every permutation, and the set of filter orderings can become +very large. For example, for a set of 5 filters, there are 120 different orderings, each of +which must be verified to determine if commutativity holds. These definitions allow to +present the two main results of this section. +4.3 +Commutativity Results +Commutativity necessarily makes certain demands on the properties that the applied +consideration filters must have. I present two results, which show that IO filters are +necessarily commutative, both in the 2-step and n-step cases. Recall the definition of +IO: +Re-Definition 1. Independence of Others A consideration filer Γ satisfies Inde- +pendence of Others (IO) if one of the following two conditions holds: +1. x ∈ Γ(A) ∀ A ∈ M(X) s.t. x ∈ A or +2. x /∈ Γ(A) ∀ A ∈ M(X) +18 + +IO maintains that each alternative is always considered when available, otherwise +else it is never considered. I now introduce the first result, which requires IO for two +filters to be commutative for any arbitrary menu. +Theorem 2. IO and Commutativity with 2 Filters +Γ1 and Γ2 are commutative for all A ∈ M(X) if and only if Γ1 and Γ2 are IO. +Proof. See appendix. +I wish to place emphasis on the fact that, while IO is necessary for two filters to be +commutative for any arbitrary menu, two filters can be commutative for a given menu, +while not being commutative for all menus. A simple possible example is if the menu is +the null set ∅. Any two filters are commutative for this menu, as the consideration set +remains null, while, if they are not IO, order reversal could change the consideration +set generated from a non-empty menu. +Theorem 3. IO and Commutativity with N Filters +Γ1, Γ2, ..., Γn are commutative for all A ∈ M(X) if and only if Γ1, Γ2, ..., Γn are +IO. +Proof. See appendix. +This extends Theorem 2 to cases with potentially more then 2 filters, although the +degenerate case of n = 2 shows us that Theorem 3 nests Theorem 2, trivially. The proof +idea, executed in the appendix, is to use the 2-step proof as a base case for induction. +4.4 +Summary +This section introduces the idea of sequential consideration, the idea that multiple +filters can be applied to a given menu, each of which entails a new contraction of the +consideration set. Filters come from the larger space of filters γ, and may satisfy any +number of properties. I then introduce the concept of commutativity, which specifies +cases in which 2, or any countable number of filters generate the same consideration +set regardless of the order in which they are applied. Commutativity will always hold +among any number of IO filters, and can hold among non-IO filters in more limited +cases. Generally, commutativity is a useful benchmark in thinking about consideration +within settings in which information acquisition, and the decisions made from such +information, are paramount. This allows decision theorists to specify when and under +what conditions information use is unaffected by the order of its arrival. In this next +section, I use the idea that there may exist multiple filters in a decision process to +endogenize the choice of a filter within the rational-attention context. +19 + +5 +Extension 2: Preferences over Filters +To this point, the concept of consideration filters has been well-developed, in particular +an outline of their potential properties and interrelations. In this section, I model +individuals as having preferences not simply over alternatives, but over filters that +affect the set of alternatives with which they are presented. +Such a formulation is well-suited to the nuances of individual decision making. Choice +naturally involves rules of thumb — filters, as I model them — however, individuals +may apply such rules selectively across settings. The best consideration heuristic for +choosing a car will naturally differ from that which is optimal for shopping for bananas. +I there endogenize the choice of filters, allowing individuals in my model to select which +consideration filter to apply to a given menu. For simplicity, I assume in this section +that individuals only choose one filter, abstracting away from the sequential setup of the +last section. In practice, the two concepts are easy to combine; I herein wish to focus +on the preference portion to make the concept clear. I place my model in the context +of the rational attention literature, positing that there are two competing factors in +choosing a filter: the greater optionality provided by a larger consideration set, and +the costly mental strain associated with sifting through numerous options. I build a +parsimonious model to capture the substance of this idea, while omitting certain details +that risk over-complicating the setup. +5.1 +Environment and Details +I now formally characterize the space in which the definitions, axioms, and results to +follow shall operate. In Section 3.1 I outlined the environment, particularly defining the +relationship between the set of alternatives X and its constituent menus A ∈ M(X). I +will now operate in a space of filters, as in the last section: +Re-Definition 2. Space of Filters There exists a space of filters γ, comprising +constituent filters Γ ∈ γ. Γi represents the ith filter. +I allow individuals to select which filter to apply to a menu, based on the competing +factors I discuss in this section’s preamble. Preference relations and related concepts +will prove useful in setting up the environment. +Definition 18. Filter Preference Relation There exists a weak preference relation +≿γ over the set of filters Γi ∈ γ. +This preference relation ≿f allows us to formally define the individual’s preference +over the filters Γi ∈ γ. The addition of choice over filters, as opposed to the exogenously- +imposed consideration filter implicitly assumed earlier, necessitates the inclusion of this +relation. +20 + +Axiom 3. Completeness of Filter Preferences The relation ≿γ is complete. +That is, for two filters Γi and Γj ∈ γ, Γi ≿γ Γj or Γj ≿γ Γi. +This axiom states that the individual has a defined preference over every pair of +filters with which they can be presented. A point to note is that I do not model +uncertainty here: for simplicity, this model has full information over filters. +Axiom 4. Transitivity of Filter Preferences For any three filters Γi, Γj, and +Γk ∈ γ, Γi ≿γ Γj and Γj ≿γ Γj implies Γi ≿γ Γk. +Transitivity of ≿γ prevents cycling, that is, preferences that move in a circular +fashion. This is necessary as a consistency condition in order for choices to be indicative +of underlying preferences, a key stipulation for utility representation. +Axiom 5. Rationality of Filter Preferences The relation ≿γ is rational. +Rationality of ≿γ follows from completeness and transitivity.13 +Axiom 6. Utility Representation For a menu A, there exists a utility representa- +tion uf over filters Γi ∈ γ such that: +Γi ≿γ Γi ⇔ uγ(Γi) ≥ uγ(Γi) +By rationality, we know that there exists a utility representation.14 This utility +function is, by nature, ordinal in that that it captures preferences but not necessarily +their intensity. This utility function, as well as the underlying preferences, hold the menu +constant. That is, given a generic menu A ∈ M(X), preferences are then well-defined +over the space of filters γ. We can now work with the following generic function: +Definition 19. Filter Utility Individuals have filter preferences: +uγ = bγ(Γ, A) − cγ(Γ, A) +The utility function gives us the utility that the individual derives from applying a +filter Γ to menu A. The first argument, bγ(Γ, A), gives the benefit of the filter which I +will define as coming from the alternative x ∈ A that is eventually chosen. The cost, +cγ(Γ, A), is the disutility of consideration. +This basic model fits in well the rational attention literature in that it balances the +benefit of choices with the costs of attention. Additionally sense, this model extension +could be argued to be a motivation for why consideration filters are used in the first +13Mas-Colell et al. (1995) Definition 1.B.1. +14Mas-Colell et al. (1995) Definition 1.B.2 +21 + +place: the standard model, in which individuals implicitly consider all goods, induces +costs which may not be worthwhile. +One must note at this point that , to this point, this model has no empirical content. +The terms bγ(Γ, A) and cγ(Γ, A) are too general to be identified. In order to impose +some structure upon the model, I make additional specifications: +Definition 20. Choice of Alternative from Menu c(Γ(A)) selects the most +preferred element from a menu A, according to the rational preference relation ≿x. +c(A) is simply the “best” alternative in A. This allows one to more specifically define +the benefit of a particular filter, according to the alternative that it eventually generates. +This makes sense as an individual will likely evaluate a consideration criterion according +to the utility resulting from the choice that is eventually made. +Definition 21. Benefit of Consideration The benefit of consideration bγ(Γ, A) is, +equivalently, bγ(c(Γ(A))). +The benefit of a consideration filter Γ is a function of the the best alternative in the +generated consideration set, because that is the alternative which is chosen. +I now define the necessary elements in order to represent the cost of consideration. +Definition 22. Cardinality of Menu The cardinality of a menu A, |A|, the number +of alternatives x ∈ A. +Cardinality is the necessary concept to define the cost of consideration, as being a +function of the cardinality of the consideration set. +Definition 23. Cost of Consideration The cost of consideration cγ(Γ, A) is, equiv- +alently, cγ(|Γ(A)|). +I define the disutility of consideration as direct function of the number of alternatives +in the consideration set. This mirrors the well-known costly attention framework: there +exists some cognitive cost of attention (time, mental strain, etc.) that induces negative +utility coming from the sheer number of alternatives considered. I place more structure +on the cost function: +Axiom 7. Convex Cost of Consideration The cost of consideration cγ(|Γ(A)|) is +globally convex. +The more alternatives considered, the greater the disutility, and this disutility +increases marginally: +∂c(|Γ(A)|) +∂|Γ(A)| +> 0, ∂2c(|Γ(A)|) +∂|Γ(A)|2 +> 0 +Now that all arguments have been defined, I arrive at the following specification for +the utility representation of preferences over filters: +22 + +Definition 24. Specified Filter Utility Function Individuals have filter prefer- +ences: +uγ = bγ(c(Γ(A))) − cγ(|Γ(A)|) +Filter utility uγ is broken down into two portions. As specified above, uc(Γ(A))) +denotes the utility derived from the alternative eventually chosen in accordance with +rational preferences. A filter is only as good as the choice it leads to. The cost, cγ(|Γ(A)|) +is a convex function of the number of alternatives an individual considers before making +a decision. +Definition 25. Filter Choice Rule Individuals choose filters Γ ∈ γ so as to +maximize: +c(γ) = arg max +Γ∈γ uγ = bγ(c(Γ(A))) − cγ(|Γ(A)|) +This formally specifies the objective. +Axiom 8. Filter Choice Mandate c(γ) ̸= ∅ +The individual must choose a filter. This prevents the convex cost function from +inducing null choice sets. Equipped with this model setup, I now present formal results. +5.2 +Results +Below, I provide a remark on a cost-induced property of the filter choice process, as +well two boundary results on the the number of alternatives an individual will consider. +Definition 26. Preference for Flexibility A filter choice rule c(γ) represents a +preference for flexibility if Γ1(A) ⊇ Γ2(A) implies that cγ(Γ1, Γ2) = Γ1. +Choice rules satisfying this Preference for Flexibility — a classic property in the +decision theory literature — will induce individuals to choose filters that generate +largest possible consideration sets. +This is relevant in cases in which attention is +costless, meaning individuals cannot be made worse off by more options given free +disposal. Costly attention of the filter objective function leads Preference for Flexibility +to fail in the model: +Remark 1. The filter choice rule cγ does not represent a preference for flexibility. +The logic for this result follows directly from the existence of the cost function +c(|Γ(A)|). It may be the case that, while more alternatives may lead to better choices, +the magnitude of the increased benefit may be outweighed by the cost of attention. +I now move into the two key theorems, detailing edge cases on filter choices. +23 + +Theorem 4. Costless Consideration Implies Full Consideration +If cγ|Γ(A)| = 0 for all Γi ∈ γ, the individual considers all alternatives x ∈ A. Γ(A) = A. +Proof. See appendix. +This result is intuitive. If attention is costless, then there is no reason to not consider +all alternatives as the upside is potentially limitless with no downside cost. +Theorem 5. Worthless Consideration If bγ(Γ(A)) is equal among all Γi ∈ γ, the +individual chooses the filter Γi ∈ γ so as to minimize cγ|Γ(A)|. +Proof. See appendix. +In Theorem 4, I shut down heterogeneity in cost by setting the cost function globally +to zero. Theorem 5 can be seen as a reversal - here, eliminate heterogeneity in “rewards” +by equating benefits from choices across all consideration sets. This result is similarly +intuitive. If there is no potential benefit of a larger consideration set, the individual is +justified in only considering one alternative, as there is assurance that they could not +have improved their condition through increased consideration.15 This result thus can +be seen as a “no better off” theorem. +These two theorems complete my analysis of filter preferences by defining the edges of +possible choices: at one extreme, individuals consider all alternatives available costlessly; +at the other, individuals consider the minimum number of alternatives so as to simply +satisfy the axiom that at least one alternative must be selected. +5.3 +Summary +In this section, I extended the basic consideration model to a setting in which individuals +may choose which consideration filters they apply to menus. In doing so, I provide a +rational-attention model of limited consideration, modeling individuals as weighing the +benefit of a larger consideration set with the convex costs of increased consideration. I +provide two boundary results detailing cases in which an individual considers either all +alternatives, or the minimum number possible. The next section outlines the viability +of utility representation in a limited consideration setting. +15As a nod to contract theory, this result mirrors the well-known result that, in the classic principal-agent +setting, setting equal wages for high output and low output will induce shirking on the part of the agent. +24 + +6 +Utility Representation +The ability to construct utility functions from observed choices — utility representation +— relies on rational underlying preferences. This means that preferences must be +complete and transitive. Under limited consideration, completeness naturally does not +hold because the individual decision-maker does not necessarily consider every available +alternative. Moreover, transitivity can also fail in the event that incomplete preferences +lead choices to cycle. +As a result, limited consideration poses a fundamental threat to utility representation. +In order to maintain the utility functions commonly assumed in structural models, +assumptions need to be made on the process of consideration employed by individuals, +who constitute the representative agents in applied literatures. +In this section, I begin by constructing a utility function that links to consideration- +mediated choices. I then show that consideration filters that satisfy Independence +of Others (IO) are sufficient for this form of utility representation. I then follow the +approach of Lleras et al. (2017) by providing a modification of the weak axiom to match +this consideration-consistent utility function. +In both the utility representation and weak axiom settings, it becomes clear that IO +poses an extremely strong condition on choices and preferences. I acknowledge this and +conjecture methods which can be used to weaken IO and preserve applicability. +6.1 +Consideration Utility Function +I present a general utility function, which I will use for the results that follow. +Definition 27. Generic Multi-Argument Utility +u(x ∈ A) = f(u1(x), u2(x), ..., un(x)) +This utility function maps each alternative x ∈ A to the real numbers according to +some amalgamation of n different arguments. In the degenerate case there may exist +only one argument. In order to capture the process of consideration, I denote the first +constituent function, u1, to be the threshold function, the naming of which will become +clear: +Definition 28. Threshold Function Within f(u1(x), u2(x), ..., un(x)), u1 is known +as the threshold function. +I require that, in order form some alternative x ∈ A to be in the consideration set +Γ(A), the value generated by u1 from x must reach a certain threshold value, k∗: +Axiom 9. Threshold k∗ x ∈ Γ(A) if and only if u1(x) ≥ k∗ +25 + +Where k finds its value among the real numbers: +Axiom 10. K is real k∗ ∈ R +The consideration set from a menu A thus consists of its alternatives that meet the +threshold: +Definition 29. Threshold k∗ Consideration Set +Γ(A) = {x ∈ A : u1 ≥ k∗} +The individual forms their consideration set from the alternatives x ∈ A that meet +a threshold condition that u1(x) ≥ k∗. +This now provides a way to demonstrate +consideration in the space of real numbers rather than purely through set theory. +After completing this setup, the overall choice function now becomes: +Definition 30. Threshold Choice Function +c(A) = arg max +x∈Γ(A) f(u1(x), u2(x), ..., un(x)) +Where Γ(A) = {x ∈ A : u1 ≥ k∗} +In words, the above choice function specifies the process: +1. Individual is presented with menu A +2. Individual narrows menu A to consideration set Γ(A) by only considering alterna- +tives x ∈ A for which u1(x) ≥ k∗ +3. An alternative x is chosen from the consideration set Γ(A) according to some +rational preference relation. +As stated above, this is one among many potential examples of how the set of real +numbers can be used to facilitate the modeling of consideration. The setup I have +develop allows me to present a utility representation result. I show that any choice +process consistent with the above formulation must only involve consideration filters +that satisfy IO: +Theorem 6. IO Utility Representation Γ satisfies IO if and only if ∃k∗ and +u1(x) : X �→ R such that Γ(A) = {x ∈ A : u1 ≥ k∗}. +Proof. See appendix. +Any filter used in the specified choice process must be IO. Recall that IO is equivalent +to the joint presence of Sen’s α and Condition τ, and so these two conditions may also +substitute into the result. +26 + +The idea of the proof, found in the appendix, is to fix the threshold k∗ to 1, and +define k∗ as 1 for all alternatives within the IO-generated consideration set, and 0 for +those without. By demonstrating that the set of alternatives meeting the threshold are +also those within the consideration set, the proof is completed. +The above exercise shows an example of how consideration can be nested within +utility function once properties of the consideration filters are specified. I now extend +this exercise to the weak axiom, using IO as a proof of concept as to how the weak +axiom can be modified to match the limited consideration setting. +6.2 +Weak Axiom Modifications +The Weak Axiom of Revealed Preference (WARP) is a consistency condition that aligns +choices with rational underlying preferences. The weak axiom, when it holds, mandates +that if some alternative x1 is chosen over x2, then the reverse cannot happen when both +are available in some other menu. In essence, preferences cannot “reverse” across two +different menus. The weak axiom forms the basis for choice theory and, by extension, +underpins utility representation. +Despite the ubiquity of the weak axiom, it does not naturally account for limited +consideration. If, in the second menu, x1 were not considered, then it is quite plausible +for x2 to be chosen, given that the individual may not even have been aware of the +presence of x1 in the menu. Therefore, the weak axiom needs to be modified to match +the limited consideration setting. +This has been done before. Lleras et al. (2017) provide a modification of the weak +axiom to account for the phenomenon of “choice overload.” Choice overload refers +to situations in which individuals consider certain alternatives in small menus, but +somehow lose track of these alternatives when presented with a much larger menu that +includes them. The rationale is that the overwhelming number of alternatives can be +cognitively challenging and may induce forgetfulness or similar mental lapses. The +choice overload WARP modification requires some simple notation. Refer to S and T +as two menus within M(X) and call b an alternative in X. The choice overload WARP +modification is: +Axiom 11. WARP Choice Overload (WARP-CO) For any nonempty S, there +exists b∗ ∈ S such that for any T including b∗, if: +1. c(T) ∈ S and +2. b∗ = c(T +′) for some T +′ ⊃ T +then c(T) = b∗ +27 + +By requiring that the chosen alternative b is considered in the larger set, WARP-CO +“closes the hole” punctured by choice overload, allowing choices to again be consistent +with rational preferences. Across many settings in which limited consideration may +jeopardize completeness, it behoves the decision theorist to consider WARP modifications +that are appropriate to the application of interest. As an example, I now provide a +WARP modification that matches Independence of Others (IO), using the same notation +as that of WARP-CO: +Axiom 12. WARP-IO For any nonempty S, there exists b∗ ∈ S such that for any T +including b∗, if: +1. C(S) = b* +2. C(T) ∈ S, and +3. C(T) = b* if and only if c(Q) = b*, where Q = {b*, x}, ∀ x ∈ T such that ∃ J ∈ +X such that x = C(J) +then c(T) = b∗. In addition: +1. consider B ⊂ X, where B = {b, ∅} +2. if c(B) = ∅, then c(J) ̸= b for all menus J ∈ X +Recall the definition of IO: +Re-Definition 3. Independence of Others A consideration filer Γ satisfies Inde- +pendence of Others (IO) if one of the following two conditions holds: +1. x ∈ Γ(A) ∀ A ∈ M(X) s.t. x ∈ A or +2. x /∈ Γ(A) ∀ A ∈ M(X) +The two conditions I provide in WARP-IO correspond to the two portions of the +IO definition, respectively. In the first case, WARP-IO mandates that any choice must +pairwise beat every other alternative in the menu. This matches full consideration in +that there cannot be a case in which an alternative is selected despite not being preferred +to some other alternative, which may happen if limited consideration restricts the scope +of the consideration set. The second branch of WARP-IO concerns alternatives that +are not chosen when they are the only alternative available. Naturally, this must mean +that these alternatives, for some reason, were not considered, given that they are in the +available set16. In this case, they are never chosen, as IO states that alternatives that +are not considered in some instance are never considered in any menu. +16I again remind the reader that, throughout the paper, I assume that the available set only includes +alternatives that are within the individual’s budget set. +28 + +I have presented a modification to the Weak Axiom of Revealed Preference (WARP) +to align with consideration heuristics that are modeled by IO filters. In doing so, I +follow in the vein of Lleras et al. (2017), who also devise a WARP modification to +account for the nuances of consideration-mediated choices. Similar modifications are, +in principle, possible for any property of consideration filters, and future literature can +make large strides by developing the appropriate modifications to match the common +behavioral processes most commonly observed in real-world economic settings. +6.3 +Summary +In section, I have explored the implications of limited consideration on utility repre- +sentation. Choices made under limited consideration may appear to reflect underlying +preferences that are neither complete nor transitive, presenting a threat to the ability to +use utility functions to represent consideration-mediated choices. In order to preserve +utility representation, assumptions need to be made on the properties of consideration +filters. As an example, I show an example of a utility function which captures choices +made using an IO consideration filter, showing that IO is sufficient to model choices +made via the objective function I set up. +I also address the Weak Axiom of Revealed Preference (WARP), which implies full +consideration. I follow Lleras et al. (2017) by modifying the weak axiom to match a +property of consideration filters, providing an example for IO. In future, as I discuss in +the next section, the characterizations I provide ought to be extended to cover filter +properties that are not as strong as IO. +7 +Future Directions +In this analysis I provide a language for discussing limited consideration, extended +the basic model, and conjectured conditions for utility representation. Below I briefly +mention three potential avenues for future work on limited consideration. +Weakening IO. IO is clearly the strongest condition one can impose upon the +formation of consideration sets — either an alternative is always considered when +available, or else it is never considered, not even in the singleton set. IO, the foundation +for some of the results in this paper, is clearly not flexible enough to match the nuances +in individual behavior. However, the standard model, and classical revealed preference, +make a similarly strong assumption: that the available set is always the consideration +set (Γ(A) = A). Full consideration, a special case of IO,17 is therefore not realistic +17Define the IO filter rule as: every alternative x ∈ A is always considered when available. The available +set is then always equivalent to the consideration set. +29 + +either, and so weakening of IO must be explored. This will require prudence, however, +as the desired condition must be weaker than IO, while still having enough “bite” to +ensure falsifiability. +Incorporating Consideration into Structural Models. Many empirical phenomena +could be better understood using the limited consideration framework. For example, +Larcom et al. (2017) explore route choice in the London subway system before and +after a temporary shortage, finding that some commuters used different routes after +the transportation restart, meaning they were not optimizing before the strike. The +authors build a structural model of route choice, finding that daily commuters often +were not aware of routes that were faster than the ones they previously used. They +allude to limited consideration, wondering why commuters did not experiment enough +before the strike. One potential answer is limited consideration: individuals did not +consider all available routes. This could take the form of a satisficing heuristic, or it +could be modeled using the rational-attention model I developed in Section 5 in the +event that the process of analyzing routes is seen to be costly. +How Much Do We Toss Out? Related to the first two suggestions, literature across +all subfields assumes full consideration in some fashion or another. Given that this +is not a realistic axiom, it is worth examining what sorts of theoretical and empirical +results are no longer viable once one understands the salience of limited consideration. +For example, in a setting in which individuals often use consideration heuristics, welfare +analysis grounded in observed choices is likely to need adjustment. +8 +Conclusion +Revealed preference takes observed choices to be indicative of individual preferences. +For example, if, given the set {A, B, ..., Z}, an individual chooses R, revealed preference +indicates that R is preferred to all other letters. This approach to choice theory assumes +that individuals examine every available option before making a choice. In contrast, +limited consideration posits that individuals narrow menus into consideration sets before +making choices. This framework is better suited to modeling individual decision-making, +which often involves various rules of thumb that filter out certain options. +The literature on limited consideration and related processes has been well-developed +and includes theoretical models18, consumer choice analyses19, axiomatic characteriza- +tions of normative preferences, 20 and structural work in various empirical settings.21 +In this paper, I provide a general model of limited consideration to unite the literature. +18Masatlioglu et al. (2012); Masatlioglu and Nakajima (2015); Lleras et al. (2017) +19Hauser and Wernerfelt (1990); Roberts and Lattin (1991); Erdem and Keane (1996) +20Cherepanov et al. (2013) and Ridout (2021) +21Abaluck and Gruber (2016); Larcom et al. (2017); Abaluck and Adams-Prassl (2021) +30 + +I begin by outlining the main features of consideration model: individuals observe +menus, narrow them into consideration sets, and make choices from said consideration +sets. The channel by which menus are translated into consideration sets is captured by +consideration filters, functions that downsize menus into consideration sets by mapping +them to one of their subsets. Consideration filters correspond to various rules of thumb +that individuals may use in narrowing down menus. To account for the large number +of heuristics that individuals may use in practice, I introduce a number of properties +that consideration filters may have. These properties describe the manner in which a +menu begets a consideration set, and are neither mutually exclusive nor mandated to +coexist with one another. A consideration filter may satisfy one, all, or none of the +properties I introduce. The strongest condition, Independence of Others (IO), describes +consideration filters which select a certain set of alternatives in any menu in which they +appear, and never selects any alternative not in this set. IO, which closely approximates +the standard rational model, forms the basis for a number of later results in the paper. +I then extend the consideration model in two ways. First, I develop a model of +sequential consideration, which allows more than one filter to be applied to a given menu. +This matches settings in which individuals are thought to apply more than one rule of +thumb in narrowing down large choice sets. I introduce the concept of commutativity, +borrowing from algebra to describe filters which can be applied to a menu in any order +and still generate the same final consideration set. Filters satisfying IO are always +commutative. My second model extension is a rational-attention analogue, in which I +model an individual who must choose which filter to apply to a given menu. Individuals +weigh competing forces in choosing a filter: the greater optionality associated with a +larger consideration set and the examination costs associated with sifting through a +large number of alternatives. I present two boundary results, showing in which cases an +individual will consider every alternative, or the minimum number of alternatives. +I address the implications of limited consideration on utilty representation. The abil- +ity to construct utility functions corresponding to observed choices relies on underlying +preferences being both complete and transitive. In the consideration model, both condi- +tions often fail. To counteract this, I construct a utility function that accurately models +choices made using an IO consideration filter. Such a link between consideration-based +utility functions and the filter properties that may generate them may be possible for a +large array of consideration heuristics. I also provide a modification to the Weak Axiom +that corresponds to IO. In both cases, I use IO as a basic proof of concept to demonstrate +how standard choice theory can be reconciled with the limited consideration framework. +There are many potential future directions for theoretical work on limited consider- +ation. For example, IO can be weakened to find a filter property that is more realistic +yet tractable. In addition, current structural choice models used by applied economists +31 + +can be better reconciled with limited consideration, especially in empirical settings in +which choices are thought to be made from consideration sets. +In summary, I have presented a detailed characterization of limited consideration, +nesting some of the prior literature into a formal language while also developing novel +model extensions. The hope is that a complete theory of consideration, building off +this work as well as that of other scholars, will improve the robustness and applicability +of rational choice theory. +32 + +References +Abaluck, J. and A. Adams-Prassl (2021): “What do consumers consider before +they choose? Identification from asymmetric demand responses,” The Quarterly +Journal of Economics, 136, 1611–1663. +Abaluck, J. and J. Gruber (2016): “Evolving choice inconsistencies in choice of +prescription drug insurance,” American Economic Review, 106, 2145–84. +Apesteguia, J. and M. A. Ballester (2013): “Choice by sequential procedures,” +Games and Economic Behavior, 77, 90–99. +Caplin, A. and M. Dean (2011): “Search, choice, and revealed preference,” Theoret- +ical Economics, 6, 19–48. +Cherepanov, V., T. Feddersen, and A. Sandroni (2013): “Rationalization,” +Theoretical Economics, 8, 775–800. +Erdem, T. and M. P. Keane (1996): “Decision-making under uncertainty: Capturing +dynamic brand choice processes in turbulent consumer goods markets,” Marketing +science, 15, 1–20. +Hauser, J. R. and B. Wernerfelt (1990): “An evaluation cost model of considera- +tion sets,” Journal of consumer research, 16, 393–408. +Larcom, S., F. Rauch, and T. Willems (2017): “The benefits of forced experimen- +tation: striking evidence from the London underground network,” The Quarterly +Journal of Economics, 132, 2019–2055. +Lleras, J. S., Y. Masatlioglu, D. Nakajima, and E. Y. Ozbay (2017): “When +more is less: Limited consideration,” Journal of Economic Theory, 170, 70–85. +Manzini, P. and M. Mariotti (2007): “Sequentially rationalizable choice,” American +Economic Review, 97, 1824–1839. +Mas-Colell, A., M. D. Whinston, J. R. Green, et al. (1995): Microeconomic +theory, vol. 1, Oxford university press New York. +Masatlioglu, Y. and D. Nakajima (2015): “Completing incomplete revealed +preference under limited attention,” The Japanese Economic Review, 66, 285–299. +Masatlioglu, Y., D. Nakajima, and E. Y. Ozbay (2012): “Revealed Attention,” +American Economic Review, 102, 2183–2205. +33 + +Ridout, S. (2021): “Choosing for the right reasons,” Unpublished manuscript. +Roberts, J. H. and J. M. Lattin (1991): “Development and testing of a model of +consideration set composition,” Journal of Marketing Research, 28, 429–440. +Simon, H. A. (1955): “A behavioral model of rational choice,” The quarterly journal +of economics, 99–118. +Tversky, A. (1972): “Elimination by aspects: A theory of choice.” Psychological +review, 79, 281. +34 + +Appendix: Proofs of Results in Main Text +Theorem 1: Sen’s α and Condition τ ⇔ IO +For the if direction, I show that any filter Γ satisfying Sen’s α and Condition τ is also +IO. Suppose filter Γ satisfies α and τ. Further suppose that some alternative x is in the +consideration set generated by this Γ on menu A. By Sen’s α, x is in the consideration +set of the singleton menu {x} ∈ A.22 By τ, x is in the consideration set of any menu +that includes x, since any such menu is a superset of {x}.23 Since x is always considered +when available, Γ is IO. +For the only if direction, I now show that any filter Γ satisfying IO also satisfies +Sen’s α and Condition τ. Suppose some filter Γ satisfies IO. Further suppose that some +alternative x is in the consideration set generated by this Γ on menu A. By IO, x is +always considered when available. By definition, x appears in all subsets (Sen’s α)and +supersets (Condition τ) of A. Therefore Γ satisfies Sen’s α and Condition τ, as desired. +Theorem 2: IO and Commutativity with 2 Filters +I start with the if direction: if Γ1 and Γ2 are IO, then they are commutative for any +menu A ∈ M(X). This simply requires me to show that x ∈ Γ12(A) if and only if +x ∈ Γ21(A). First, assume that x ∈ Γ12(A). Recall that, if an IO filter retains some +alternative x, then it must retain x in all menus A that contain x. Therefore, x ∈ Γ12(A) +implies that x ∈ Γ1(A) for if x ∈ A, and it is also true that x ∈ Γ2(A) for if x ∈ A. +Now, recall that Γ21 applies filter Γ1 followed by Γ2. Both filters, as I have shown, +always retain x when x ∈ A, and so x ∈ Γ21. +Now, for the only if direction, which entails showing that if any two filters Γ1 and +Γ2 are commutative for any menu A, then they must be IO. This result can be proved +by inspection, noting the intuition behind IO. If in the event that filter Γ1 is not IO, +there necessarily exists a pathological case in which the consideration set generated +by Γ1’s involves a comparative selection process, i.e. alternatives are selected based +on their desirability relative to others. In that case, the consideration set generated +the successive application of Γ1 and Γ2 is clearly dependent upon the structure of the +original menu. +Theorem 3: IO and Commutativity with N Filters +I prove this result using Theorem 2 as a base case. By Theorem 2, any two filters Γ1 +and Γ2 are commutative for any menus A ∈ M(X) if and only if they are IO. Because +22This is the going down step. +23This is the going up step. +35 + +Γ1 and Γ2 are commutative, they can be collapse into one filter, since the order of their +application does not matter. Call this new filter Γc. Suppose one adds a third filter Γ3. +By Theorem 2, Γc and Γ3 are commutative if and only if they are IO. Recalling that Γc +is an amalgam of Γ1 and Γ2, filters Γ1, Γ2, and Γ3 are commutative if and only if they +are IO. +If I add a fourth filter Γ4, the same approach works by collapsing the first three +filters into one. The proof strategy scales up for any n filters. +Theorem 4: Costless Consideration Implies Full Consider- +ation +Recall the choice function +c(γ) = arg max +Γ∈γ uΓ = bγ(c(Γ(A))) − cγ(|Γ(A)|) +If consideration is costless, cγ(|Γ(A)|) = 0, therefore we have: +c(γ) = arg max +Γ∈γ uΓ = bγ(c(Γ(A))) +Which is maximized by considering all alternatives x ∈ A.24 +Theorem 5: Worthless Consideration +Recall the choice function +c(γ) = arg max +Γ∈γ uΓ = bγ(c(Γ(A))) − c(|Γ(A)|) +If the benefit of consideration bγ(c(Γ(A))) is constant across all filters, then it is +invariant to any change in the filter and this has no effect on the optimal choice. The +problem reduces to cost minimization: +c(γ) = arg max +Γ∈γ uΓ = −c(|Γ(A)|) +In order to maximize the objective, while satisfying Axiom 6, the filter choice +mandate, the individual will simply choose the filter that minimizes the cost of consid- +eration. +24This assumes the benefit of consideration bγ is non-decreasing. +36 + +Theorem 6: IO Utility Representation +The if direction, that IO filters can be represented by the threshold choice function, is +straightforward. List all alternatives x ∈ A which are in the consideration set of the IO +filter Γ. For each x ∈ Γ(A), set u1(x) = 1. For each x /∈ Γ(A), set u1(x) = 0. Then set +k∗ = 1. Therefore, all alternatives Γ1 meet the u1 threshold. +The only if direction makes us of the same technique, paired with the application +Sen’s α and Condition τ to the relevant menus. To prove this direction. Define a set +Y ⊆ X as follows +y ∈ Y iff x ∈ Γ(A) for some A. +Define +u1(x) = +� +� +� +1 +if x ∈ Y +0 +if x /∈ Y +It remains to verify that Γ(A) = {x ∈ A : u1(x) ≥ k∗} as desired. Let’s check. By +construction, the right hand side equals {x ∈ A : x ∈ Y } = A ∩ Y . So it remains to +check whether Γ(A) = A ∩ Y . There are two arguments needed: +1. If x ∈ Γ(A) then x ∈ A ∩ Y +2. If x ∈ A ∩ Y then x ∈ Γ(A). +To prove the first argument, assume that x ∈ Γ(A). Then automatically x ∈ A, and +x ∈ Y By definition of Y . +To prove the second argument, assume that x ∈ A ∩ Y . Then by definition of Y +there exists a set B such that x ∈ Γ(B). Let’s consider the set C := A ∩ B. We know +x ∈ C since both A and B contain it. First, apply Sen’s α condition to x ∈ C ⊆ B. +This implies that x ∈ Γ(C). Then apply Condition τ condition to x ∈ C ⊆ A. This +implies that x ∈ Γ(A), as desired. +37 + diff --git a/ltE5T4oBgHgl3EQfig-m/content/tmp_files/load_file.txt b/ltE5T4oBgHgl3EQfig-m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02ef0f8086c2a02b8f3627edebd35dc9d3da5089 --- /dev/null +++ b/ltE5T4oBgHgl3EQfig-m/content/tmp_files/load_file.txt @@ -0,0 +1,882 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf,len=881 +page_content='Filtering Down to Size: A Theory of Consideration∗ Tonna Emenuga† January 16, 2023 Abstract The standard rational choice model describes individuals as making choices by selecting the best option from a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A wealth of evidence instead suggests that individuals often filter menus into smaller sets — consideration sets — from which choices are then made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I provide a theoretical foundation for this phenomenon, developing a formal language of axioms to characterize how consideration sets are formed from menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I posit that consideration filters — mappings that translate a menu into one of its subsets — capture this process, and I introduce several properties that consideration filters can have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then extend this core model to provide linkages with the sequential choice and rational attention literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Finally, I explore whether utility representation is feasible under this consideration model, conjecturing necessary and sufficient conditions for consideration-mediated choices to be rationalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' ∗I am grateful to Tomasz Strzalecki for his guidance and mentorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I thank Jerry Green, Shengwu Li, David Laibson, Yannai Gonczarowski, Nathaniel Hendren, Jeff Miron, Matthew Rabin, Angie Acquatella, Sejal Aggarwal, Shani Cohen, Roberto Colarieti, Benny Goldman, Zo¨e Hitzig, Martin Koenen, Pierfrancesco Mei, Akash Nandi, Cassidy Shubatt, Chris Walker and numerous seminar and workshop participants in the Harvard Economics Department for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This work was supported by a grant from the Harvard College Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' All errors are my own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' †Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Email: tonnaemenuga@college.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='05649v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='TH] 13 Jan 2023 1 Introduction Economists’ ability to deduce individual preferences from observed choices informs much of microeconomic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In particular, the foundational concept of revealed preference asserts that an individual’s choice of an item (an alternative) from a set of options (a menu) is reflective of their underlying preference, allowing economists to determine preferences simply from a summary of choices made from various menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, if when presented with two cars of equal cost1 — one red and one gray — a consumer purchases the red car, revealed preference tells us that the they must prefer the red car over the gray car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this standard model, individuals observe menus, analyze all available options, and make choices that are most consistent with their tastes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This model, which forms the foundation of rational choice theory as well as applied analysis across a variety of subfields, relies on an assumption referred to as “full consideration.” This means that, when analyzing a menu, a decision-making individual considers every item available before making a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the car choice example, this entails assuming the individual considered the gray car, or was at least was aware of its presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Contrary to this assumption, a wealth of empirical evidence demonstrates that, often, individuals will only consider a subset of a menu before making a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In many cases, the entire menu is not fully examined: only the few alternatives which come to mind are fully considered by the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This is known as limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Rather than making a choice directly from a menu, individuals may filter menus into consideration sets, which are smaller groupings of alternatives from which choices are eventually made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This filtered decision process creates some challenges for the classic revealed pref- erence approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, if an individual is presented with a menu consisting of alternatives {x, y, z} and chooses y, can one state, as usual, that the individual necessarily prefers y over x and z?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the case of a filtered process, in which limited consideration holds, one cannot make this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Alternatives x and z may not have been in the individual’s consideration set — x and z were not examined — and thus the preference relation of y to x and z remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Limited consideration also jeopardizes the ability attain utility representation of individual preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As is well known,2 the ability to construct utility functions corresponding to observed choices relies on preferences being both complete3 and 1Throughout, I will assume all alternatives within a menu are affordable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' this is a standard assumption in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 2This is the utility representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 3Preferences are complete if there is a well-defined relation between any two alternatives in a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Given two options, an individual with complete preferences will weakly prefer one of them, otherwise they are indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 2 transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='4 In the case of limited consideration, completeness is most clearly in question, and transitivity can fail as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='5 To better understand these issues, I provide in this paper a general model that can form the theoretical basis of work aimed at reconciling the standard model with the challenges imposed by limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In doing so, I develop a formal language of axioms to characterize how individuals may not make choices from menus, but rather from consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I imagine that individuals observe menus, filter them into smaller menus, and then make rational choices from these smaller menus which are known as consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I posit that the process by which individuals go from menus to consideration sets is mediated by consideration filters,6 which are mappings that translate a given menu into one of its subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Such a model has been developed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The idea that only a subset of available options are considered dates back as far as the Simon (1955) model of satisficing and optimal stopping, whereby an individual browses options only up until an acceptable one is found;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' at that point, search ceases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Masatlioglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2012) and Masatlioglu and Nakajima (2015) develop models to capture the process of limited consideration, focusing on the ability to infer consideration sets from observed choices under limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' On a normative level Cherepanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2013) present a model in which individuals only consider alternatives that can be rationalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Ridout (2021) axiomatizes the decision-making process of an individual “choosing for the right reasons,” modeling a decision maker who only makes choices that can be justified to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Such axiomatic formulations forms a solid theoretical grounding to make sense of much of the applied work on consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In particular, Erdem and Keane (1996), Hauser and Wernerfelt (1990), and Roberts and Lattin (1991) test structural models to describe the formation of consumers’ consideration sets over goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' More recently, Abaluck and Gruber (2016) apply limited consideration to choices over healthcare plans, and Abaluck and Adams-Prassl (2021) develop a structural demand model based on limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I contribute to the literature by uniting much of the above work into a general framework for understanding limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I begin by formally outlining the main feature of the model: individuals’ choice processes exhibit two mappings, one from menus to consideration sets and another from consideration sets to eventual choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The timing of limited consideration features an individual observing a menu, considering some subset of the available alternatives (the consideration set), and then making a choice from said consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4Preferences are transitive if x ≿ y and y ≿ z implies x ≿ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 5Masatlioglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2012) provide several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 6Also referred to as consideration set mappings in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 3 I model consideration sets as generated by consideration filters, functions that map the set of menus into subsets of themselves, thereby capturing the process of “filtering out” certain alternatives according to some heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A consideration heuristic is simply a rule that determines which alternatives in a given menu are in the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, an individual intending to select a banana from a grocery store is unlikely to examine every banana in the produce section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' rather, they may simply consider those bananas which are in the front row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this case, the menu is the set of all bananas, the consideration set is the front row, and, importantly, the consideration filter is the guiding heuristic “only look at bananas in the front row.” Consideration heuristics may be intentional, in the case of the individual looking to purchase a banana, or subconscious, in the case of an individual aiming to select a fruit of any kind, and failing to see that there are apples, which they might very well prefer, in the next aisle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The list of potential consideration heuristics is clearly enormous, at least as large as the number of decision-making rules and behavioral biases that one could model an individual as having.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I outline several generic qualities that such heuristics, or consideration filter properties, may have as well a the relationships that these properties have with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' One of these properties, which I call Independence of Others (IO), is the focus of many of the exercises contained in the proceeding sections and in the paper as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A consideration filter is IO if and only if alternatives in a menu are either always considered when available, or never considered at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The sense in which this makes alternatives “independent” of each other is clear: the presence, or lack thereof, of other alternatives in a menu has no bearing on whether a particular alternative is in the final consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO closely approximates the rational model, insofar as IO filters generate choices structures that cannot cycle7 and hence can be represented by the utility function I derive in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I also present other potential filter properties corresponding to different consideration heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then extend the basic model of consideration to account for the use of filters in a sequential fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The literature on sequential consideration begins with the Tversky (1972) model of elimination by aspect, whereby one sequentially removes alternatives from the consideration set based on particular qualities — aspects — that these alternatives may or may not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Manzini and Mariotti (2007) model a decision- making process whereby an agent eliminates inferior alternatives sequentially, applying a complete and transitive preference profile to the choice set until only one alternative remains — the final choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Apesteguia and Ballester (2013) take a game-theoretic approach, using game trees to characterize which sequential choice processes of the above sort are rationalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I propose a more general model, nesting the above models 7Choices cycle if x is chosen over y and x is chosen over y, yet z is chosen over x, violating transitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4 into a general framework for understanding multi-step consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then develop a second model extension, constructing a “rational attention”-style analog of the consideration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This allows me to model preferences that decision- makers may have over consideration filters themselves, rather than simply over goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In principle, there are may be a number of different consideration heuristics that an individual may employ, and the ultimate choice in large part rests upon which of these rules is applied to the menu they are presented with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I model an individuals who chooses which filter to apply to a given menu, weighing two competing forces: the benefit of a larger consideration set (a larger choice set may raise the chance of a particularly good alternative being in the menu) and the cost of examining many items (it takes time and effort to sift through many options).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I derive two boundary conditions that give a flavor for how subsequent work can unite the process of consideration with the behavioral reality of limited attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Finally, I address the ability to rationalize choices that are made under limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The key threat to rationality, and utility representation, is completeness of preferences, which is clearly not the case when only a subset of available alternatives are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Caplin and Dean (2011) discuss this challenge in a search-model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I provide a strong condition, IO, under which consideration-mediated choices can be represented by a utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Relatedly, I also provide a modification to the Weak Axiom of Revealed Preference (WARP) that matches the consideration setting, following in the vein of Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Overall, this paper outlines a general framework for understanding the phenomenon of limited consideration, extends the basic model in novel ways, and discusses the implications of limited consideration for utility representation and rational choice theory more broadly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 2 outlines the basic model of consideration filters and resultant consideration sets, proving a language through which the ideas of the papers can be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 3 provides a number of qualities which consideration filters may have, in particular IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sections 4 and 5 represent extensions to the base model: Sections 4 allows for the use of more than one filter, and Section 5 models individuals as having preferences over filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 6 discusses the potential for utility representation of consideration-mediated choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 7 discusses the limitations of my results and future directions that the literature can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 8 concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' An appendix8, containing proofs of results, is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 5 2 General Theory of Consideration This section outlines the overall theory of limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I introduce the key players — consideration sets and consideration filters — while providing the surrounding definitions and axioms to close the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 Overview of Limited Consideration The basic theory of limited consideration is that an individual will not pay attention to every option they are presented with before making a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The simple translation of menus to choices is inadequate to capture a key nuance of human decision-making, namely that attention tends to be “costly” and thus individuals consider only a small set number of alternatives in the choice process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Between menus and choices, there exists an intermediate step known as consideration, which defines the smaller set of alternatives that an individual will examine and ultimately choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Rather than making choices from menus, individuals make choices from consideration sets, which are nested within the original menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 makes clear the relationships between menus, consideration sets, and choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1: Menu, Consideration Set, and Choice I now provide the formal definitions and axioms to set up the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 Definitions and Axioms Equipped with an intuitive notion of the idea of consideration sets, I now define the mathematical environment in more concrete terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This section will use concepts that are quite familiar to the decision theorist and discrete mathematician alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 6 Menu Consideration Set Choice2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 Set Notation I use basic set-theoretic concepts to define the terms used in the process consideration- mediated choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The set of alternatives is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The set of alternatives represents the total set of alternatives which may be presented to an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For an individual browsing cars at a dealership, X is the total set of cars in the lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The set of cars consists of individual cars, which are its constituent alternatives: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The constituent alternatives within X are known as xi, where xi ∈ X for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The individual’s final choice will be some alternative xi from the universe X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I henceforth abuse notation by omitting the subscript i to refer to alternatives by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As discussed, individuals must first observe a menu — some subset of the available alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The set of menus is M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The set of menus constitutes every combination of alternatives {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', xn} that the individual may be presented with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Axiom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There exist 2|X| menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Following from the definition of the power set, the set of menus M(X) will necessarily contain 2|X| menus, where |X| is the cardinality of X, the number of alternatives contained in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This includes both the full set, containing every alternative, as well as the null set ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Generic menus are denoted A and B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A ⊆ B ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I define two generic menus A and B, which will be used in proceeding sections when discussing properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A is a subset of B — every alternative in A also finds itself in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I later sections, I will at times make use of other generic menus, such as Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' in each instance, take Q to be some generic menu with the same properties as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 Mappings and Consideration Sets I now define consideration filters and consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The consideration model assumes that individuals observe menus, filter them into consideration sets, and then make choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The first step in the process is the mapping from menus to consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Formally, a consideration filter is a mapping Γ : M(X) �→ M(X) that takes the 7 set of menus and returns corresponding subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, a consideration filter takes a menu and returns one of its subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For the menu A, the filter returns Γ(A), where Γ(A) ⊆ A for all menus A ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recall the generic menus A and B, keeping in mind that A is a subset of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A consideration filter, in the most general sense, takes B and returns some A: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ : B �→ A is a consideration filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Any function that takes a set and returns one its subsets qualifies as a consideration filter, capturing the phenomenon of the limited consideration that motivates the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Naturally, such filters can take various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, Masatlioglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2012) restrict their analysis to those filters classified as “attention filters,” while Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017) study the case of “choice overload.” Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ(B) is the consideration set of menu B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The result of the consideration filter, applied to a menu, is the consideration set, represented by the red region of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Choices, ultimately, are made from consideration sets rather than menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Axiom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' c(B) ∈ Γ(B) The above axiom simply establishes that the choice from a menu must also be in that menu’s consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In essence, choices are made from consideration sets, not menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This provides a starting point for any attempts to identify consideration sets from observed choice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If an alternative is chosen, one can state with certainty that said alternative must have been considered, and thus finds itself within the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='8 In order to determine the structure consideration sets beyond the chosen alternative, assumptions must be made on the process of consideration, necessitating some language for describing axioms of consideration filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The following section provides such a framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 3 Properties of Consideration Filters This section introduces various properties that consideration filters may have, providing a language through which consideration filters and consideration sets can be described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 re-states the mathematical preliminaries, outlining a rigorous vocabulary for describing the environment of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 lists properties that consideration filters may have, providing examples, descriptions, and formal results regarding their interrelations when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 8Note that, if an alternative is not chosen, one cannot make any inference on whether it was considered, without making additional assumptions on the consideration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 Environment I reformulate the mathematical environment, using basic notions of sets and set rela- tionships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I begin with an individual, who goes through a two-step process to make a choice: the individual is presented with a menu of items, filters it down to a weakly smaller consideration set, and then makes a final choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this section, I focus on the first portion of the process: that is, the transformation of menus into consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There is a set of alternatives, X, which captures the universe of goods that the individual may potentially observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The space X is non-empty, instead having con- stituent elements x ∈ X, each of which is a particular alternative which could be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' However, individuals do not ordinarily observe the entire set of alternatives X: while this is possible, individuals more commonly are presented with menus, smaller collections of alternatives which are drawn from the larger menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Formally, M(X) denotes the set of menus which can be derived from X, each of which is a subset of X: for a menu A, A ∈ M(X) ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There are 2X potential menus, including the full set X and the null set ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I refer to A as being a generic menu in M(X) and also make use of a generic menu B representing a menu that constitutes a superset of menu A: A ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Individuals observe a menu and consider a subset of its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To describe this, I introduce a mapping, a consideration filter, which takes an observed menu A ∈ M(X) and produces another menu Γ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Because an individual can only consider alternatives which are in the original menu, the consideration set is a subset of the original menu: Γ(A) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The consideration filter is, then: Γ : M(X) �→ M(X) The following section, a listing of properties of consideration filters, puts more structure on the nature of consideration filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 List of Properties The following list of properties describes various features that a consideration filter may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A given consideration filter may satisfy some number of these properties, all of them, or none at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='. Note that, while these properties are distinct, they are not necessarily mutually exclusive, nor are they, as a rule, concomitant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The first two properties represent adaptations of well-known axioms in rational choice theory, translated here to axiomatize consideration filters rather than choice functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the standard choice literature, axioms are frequently presented in the following style: 9 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A Generic Choice Axiom If x is chosen from menu A, it must be chosen from menu B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The weak axiom, for example, takes this basic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The idea behind this approach is that, by making assumptions on the details of the decision process, one can make claims about which alternatives from menu are chosen from another menu, simply from taking note of the latter menu’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this section, I adapt this familiar axiomatic style to accommodate consideration rather than the final choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, I present a group of axioms that, under certain conditions, give clarity on which alternatives from a given menu are in the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In doing so, I extend the rigour of axiomatic choice theory into the realm of consideration, providing the literature with a set of sample axioms to form a basis for further exploration and applied work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Each axiom I present represents a property that a consideration filter may have, outlining a condition under which the consideration set of a menu can, at least in part, be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I pair each axiom with an accompanying example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Throughout these examples, I make the following assumptions: The set of alternatives is X : {1, 2, 3, 4, 5, 6, 7, 8, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The set of menus M(X) contains 29 = 512 menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A is a generic menu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A : {1, 2, 3} B is a generic menu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' B : {1, 2, 3, 4, 5} These assumptions preserve generality and are made for the purpose of concretizing each axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I also introduce generic menus other than A and B is specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I begin with Sen’s α: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sen’s α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ satisfies Sen’s α if x ∈ Γ(B) and x ∈ A ⊆ B implies x ∈ Γ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This is the well-known Sen’s α9 axiom, adapted to the consideration setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In short, Sen’s α asserts that, whenever x is considered from a set B, x will also be considered from all of B’s subsets in which it is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Accordingly, I call this a “large-to-small” condition, guaranteeing that any alternative considered from a larger set will also be considered in its subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sen’s α example Suppose Γ satisfies Sen’s α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose there is a menu B = {1, 2, 3, 4, 5} with consideration set Γ(B) = {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If A : {1, 2, 3}, then 2, 3 ∈ Γ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 9Also known as Chernoff’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 10 This example shows how Sen’s α, when assumed to hold with respect to a given consideration filter Γ, allows an observer to determine which alternatives are in the consideration set of another menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' One salient point to note is that the consideration set of A, in this case, is not limited to 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' It is possible that 1 is also in the consideration set of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' However, given Sen’s α, we can only say for sure that 2 and 3 are in the consideration set of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' These axioms provide a lower bound on the alternatives within a menu’s consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sen’s β is another familiar result, here reformulated to match the consideration filter environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sen’s β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ satisfies Sen’s β if x1, x2 ∈ A ⊆ B and x2 ∈ Γ(B) implies x1 ∈ Γ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sen’s β is another classical axiom of decision theory that I analogize to the consider- ation setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The axiom asserts what one might call the “take-with-me” property: if two alternatives are considered from a menu, and then available in a larger menu, then one cannot be considered without the other10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sen’s β example Suppose Γ satisfies Sen’s β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose there is a menu A = {1, 2, 3} with consideration set Γ(A) = {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If 2 ∈ Γ(B), where B ⊇ A, then 3 ∈ Γ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The next property is a novel axiom I present as a counterpart to Sen’s α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Condition τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ satisfies Condition τ if x ∈ Γ(A) and A ⊆ B implies x ∈ Γ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Condition τ “reverses” Sen’s α, considering a “small-to-large case” rather than a large-to-small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Condition τ states that if an alternative x is in the consideration set of a menu A, then it is in the consideration set of every superset of A in which it is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This gives a characterization of the preservation of consideration sets under “expansion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' When Condition τ holds, no alternatives are dropped from consideration during this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Note that filters satisfying this Condition τ preclude the possibility of choice overload, the “more is less” phenomenon described by Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The concept of choice overload posits that adding items to a menu may result in previously-considered alternatives ceasing to be a part of the new consideration set, mirroring the well-known literature of “overwhelmed” consumers losing track of alternatives and thus dropping goods from consideration when confronted with larger menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Condition τ is a direct remedy for this, although it does not not apply realistically to every setting of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 10Of course, this abstracts away from externally-imposed limits on the size of consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I address this in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 11 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Condition τ example Suppose Γ satisfies Condition τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose there is a menu A = {1, 2, 3} with consideration set Γ(A) = {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For menu B ⊇ A, where B = {1, 2, 3, 4, 5}, 2, 3 ∈ Γ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Independence of Others Γ satisfies Independence of Others (IO) if the following condition holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x ∈ Γ(A) ∀ A ∈ M(X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x ∈ A or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x /∈ Γ(A) ∀ A ∈ M(X) Independence of Others, when true, states that every alternative x ∈ X is either always considered whenever it is available (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' in the menu), otherwise it is never considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This gives an intuition for the name: an alternative’s inclusion in the consideration set is purely independent of the the presence of all other alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There is no comparative process at this stage of the decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Naturally, IO corresponds to cases in which there exist certain properties of goods that are not subject to change, and individuals form consideration sets based on these unchangeable qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, an individual deciding which car to purchase at a dealership, but restricting their consideration to cars that travel at least 30 miles per gallon, chooses to consider cars in a way that is IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Any car in the available set with at least 30 MPG is in the consideration set, independent of whether any other car is in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Most consideration heuristics are not IO, particularly those which make consideration dependent on comparison or position in some order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, the heuristic “when shopping for fruits, only consider bananas in the front row” is not IO because moving a banana’s position affects whether or not it is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO consideration heuristics can, more so than other rules, generate null consideration sets in the event that no alternative in the menu satisfies the criterion of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO matches the standard rational choice model in the sense that revealed preference analysis is always possible over alternatives that are ever chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recall that the main threat to revealed preference is incomplete preferences caused by lack of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO largely closes this hole by guaranteeing that any alternative that chosen from any menu is considered across all menus in which it is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This gives greater scope to infer preferences from observed choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Of course, IO is not realistic in every setting, and this exposes a weakness in the standard rational choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Any preference ordering taken directly from observed choices implicitly assumes full consideration, which IO approximates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Thus, this presents a challenge for economists who maintain the usefulness of the standard revealed preference framework while naturally disputing the accuracy of IO in describing real-world decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 12 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO example Suppose Γ satisfies IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose there is a menu A = {1, 2, 3} with con- sideration set Γ(A) = {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If we have menus B = {1, 2, 3, 4, 5}, Q = {1, 4, 7}, and S{1, 6, 8}, 1 ∈ Γ(B), then 1 ∈ Γ(Q), and 1 ∈ Γ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I now introduce a result regarding the relationship between IO and two preceding properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (Sen’s α and Condition τ ⇔ IO) Γ satisfies Sen’s α and Condition τ if and only if it satisfies IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' See appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Intuitively, Sen’s α and Condition τ jointly provide both a “large to small” as well as a “small to large” condition on filter Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The proof strategy relies on the singleton set {x} being a subset of any menu A that includes x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If x is in the consideration set of A, Γ(A), one can guarantee consideration of x in any menu that includes by going “down and up,” a technique I demonstrate in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The next property, Dynamic Independence of Others (DIO) 11, extends IO, and the process of consideration, into a dynamic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' DIO states that, when alternatives in a menu are filtered in some order, the order does not affect the final consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To characterize DIO formally, fix the elements of menu A ∈ M(X) so as to consider every possible ordering of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, the A = {1, 2} can equivalently be represented as A = {2, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For a menu A with |A| elements, there exist |A|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' possible orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Call the first ordering A1 and the second A2, giving us a general notation where An denotes the nth ordering of menu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For a menu A, imagine an individual is endowed with |A| units of time, during each of which a singular element is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, in period 1, the individual considers the first element, then in period 2 the second element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' until the final element is considered in period |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Dynamic Independence of Others asserts that the final consideration set will be equal among each such ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Dynamic Independence of Others Γ satisfies Dynamic Indepen- dence of Others (DIO) if Γ(A1) = Γ(A2) = · · · = Γ(An) where n = |A|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='. DIO rules out the possibility behavioral biases that make choices a function of the manner in which alternatives are presented, rather than simply on the combination of alternatives which constitute the observed menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For this reason, DIO represents a very strong condition on filtering in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 11Sejal Aggarwal coined the name of this property, helping me replace a previous term that was not as clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 13 Consideration heuristics satisfying DIO are incompatible with satisficing behavior of the sort described by Simon (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In Simon’s model, consumers view alternatives x ∈ X in a set order, and make a choice once some alternative meeting quality threshold is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Clearly, order matters here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, in a setting with 10 alternatives where 5 are “satisfying” but only 3 are to be considered, the 3 out of 5 satisfying alternatives which come first will be considered, while 2 “miss out.” DIO precludes this from occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Therefore, it is not applicable to satisficing scenarios and perhaps a wide array of models in which one assumes a similar “first-mover” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' However, DIO gives the standard consideration filter model a benchmark against which such dynamic behavioral frameworks can be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' DIO example Suppose Γ satisfies DIO and consideration is made in a ordered fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose there are menus A1 = {1, 2, 3}, A2 = {1, 3, 2}, and A3 = {3, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ(A1) = Γ(A2) = Γ(A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The next property, Constant Number, makes use of the notion of cardinality, the number of elements in a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As stated earlier, call |M(X)| the number of elements (cardinality) of a menu M(X), by virtue of which |Γ(A)| is the number of elements considered from a menu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, if A = {1, 2}, then |A| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Constant Number (CN) Γ satisfies Constant Number if |Γ(A)| = n for all A ∈ such that |A| ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A constant number consideration heuristic always considers the same number of alternatives from a menu, provided that the menu has that number of alternatives available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This requires one to fix some n ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='12 Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Constant Number example Suppose Γ satisfies CN, with n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose there is a menu A = {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Gamma(A) must be {1, 2}, {1, 3}, or {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='3 Summary This section restated the formal language of consideration sets and consideration filters, allowing me to define various properties which consideration filters can potentially satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' These properties are neither mutually exclusive as a rule nor necessarily concomitant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The first two properties, Sen’s α and Sen’s β, are adaptations of well-known choice axioms, herein translated to match the setting of interest: mappings from menus to 12W is the set of natural numbers and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 14 consideration sets, rather than the standard mapping from menus to choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then introduce several novel properties that consideration filters may have: Condition τ Independence of Others (IO), Dynamic Independence of Others (DIO), and Constant Number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Condition τ reverses Sen’s α to preclude the possibility of the choice overload phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO is a very strong condition, mandating that an alternative is either always considered when available, or never considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' DIO extends IO to a dynamic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As will be shown is later sections, IO matches the rational model and, while perhaps unrealistic, is necessary for utility representation under the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Constant Number makes all consideration sets equal in cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the next section, I extend the basic consideration model to allow for the use of more than one consideration filter on a given menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4 Extension 1: Sequential Consideration Imagine an individual walks into a store looking for a new pair of shoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In narrowing down options, they focuses their attention on shoes in the first aisle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' They then only consider shoes of size 10, further narrowing down their options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' How might one model this process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As it stands, the current consideration model can only account for one consideration filter, however this example requires two: one from all shoes to those in the first aisle, and then another from those in the first aisle to those which are also size 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A extension on the basic framework is needed to account for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this section, I propose and develop a model of sequential consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' While the idea of consideration sets has been explored in the literature, two-step, or even n-step (with more than 2 rounds of narrowing down), has not been formally explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Multiple rounds of consideration may better match real-world settings, by covering scenarios in which individuals narrow down large sets based on multiple criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recall that, in the original case, we have the following mapping from menus to consideration sets to eventual choices: A �→ Γ(A) �→ c(Γ(A)) Whereas there could exist more than one consideration filter: M(X) �→ Γ1(A) �→ Γ2(A) �→ c(Γ12(A)) One might imagine Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1, in this case, having series of circles within the original one, each of which represents a new downsizing of the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I provide a formal model for this, first by constructing a space of filters that contains any of 15 the many consideration filters an individual may use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' These consideration filters may satisfy different properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, one filter may be IO and another may be CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Individuals can apply any number of filters to a given menu, narrowing it down to a final consideration set in multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This framework has precedent in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Tversky (1972) introduced the well-known process of elimination by aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In his model, menus contain alternatives, each which has or does not have some aspect — a desirable features of goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Individuals make choices in a multi-step process, where each step involves eliminating all alternatives which do not have a particular aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The process continues until only one alternative is left, which becomes the final choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Manzini and Mariotti (2007) imagine an individual using multiple “rationales” — complete and transitive preference relations — to a given menu, applying such rationale in a fixed order to arrive at a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The authors then evaluate which sorts of choice functions are consistent with the unique alternative selected by such a process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Apesteguia and Ballester (2013) provide a taxonomy for the sort of model specified by Manzini and Mariotti (2007), using game trees to formalize the idea of sequential rationalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I propose a more general model, nesting the above models into a general framework for understanding consideration in multiple steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, elimination by aspects can be modeled as applying a set of IO filters to a given menu, given that each “aspect” represents an immutable characteristic of an alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' My model differs from that of Manzini and Mariotti (2007) in that I do not make use of rational preferences until after the consideration set has been formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I take a different approach from Apesteguia and Ballester (2013), grounding my analysis in the standard choice framework rather than the style of game-theoretic decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In addition, I introduce a property, commutativity, borrowing the term from algebra to characterize relations between two or more filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I say that two or more filters are commutative if their successive application to a menu produces the same consideration set, regardless of the order in which the filters are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Commutativity, as a concept, has applications to any decision setting in which choices may be contingent on the manner in which information is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I show that any number of IO filters are always commutative, proving two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 Definitions I begin by defining the space of filters, which contains the mass of consideration filters that can be applied to a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Space of Filters There exists a space of filters γ, comprising con- stituent filters Γi ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γi represents the ith filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 16 This defines the space of filters that are available to a decision maker, each of which narrows a menu according to whatever properties it has and the heuristic that is implicitly associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Any filter that is applied to a menu comes from γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The example above involves Γ1 and Γ2, two are consideration filters which come from γ and are applied to menu A ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There are multiple ways to represent two filters applied to a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Representation of 2-Step Consideration If Γ1, and then Γ2, are applied to menu A, the resultant consideration set is Γ2(Γ1(A) or Γ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Notice that the nested notation, Γ2(Γ1(A), reads from right to left, whereas the reduced form Γ12 reads from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The timing of 2-step consideration proceeds as follows: an individual observes a menu, applied a consideration filter to that menu to get a consideration set, and then applies another filter the consideration set to end with a final consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As such, consideration filters can be analogized to contracting mappings in dynamic programming: each one, applied in succession, takes the current menu and returns a smaller menu, which is still in the space of menus M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Similar notation and intuitions hold for n-step consideration, in which any finite number of consideration filters within γ are applied to a menu in succession: Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Representation of N -Step Consideration If Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', Γn are applied to menu A, the resultant consideration set is Γn(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='(Γ2(Γ1(A)))) or Γ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Here, more than one filter can be applied to a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The timing of this process is principle the same as that of 2-step consideration, albeit with a potentially large number of new, smaller considerations sets being formed with each application of a filter Γi from γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Equipped with notation to describe sequential consideration, I now introduce the notion of commutativity, extending the framework to detail the importance of the order in which a set of filters is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 Commutative Filters When different filters are applied sequentially to a menu, does order matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, can one apply them in any order and expect to get the same final consideration set?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As a concrete example, imagine an individual asking a librarian for help selecting a new book to read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Alarmed by the incalculably-high number of books to choose from, they use two rules of thumb: they want to read fiction, and also want to read one of the first books that comes to the librarian’s mind when asked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The individual will make both of these requests, but could either ask for fiction and then for the first few books that 17 come to mind, or ask for the first few books that come to mind, and then ask which of these books are fiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This section addresses whether the consideration set from which they end up choosing the will be the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In order to answer these questions, I introduce a new concept: commutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I adapt the basic axiom of algebra here to describe cases in which consideration filters can be applied in any order to a menu, without the consideration set changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Simply put, order does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As one may expect, this is a rather strong condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In particular, its viability will often depend on whether Independence of Others (IO) is satisfied by the filters in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Before introducing results, I define commutativity in both the 2-step and n-step cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Without loss of generality, Γ1 and Γ2 refer to two, distinct, arbitrary filters in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 2 Commutative Filters Γ1 and Γ2 are commutative if Γ12(A) = Γ21(A) for all A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, x ∈ Γ12 iff x ∈ Γ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Two filters are commutative if their application to a menu A generates the same final consideration, regardless of the order in which they are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This provides language to describe consideration sets that are invariant to the order of 2 filters that generated them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I extend this definition to the n-step case, which is a more demanding condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' N Commutative Filters N filters Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', Γn are commutative if Γ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='n(A) is invariant to permutations in the order of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Commutativity for n filters works in the same way that it does for 2 filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' However this DIO requires checking every permutation, and the set of filter orderings can become very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, for a set of 5 filters, there are 120 different orderings, each of which must be verified to determine if commutativity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' These definitions allow to present the two main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='3 Commutativity Results Commutativity necessarily makes certain demands on the properties that the applied consideration filters must have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I present two results, which show that IO filters are necessarily commutative, both in the 2-step and n-step cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recall the definition of IO: Re-Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Independence of Others A consideration filer Γ satisfies Inde- pendence of Others (IO) if one of the following two conditions holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x ∈ Γ(A) ∀ A ∈ M(X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x ∈ A or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x /∈ Γ(A) ∀ A ∈ M(X) 18 IO maintains that each alternative is always considered when available, otherwise else it is never considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I now introduce the first result, which requires IO for two filters to be commutative for any arbitrary menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO and Commutativity with 2 Filters Γ1 and Γ2 are commutative for all A ∈ M(X) if and only if Γ1 and Γ2 are IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' See appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I wish to place emphasis on the fact that, while IO is necessary for two filters to be commutative for any arbitrary menu, two filters can be commutative for a given menu, while not being commutative for all menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A simple possible example is if the menu is the null set ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Any two filters are commutative for this menu, as the consideration set remains null, while, if they are not IO, order reversal could change the consideration set generated from a non-empty menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO and Commutativity with N Filters Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', Γn are commutative for all A ∈ M(X) if and only if Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', Γn are IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' See appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This extends Theorem 2 to cases with potentially more then 2 filters, although the degenerate case of n = 2 shows us that Theorem 3 nests Theorem 2, trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The proof idea, executed in the appendix, is to use the 2-step proof as a base case for induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='4 Summary This section introduces the idea of sequential consideration, the idea that multiple filters can be applied to a given menu, each of which entails a new contraction of the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Filters come from the larger space of filters γ, and may satisfy any number of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then introduce the concept of commutativity, which specifies cases in which 2, or any countable number of filters generate the same consideration set regardless of the order in which they are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Commutativity will always hold among any number of IO filters, and can hold among non-IO filters in more limited cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Generally, commutativity is a useful benchmark in thinking about consideration within settings in which information acquisition, and the decisions made from such information, are paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This allows decision theorists to specify when and under what conditions information use is unaffected by the order of its arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this next section, I use the idea that there may exist multiple filters in a decision process to endogenize the choice of a filter within the rational-attention context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 19 5 Extension 2: Preferences over Filters To this point, the concept of consideration filters has been well-developed, in particular an outline of their potential properties and interrelations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this section, I model individuals as having preferences not simply over alternatives, but over filters that affect the set of alternatives with which they are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Such a formulation is well-suited to the nuances of individual decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Choice naturally involves rules of thumb — filters, as I model them — however, individuals may apply such rules selectively across settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The best consideration heuristic for choosing a car will naturally differ from that which is optimal for shopping for bananas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I there endogenize the choice of filters, allowing individuals in my model to select which consideration filter to apply to a given menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For simplicity, I assume in this section that individuals only choose one filter, abstracting away from the sequential setup of the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In practice, the two concepts are easy to combine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I herein wish to focus on the preference portion to make the concept clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I place my model in the context of the rational attention literature, positing that there are two competing factors in choosing a filter: the greater optionality provided by a larger consideration set, and the costly mental strain associated with sifting through numerous options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I build a parsimonious model to capture the substance of this idea, while omitting certain details that risk over-complicating the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 Environment and Details I now formally characterize the space in which the definitions, axioms, and results to follow shall operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 I outlined the environment, particularly defining the relationship between the set of alternatives X and its constituent menus A ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I will now operate in a space of filters, as in the last section: Re-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Space of Filters There exists a space of filters γ, comprising constituent filters Γ ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γi represents the ith filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I allow individuals to select which filter to apply to a menu, based on the competing factors I discuss in this section’s preamble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Preference relations and related concepts will prove useful in setting up the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Filter Preference Relation There exists a weak preference relation ≿γ over the set of filters Γi ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This preference relation ≿f allows us to formally define the individual’s preference over the filters Γi ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The addition of choice over filters, as opposed to the exogenously- imposed consideration filter implicitly assumed earlier, necessitates the inclusion of this relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 20 Axiom 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Completeness of Filter Preferences The relation ≿γ is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, for two filters Γi and Γj ∈ γ, Γi ≿γ Γj or Γj ≿γ Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This axiom states that the individual has a defined preference over every pair of filters with which they can be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A point to note is that I do not model uncertainty here: for simplicity, this model has full information over filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Axiom 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Transitivity of Filter Preferences For any three filters Γi, Γj, and Γk ∈ γ, Γi ≿γ Γj and Γj ≿γ Γj implies Γi ≿γ Γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Transitivity of ≿γ prevents cycling, that is, preferences that move in a circular fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This is necessary as a consistency condition in order for choices to be indicative of underlying preferences, a key stipulation for utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Axiom 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Rationality of Filter Preferences The relation ≿γ is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Rationality of ≿γ follows from completeness and transitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='13 Axiom 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Utility Representation For a menu A, there exists a utility representa- tion uf over filters Γi ∈ γ such that: Γi ≿γ Γi ⇔ uγ(Γi) ≥ uγ(Γi) By rationality, we know that there exists a utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='14 This utility function is, by nature, ordinal in that that it captures preferences but not necessarily their intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This utility function, as well as the underlying preferences, hold the menu constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' That is, given a generic menu A ∈ M(X), preferences are then well-defined over the space of filters γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' We can now work with the following generic function: Definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Filter Utility Individuals have filter preferences: uγ = bγ(Γ, A) − cγ(Γ, A) The utility function gives us the utility that the individual derives from applying a filter Γ to menu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The first argument, bγ(Γ, A), gives the benefit of the filter which I will define as coming from the alternative x ∈ A that is eventually chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The cost, cγ(Γ, A), is the disutility of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This basic model fits in well the rational attention literature in that it balances the benefit of choices with the costs of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Additionally sense, this model extension could be argued to be a motivation for why consideration filters are used in the first 13Mas-Colell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (1995) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 14Mas-Colell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (1995) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 21 place: the standard model, in which individuals implicitly consider all goods, induces costs which may not be worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' One must note at this point that , to this point, this model has no empirical content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The terms bγ(Γ, A) and cγ(Γ, A) are too general to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In order to impose some structure upon the model, I make additional specifications: Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Choice of Alternative from Menu c(Γ(A)) selects the most preferred element from a menu A, according to the rational preference relation ≿x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' c(A) is simply the “best” alternative in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This allows one to more specifically define the benefit of a particular filter, according to the alternative that it eventually generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This makes sense as an individual will likely evaluate a consideration criterion according to the utility resulting from the choice that is eventually made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Benefit of Consideration The benefit of consideration bγ(Γ, A) is, equivalently, bγ(c(Γ(A))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The benefit of a consideration filter Γ is a function of the the best alternative in the generated consideration set, because that is the alternative which is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I now define the necessary elements in order to represent the cost of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Cardinality of Menu The cardinality of a menu A, |A|, the number of alternatives x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Cardinality is the necessary concept to define the cost of consideration, as being a function of the cardinality of the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Cost of Consideration The cost of consideration cγ(Γ, A) is, equiv- alently, cγ(|Γ(A)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I define the disutility of consideration as direct function of the number of alternatives in the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This mirrors the well-known costly attention framework: there exists some cognitive cost of attention (time, mental strain, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=') that induces negative utility coming from the sheer number of alternatives considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I place more structure on the cost function: Axiom 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Convex Cost of Consideration The cost of consideration cγ(|Γ(A)|) is globally convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The more alternatives considered, the greater the disutility, and this disutility increases marginally: ∂c(|Γ(A)|) ∂|Γ(A)| > 0, ∂2c(|Γ(A)|) ∂|Γ(A)|2 > 0 Now that all arguments have been defined, I arrive at the following specification for the utility representation of preferences over filters: 22 Definition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Specified Filter Utility Function Individuals have filter prefer- ences: uγ = bγ(c(Γ(A))) − cγ(|Γ(A)|) Filter utility uγ is broken down into two portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As specified above, uc(Γ(A))) denotes the utility derived from the alternative eventually chosen in accordance with rational preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A filter is only as good as the choice it leads to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The cost, cγ(|Γ(A)|) is a convex function of the number of alternatives an individual considers before making a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Filter Choice Rule Individuals choose filters Γ ∈ γ so as to maximize: c(γ) = arg max Γ∈γ uγ = bγ(c(Γ(A))) − cγ(|Γ(A)|) This formally specifies the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Axiom 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Filter Choice Mandate c(γ) ̸= ∅ The individual must choose a filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This prevents the convex cost function from inducing null choice sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Equipped with this model setup, I now present formal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 Results Below, I provide a remark on a cost-induced property of the filter choice process, as well two boundary results on the the number of alternatives an individual will consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Preference for Flexibility A filter choice rule c(γ) represents a preference for flexibility if Γ1(A) ⊇ Γ2(A) implies that cγ(Γ1, Γ2) = Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Choice rules satisfying this Preference for Flexibility — a classic property in the decision theory literature — will induce individuals to choose filters that generate largest possible consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This is relevant in cases in which attention is costless, meaning individuals cannot be made worse off by more options given free disposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Costly attention of the filter objective function leads Preference for Flexibility to fail in the model: Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The filter choice rule cγ does not represent a preference for flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The logic for this result follows directly from the existence of the cost function c(|Γ(A)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' It may be the case that, while more alternatives may lead to better choices, the magnitude of the increased benefit may be outweighed by the cost of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I now move into the two key theorems, detailing edge cases on filter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 23 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Costless Consideration Implies Full Consideration If cγ|Γ(A)| = 0 for all Γi ∈ γ, the individual considers all alternatives x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Γ(A) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' See appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This result is intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If attention is costless, then there is no reason to not consider all alternatives as the upside is potentially limitless with no downside cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Worthless Consideration If bγ(Γ(A)) is equal among all Γi ∈ γ, the individual chooses the filter Γi ∈ γ so as to minimize cγ|Γ(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' See appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In Theorem 4, I shut down heterogeneity in cost by setting the cost function globally to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 5 can be seen as a reversal - here, eliminate heterogeneity in “rewards” by equating benefits from choices across all consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This result is similarly intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If there is no potential benefit of a larger consideration set, the individual is justified in only considering one alternative, as there is assurance that they could not have improved their condition through increased consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='15 This result thus can be seen as a “no better off” theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' These two theorems complete my analysis of filter preferences by defining the edges of possible choices: at one extreme, individuals consider all alternatives available costlessly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' at the other, individuals consider the minimum number of alternatives so as to simply satisfy the axiom that at least one alternative must be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='3 Summary In this section, I extended the basic consideration model to a setting in which individuals may choose which consideration filters they apply to menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In doing so, I provide a rational-attention model of limited consideration, modeling individuals as weighing the benefit of a larger consideration set with the convex costs of increased consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I provide two boundary results detailing cases in which an individual considers either all alternatives, or the minimum number possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The next section outlines the viability of utility representation in a limited consideration setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 15As a nod to contract theory, this result mirrors the well-known result that, in the classic principal-agent setting, setting equal wages for high output and low output will induce shirking on the part of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 24 6 Utility Representation The ability to construct utility functions from observed choices — utility representation — relies on rational underlying preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This means that preferences must be complete and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Under limited consideration, completeness naturally does not hold because the individual decision-maker does not necessarily consider every available alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Moreover, transitivity can also fail in the event that incomplete preferences lead choices to cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As a result, limited consideration poses a fundamental threat to utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In order to maintain the utility functions commonly assumed in structural models, assumptions need to be made on the process of consideration employed by individuals, who constitute the representative agents in applied literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this section, I begin by constructing a utility function that links to consideration- mediated choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then show that consideration filters that satisfy Independence of Others (IO) are sufficient for this form of utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then follow the approach of Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017) by providing a modification of the weak axiom to match this consideration-consistent utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In both the utility representation and weak axiom settings, it becomes clear that IO poses an extremely strong condition on choices and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I acknowledge this and conjecture methods which can be used to weaken IO and preserve applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='1 Consideration Utility Function I present a general utility function, which I will use for the results that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Definition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Generic Multi-Argument Utility u(x ∈ A) = f(u1(x), u2(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', un(x)) This utility function maps each alternative x ∈ A to the real numbers according to some amalgamation of n different arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the degenerate case there may exist only one argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In order to capture the process of consideration, I denote the first constituent function, u1, to be the threshold function, the naming of which will become clear: Definition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Threshold Function Within f(u1(x), u2(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', un(x)), u1 is known as the threshold function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I require that, in order form some alternative x ∈ A to be in the consideration set Γ(A), the value generated by u1 from x must reach a certain threshold value, k∗: Axiom 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Threshold k∗ x ∈ Γ(A) if and only if u1(x) ≥ k∗ 25 Where k finds its value among the real numbers: Axiom 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' K is real k∗ ∈ R The consideration set from a menu A thus consists of its alternatives that meet the threshold: Definition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Threshold k∗ Consideration Set Γ(A) = {x ∈ A : u1 ≥ k∗} The individual forms their consideration set from the alternatives x ∈ A that meet a threshold condition that u1(x) ≥ k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This now provides a way to demonstrate consideration in the space of real numbers rather than purely through set theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' After completing this setup, the overall choice function now becomes: Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Threshold Choice Function c(A) = arg max x∈Γ(A) f(u1(x), u2(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', un(x)) Where Γ(A) = {x ∈ A : u1 ≥ k∗} In words, the above choice function specifies the process: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Individual is presented with menu A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Individual narrows menu A to consideration set Γ(A) by only considering alterna- tives x ∈ A for which u1(x) ≥ k∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' An alternative x is chosen from the consideration set Γ(A) according to some rational preference relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As stated above, this is one among many potential examples of how the set of real numbers can be used to facilitate the modeling of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The setup I have develop allows me to present a utility representation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I show that any choice process consistent with the above formulation must only involve consideration filters that satisfy IO: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO Utility Representation Γ satisfies IO if and only if ∃k∗ and u1(x) : X �→ R such that Γ(A) = {x ∈ A : u1 ≥ k∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' See appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Any filter used in the specified choice process must be IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recall that IO is equivalent to the joint presence of Sen’s α and Condition τ, and so these two conditions may also substitute into the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 26 The idea of the proof, found in the appendix, is to fix the threshold k∗ to 1, and define k∗ as 1 for all alternatives within the IO-generated consideration set, and 0 for those without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By demonstrating that the set of alternatives meeting the threshold are also those within the consideration set, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The above exercise shows an example of how consideration can be nested within utility function once properties of the consideration filters are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I now extend this exercise to the weak axiom, using IO as a proof of concept as to how the weak axiom can be modified to match the limited consideration setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='2 Weak Axiom Modifications The Weak Axiom of Revealed Preference (WARP) is a consistency condition that aligns choices with rational underlying preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The weak axiom, when it holds, mandates that if some alternative x1 is chosen over x2, then the reverse cannot happen when both are available in some other menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In essence, preferences cannot “reverse” across two different menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The weak axiom forms the basis for choice theory and, by extension, underpins utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Despite the ubiquity of the weak axiom, it does not naturally account for limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If, in the second menu, x1 were not considered, then it is quite plausible for x2 to be chosen, given that the individual may not even have been aware of the presence of x1 in the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Therefore, the weak axiom needs to be modified to match the limited consideration setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This has been done before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017) provide a modification of the weak axiom to account for the phenomenon of “choice overload.” Choice overload refers to situations in which individuals consider certain alternatives in small menus, but somehow lose track of these alternatives when presented with a much larger menu that includes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The rationale is that the overwhelming number of alternatives can be cognitively challenging and may induce forgetfulness or similar mental lapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The choice overload WARP modification requires some simple notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Refer to S and T as two menus within M(X) and call b an alternative in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The choice overload WARP modification is: Axiom 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' WARP Choice Overload (WARP-CO) For any nonempty S, there exists b∗ ∈ S such that for any T including b∗, if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' c(T) ∈ S and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' b∗ = c(T ′) for some T ′ ⊃ T then c(T) = b∗ 27 By requiring that the chosen alternative b is considered in the larger set, WARP-CO “closes the hole” punctured by choice overload, allowing choices to again be consistent with rational preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Across many settings in which limited consideration may jeopardize completeness, it behoves the decision theorist to consider WARP modifications that are appropriate to the application of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As an example, I now provide a WARP modification that matches Independence of Others (IO), using the same notation as that of WARP-CO: Axiom 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' WARP-IO For any nonempty S, there exists b∗ ∈ S such that for any T including b∗, if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' C(S) = b* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' C(T) ∈ S, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' C(T) = b* if and only if c(Q) = b*, where Q = {b*, x}, ∀ x ∈ T such that ∃ J ∈ X such that x = C(J) then c(T) = b∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In addition: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' consider B ⊂ X, where B = {b, ∅} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' if c(B) = ∅, then c(J) ̸= b for all menus J ∈ X Recall the definition of IO: Re-Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Independence of Others A consideration filer Γ satisfies Inde- pendence of Others (IO) if one of the following two conditions holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x ∈ Γ(A) ∀ A ∈ M(X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x ∈ A or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' x /∈ Γ(A) ∀ A ∈ M(X) The two conditions I provide in WARP-IO correspond to the two portions of the IO definition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the first case, WARP-IO mandates that any choice must pairwise beat every other alternative in the menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This matches full consideration in that there cannot be a case in which an alternative is selected despite not being preferred to some other alternative, which may happen if limited consideration restricts the scope of the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The second branch of WARP-IO concerns alternatives that are not chosen when they are the only alternative available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Naturally, this must mean that these alternatives, for some reason, were not considered, given that they are in the available set16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In this case, they are never chosen, as IO states that alternatives that are not considered in some instance are never considered in any menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 16I again remind the reader that, throughout the paper, I assume that the available set only includes alternatives that are within the individual’s budget set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 28 I have presented a modification to the Weak Axiom of Revealed Preference (WARP) to align with consideration heuristics that are modeled by IO filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In doing so, I follow in the vein of Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017), who also devise a WARP modification to account for the nuances of consideration-mediated choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Similar modifications are, in principle, possible for any property of consideration filters, and future literature can make large strides by developing the appropriate modifications to match the common behavioral processes most commonly observed in real-world economic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='3 Summary In section, I have explored the implications of limited consideration on utility repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Choices made under limited consideration may appear to reflect underlying preferences that are neither complete nor transitive, presenting a threat to the ability to use utility functions to represent consideration-mediated choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In order to preserve utility representation, assumptions need to be made on the properties of consideration filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' As an example, I show an example of a utility function which captures choices made using an IO consideration filter, showing that IO is sufficient to model choices made via the objective function I set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I also address the Weak Axiom of Revealed Preference (WARP), which implies full consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I follow Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017) by modifying the weak axiom to match a property of consideration filters, providing an example for IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In future, as I discuss in the next section, the characterizations I provide ought to be extended to cover filter properties that are not as strong as IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 7 Future Directions In this analysis I provide a language for discussing limited consideration, extended the basic model, and conjectured conditions for utility representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Below I briefly mention three potential avenues for future work on limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Weakening IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO is clearly the strongest condition one can impose upon the formation of consideration sets — either an alternative is always considered when available, or else it is never considered, not even in the singleton set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO, the foundation for some of the results in this paper, is clearly not flexible enough to match the nuances in individual behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' However, the standard model, and classical revealed preference, make a similarly strong assumption: that the available set is always the consideration set (Γ(A) = A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Full consideration, a special case of IO,17 is therefore not realistic 17Define the IO filter rule as: every alternative x ∈ A is always considered when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The available set is then always equivalent to the consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 29 either, and so weakening of IO must be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This will require prudence, however, as the desired condition must be weaker than IO, while still having enough “bite” to ensure falsifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Incorporating Consideration into Structural Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Many empirical phenomena could be better understood using the limited consideration framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, Larcom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017) explore route choice in the London subway system before and after a temporary shortage, finding that some commuters used different routes after the transportation restart, meaning they were not optimizing before the strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The authors build a structural model of route choice, finding that daily commuters often were not aware of routes that were faster than the ones they previously used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' They allude to limited consideration, wondering why commuters did not experiment enough before the strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' One potential answer is limited consideration: individuals did not consider all available routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This could take the form of a satisficing heuristic, or it could be modeled using the rational-attention model I developed in Section 5 in the event that the process of analyzing routes is seen to be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' How Much Do We Toss Out?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Related to the first two suggestions, literature across all subfields assumes full consideration in some fashion or another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Given that this is not a realistic axiom, it is worth examining what sorts of theoretical and empirical results are no longer viable once one understands the salience of limited consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, in a setting in which individuals often use consideration heuristics, welfare analysis grounded in observed choices is likely to need adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 8 Conclusion Revealed preference takes observed choices to be indicative of individual preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, if, given the set {A, B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', Z}, an individual chooses R, revealed preference indicates that R is preferred to all other letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This approach to choice theory assumes that individuals examine every available option before making a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In contrast, limited consideration posits that individuals narrow menus into consideration sets before making choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This framework is better suited to modeling individual decision-making, which often involves various rules of thumb that filter out certain options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The literature on limited consideration and related processes has been well-developed and includes theoretical models18, consumer choice analyses19, axiomatic characteriza- tions of normative preferences, 20 and structural work in various empirical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='21 In this paper, I provide a general model of limited consideration to unite the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 18Masatlioglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Masatlioglu and Nakajima (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Lleras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017) 19Hauser and Wernerfelt (1990);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Roberts and Lattin (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Erdem and Keane (1996) 20Cherepanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2013) and Ridout (2021) 21Abaluck and Gruber (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Larcom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Abaluck and Adams-Prassl (2021) 30 I begin by outlining the main features of consideration model: individuals observe menus, narrow them into consideration sets, and make choices from said consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The channel by which menus are translated into consideration sets is captured by consideration filters, functions that downsize menus into consideration sets by mapping them to one of their subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Consideration filters correspond to various rules of thumb that individuals may use in narrowing down menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To account for the large number of heuristics that individuals may use in practice, I introduce a number of properties that consideration filters may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' These properties describe the manner in which a menu begets a consideration set, and are neither mutually exclusive nor mandated to coexist with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A consideration filter may satisfy one, all, or none of the properties I introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The strongest condition, Independence of Others (IO), describes consideration filters which select a certain set of alternatives in any menu in which they appear, and never selects any alternative not in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' IO, which closely approximates the standard rational model, forms the basis for a number of later results in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I then extend the consideration model in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' First, I develop a model of sequential consideration, which allows more than one filter to be applied to a given menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This matches settings in which individuals are thought to apply more than one rule of thumb in narrowing down large choice sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I introduce the concept of commutativity, borrowing from algebra to describe filters which can be applied to a menu in any order and still generate the same final consideration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Filters satisfying IO are always commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' My second model extension is a rational-attention analogue, in which I model an individual who must choose which filter to apply to a given menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Individuals weigh competing forces in choosing a filter: the greater optionality associated with a larger consideration set and the examination costs associated with sifting through a large number of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I present two boundary results, showing in which cases an individual will consider every alternative, or the minimum number of alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I address the implications of limited consideration on utilty representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The abil- ity to construct utility functions corresponding to observed choices relies on underlying preferences being both complete and transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In the consideration model, both condi- tions often fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To counteract this, I construct a utility function that accurately models choices made using an IO consideration filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Such a link between consideration-based utility functions and the filter properties that may generate them may be possible for a large array of consideration heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' I also provide a modification to the Weak Axiom that corresponds to IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In both cases, I use IO as a basic proof of concept to demonstrate how standard choice theory can be reconciled with the limited consideration framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There are many potential future directions for theoretical work on limited consider- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For example, IO can be weakened to find a filter property that is more realistic yet tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In addition, current structural choice models used by applied economists 31 can be better reconciled with limited consideration, especially in empirical settings in which choices are thought to be made from consideration sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In summary, I have presented a detailed characterization of limited consideration, nesting some of the prior literature into a formal language while also developing novel model extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The hope is that a complete theory of consideration, building off this work as well as that of other scholars, will improve the robustness and applicability of rational choice theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 32 References Abaluck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Adams-Prassl (2021): “What do consumers consider before they choose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Identification from asymmetric demand responses,” The Quarterly Journal of Economics, 136, 1611–1663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Abaluck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Gruber (2016): “Evolving choice inconsistencies in choice of prescription drug insurance,” American Economic Review, 106, 2145–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Apesteguia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Ballester (2013): “Choice by sequential procedures,” Games and Economic Behavior, 77, 90–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Caplin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Dean (2011): “Search, choice, and revealed preference,” Theoret- ical Economics, 6, 19–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Cherepanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Feddersen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Sandroni (2013): “Rationalization,” Theoretical Economics, 8, 775–800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Erdem, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Keane (1996): “Decision-making under uncertainty: Capturing dynamic brand choice processes in turbulent consumer goods markets,” Marketing science, 15, 1–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Hauser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Wernerfelt (1990): “An evaluation cost model of considera- tion sets,” Journal of consumer research, 16, 393–408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Larcom, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Rauch, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Willems (2017): “The benefits of forced experimen- tation: striking evidence from the London underground network,” The Quarterly Journal of Economics, 132, 2019–2055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Lleras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Masatlioglu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Nakajima, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Ozbay (2017): “When more is less: Limited consideration,” Journal of Economic Theory, 170, 70–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Manzini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Mariotti (2007): “Sequentially rationalizable choice,” American Economic Review, 97, 1824–1839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Mas-Colell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Whinston, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Green, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (1995): Microeconomic theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 1, Oxford university press New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Masatlioglu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Nakajima (2015): “Completing incomplete revealed preference under limited attention,” The Japanese Economic Review, 66, 285–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Masatlioglu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Nakajima, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Ozbay (2012): “Revealed Attention,” American Economic Review, 102, 2183–2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 33 Ridout, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (2021): “Choosing for the right reasons,” Unpublished manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Roberts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Lattin (1991): “Development and testing of a model of consideration set composition,” Journal of Marketing Research, 28, 429–440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Simon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (1955): “A behavioral model of rational choice,” The quarterly journal of economics, 99–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Tversky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' (1972): “Elimination by aspects: A theory of choice.” Psychological review, 79, 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 34 Appendix: Proofs of Results in Main Text Theorem 1: Sen’s α and Condition τ ⇔ IO For the if direction, I show that any filter Γ satisfying Sen’s α and Condition τ is also IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Suppose filter Γ satisfies α and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose that some alternative x is in the consideration set generated by this Γ on menu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By Sen’s α, x is in the consideration set of the singleton menu {x} ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='22 By τ, x is in the consideration set of any menu that includes x, since any such menu is a superset of {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='23 Since x is always considered when available, Γ is IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For the only if direction, I now show that any filter Γ satisfying IO also satisfies Sen’s α and Condition τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Suppose some filter Γ satisfies IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Further suppose that some alternative x is in the consideration set generated by this Γ on menu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By IO, x is always considered when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By definition, x appears in all subsets (Sen’s α)and supersets (Condition τ) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Therefore Γ satisfies Sen’s α and Condition τ, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 2: IO and Commutativity with 2 Filters I start with the if direction: if Γ1 and Γ2 are IO, then they are commutative for any menu A ∈ M(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This simply requires me to show that x ∈ Γ12(A) if and only if x ∈ Γ21(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' First, assume that x ∈ Γ12(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recall that, if an IO filter retains some alternative x, then it must retain x in all menus A that contain x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Therefore, x ∈ Γ12(A) implies that x ∈ Γ1(A) for if x ∈ A, and it is also true that x ∈ Γ2(A) for if x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Now, recall that Γ21 applies filter Γ1 followed by Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Both filters, as I have shown, always retain x when x ∈ A, and so x ∈ Γ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Now, for the only if direction, which entails showing that if any two filters Γ1 and Γ2 are commutative for any menu A, then they must be IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This result can be proved by inspection, noting the intuition behind IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If in the event that filter Γ1 is not IO, there necessarily exists a pathological case in which the consideration set generated by Γ1’s involves a comparative selection process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' alternatives are selected based on their desirability relative to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' In that case, the consideration set generated the successive application of Γ1 and Γ2 is clearly dependent upon the structure of the original menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 3: IO and Commutativity with N Filters I prove this result using Theorem 2 as a base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By Theorem 2, any two filters Γ1 and Γ2 are commutative for any menus A ∈ M(X) if and only if they are IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Because 22This is the going down step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 23This is the going up step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 35 Γ1 and Γ2 are commutative, they can be collapse into one filter, since the order of their application does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Call this new filter Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Suppose one adds a third filter Γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By Theorem 2, Γc and Γ3 are commutative if and only if they are IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Recalling that Γc is an amalgam of Γ1 and Γ2, filters Γ1, Γ2, and Γ3 are commutative if and only if they are IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If I add a fourth filter Γ4, the same approach works by collapsing the first three filters into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The proof strategy scales up for any n filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Theorem 4: Costless Consideration Implies Full Consider- ation Recall the choice function c(γ) = arg max Γ∈γ uΓ = bγ(c(Γ(A))) − cγ(|Γ(A)|) If consideration is costless, cγ(|Γ(A)|) = 0, therefore we have: c(γ) = arg max Γ∈γ uΓ = bγ(c(Γ(A))) Which is maximized by considering all alternatives x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content='24 Theorem 5: Worthless Consideration Recall the choice function c(γ) = arg max Γ∈γ uΓ = bγ(c(Γ(A))) − c(|Γ(A)|) If the benefit of consideration bγ(c(Γ(A))) is constant across all filters, then it is invariant to any change in the filter and this has no effect on the optimal choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The problem reduces to cost minimization: c(γ) = arg max Γ∈γ uΓ = −c(|Γ(A)|) In order to maximize the objective, while satisfying Axiom 6, the filter choice mandate, the individual will simply choose the filter that minimizes the cost of consid- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 24This assumes the benefit of consideration bγ is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 36 Theorem 6: IO Utility Representation The if direction, that IO filters can be represented by the threshold choice function, is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' List all alternatives x ∈ A which are in the consideration set of the IO filter Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For each x ∈ Γ(A), set u1(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' For each x /∈ Γ(A), set u1(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Then set k∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Therefore, all alternatives Γ1 meet the u1 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' The only if direction makes us of the same technique, paired with the application Sen’s α and Condition τ to the relevant menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To prove this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Define a set Y ⊆ X as follows y ∈ Y iff x ∈ Γ(A) for some A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Define u1(x) = � � � 1 if x ∈ Y 0 if x /∈ Y It remains to verify that Γ(A) = {x ∈ A : u1(x) ≥ k∗} as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Let’s check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' By construction, the right hand side equals {x ∈ A : x ∈ Y } = A ∩ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' So it remains to check whether Γ(A) = A ∩ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' There are two arguments needed: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If x ∈ Γ(A) then x ∈ A ∩ Y 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' If x ∈ A ∩ Y then x ∈ Γ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To prove the first argument, assume that x ∈ Γ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Then automatically x ∈ A, and x ∈ Y By definition of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' To prove the second argument, assume that x ∈ A ∩ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Then by definition of Y there exists a set B such that x ∈ Γ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Let’s consider the set C := A ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' We know x ∈ C since both A and B contain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' First, apply Sen’s α condition to x ∈ C ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This implies that x ∈ Γ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' Then apply Condition τ condition to x ∈ C ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' This implies that x ∈ Γ(A), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE5T4oBgHgl3EQfig-m/content/2301.05649v1.pdf'} diff --git a/n9E2T4oBgHgl3EQfzwgf/content/tmp_files/2301.04133v1.pdf.txt b/n9E2T4oBgHgl3EQfzwgf/content/tmp_files/2301.04133v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5ed246e024618ec1b083f9f3dbc9c1e46870d4c --- /dev/null +++ b/n9E2T4oBgHgl3EQfzwgf/content/tmp_files/2301.04133v1.pdf.txt @@ -0,0 +1,2233 @@ +Astronomy & Astrophysics manuscript no. aanda +©ESO 2023 +January 11, 2023 +Clustering dependence on Lyα luminosity from MUSE surveys at +3 < z < 6 +Yohana Herrero Alonso,1 T. Miyaji,2 L. Wisotzki,1 M. Krumpe,1 J. Matthee,4 J. Schaye,3 H. Aceves,2 H. Kusakabe,5 +and T. Urrutia1 +1 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany +e-mail: yherreroalonso@aip.de +2 Universidad Nacional Autónoma de México, Instituto de Astronomía (IA-UNAM-E), AP 106, Ensenada 22860, BC, México +3 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA, Leiden, The Netherlands +4 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland +5 Observatoire de Gèneve, Université de Gèneve, 51 Chemin de Pégase, 1290 Versoix, Switzerland +Received xxx/Accepted xxx +ABSTRACT +We investigate the dependence of Lyα emitter (LAE) clustering on Lyα luminosity and connect the clustering properties of ≈ L⋆ +LAEs with those of much fainter ones, namely, ≈ 0.04L⋆. We use 1030 LAEs from the MUSE-Wide survey, 679 LAEs from +MUSE-Deep, and 367 LAEs from the to-date deepest ever spectroscopic survey, the MUSE Extremely Deep Field. All objects have +spectroscopic redshifts of 3 < z < 6 and cover a large dynamic range of Lyα luminosities: 40.15 < log(LLyα/erg s−1) < 43.35. +We apply the Adelberger et al. K-estimator as the clustering statistic and fit the measurements with state-of-the-art halo occu- +pation distribution (HOD) models. We find that the large-scale bias factor increases weakly with an increasing line luminosity. +For the low-luminosity (log⟨LLyα/[erg s−1]⟩ = 41.22) and intermediate-luminosity (log⟨LLyα/[erg s−1]⟩ = 41.64) LAEs, we com- +pute consistent bias factors blow = 2.43+0.15 +−0.15 and binterm. = 2.42+0.10 +−0.09, whereas for the high-luminosity (log⟨LLyα/[erg s−1]⟩ = 42.34) +LAEs we calculated bhigh = 2.65+0.13 +−0.11. Consequently, high-luminosity LAEs occupy dark matter halos (DMHs) with typical masses +of log(Mh/[h−1M⊙]) = 11.09+0.10 +−0.09, while low-luminosity LAEs reside in halos of log(Mh/[h−1M⊙]) = 10.77+0.13 +−0.15. The minimum +masses to host one central LAE, Mmin, and (on average) one satellite LAE, M1, also vary with Lyα luminosity, growing from +log(Mmin/[h−1M⊙]) = 10.3+0.2 +−0.3 and log(M1/[h−1M⊙]) = 11.7+0.3 +−0.2 to log(Mmin/[h−1M⊙]) = 10.7+0.2 +−0.3 and log(M1/[h−1M⊙]) = 12.4+0.4 +−0.6 +from low- to high-luminosity samples, respectively. The satellite fractions are ≲ 10% (≲ 20%) at 1σ (3σ) confidence level, sup- +porting a scenario in which DMHs typically host one single LAE. We next bisected the three main samples into disjoint subsets to +thoroughly explore the dependence of the clustering properties on LLyα. We report a strong (8σ) clustering dependence on Lyα lumi- +nosity, not accounting for cosmic variance effects, where the highest luminosity LAE subsample (log(LLyα/erg s−1) ≈ 42.53) clusters +more strongly (bhighest = 3.13+0.08 +−0.15) and resides in more massive DMHs (log(Mh/[h−1M⊙]) = 11.43+0.04 +−0.10) than the lowest luminosity one +(log(LLyα/erg s−1) ≈ 40.97), which presents a bias of blowest = 1.79+0.08 +−0.06 and occupies log(Mh/[h−1M⊙]) = 10.00+0.12 +−0.09 halos. We discuss +the implications of these results for evolving Lyα luminosity functions, halo mass dependent Lyα escape fractions, and incomplete +reionization signatures. +Key words. large-scale structure – high-redshift galaxies – HOD models – dark matter halo – satellite galaxies +1. Introduction +Dark matter halos (DMHs) serve as sites of galaxy formation but +their co-evolution is still a matter of investigation. Observations +deliver snapshots of the luminosities of galaxies at given red- +shifts, while numerical analyses succeed at simulating the evolu- +tion and copiousness of DMHs. Linking these two constituents is +not straightforward but, because the spatial distribution of bary- +onic matter is biased against that of dark matter (DM), the former +indirectly traces the latter. The evolutionary stage of the two dis- +tributions depends on both the epoch of galaxy formation and the +physical properties of galaxies (see Wechsler & Tinker 2018 for +a review). Thus, studying the dependence of the baryonic-DM +relation on galaxy properties is essential for better understand- +ing the evolution of the two components. +Exploring the spatial distribution of high-redshift (z > 2) +galaxies and its dependence on physical properties provides an +insight into the early formation and evolution of the galaxies +we observe today. Clustering statistics yield observational con- +straints on the relationship between galaxies and DMHs, as well +as on their evolution. Traditional studies of high-z galaxies (Stei- +del et al. 1996; Hu et al. 1998; Ouchi et al. 2003; Gawiser et +al. 2007; Ouchi et al. 2010; Khostovan et al. 2019) model the +large-scale (R ≳ 1 − 2 h−1cMpc) clustering statistics with a +two parameter power-law correlation function that takes the form +ξ = (r/r0)−γ (Davis & Peebles 1983) to derive the large-scale lin- +ear galaxy bias and the associated typical DMH mass. To make +full use of the clustering measurements, the smaller separations +of the nonlinear regime (R ≲ 1 − 2 h−1cMpc) are modeled by +relating galaxies to DMHs within the nonlinear framework of +halo occupation distribution (HOD) modeling. In this context, +the mean number of galaxies in the DMH is modeled as a func- +tion of DMH mass, further assessing whether these galaxies oc- +cupy the centers of the DMHs or whether they are satellite galax- +ies. +Article number, page 1 of 17 +arXiv:2301.04133v1 [astro-ph.GA] 10 Jan 2023 + +A&A proofs: manuscript no. aanda +Although clustering studies of high-redshift galaxies are +plentiful, HOD modeling has been rarely used to interpret the re- +sults. While several works have focused on Lyman-break galaxy +(LBG) surveys, only one study fit a sample of Lyman-α emitters +(LAEs) with HOD models (Ouchi et al. 2018). Durkalec et al. +(2014); Malkan et al. (2017); Hatfield et al. (2018); Harikane et +al. (2018) applied the full HOD framework to sets of LBGs to +put constraints on the central and satellite galaxy populations, +while Ouchi et al. (2018) partially exploited the power of HOD +models in a sample of LAEs to infer the threshold DMH mass +for central galaxies. +The number of studies that have investigated the correlations +between clustering strength and physical properties of high- +redshift galaxies is slightly higher. In [O ii] and [O iii] emission- +line-selected galaxy samples, Khostovan et al. (2018) found a +strong halo mass dependence on the line luminosity and stel- +lar mass. Durkalec et al. (2018) also observed a correlation with +stellar mass, together with a further dependence on UV lumi- +nosity, in a sample of LBGs. However, these correlations be- +come somewhat unclear near the epoch of reionization (z ≈ 6). +Based on LAEs surveys, Ouchi et al. (2003); Bielby et al. (2016); +Kusakabe et al. (2018) revealed tentative trends (≈ 1σ) between +luminosity (both UV and Lyα) and clustering strength, while +only Khostovan et al. (2019) reported a clear (5σ) correlation +between inferred DMH mass and Lyα luminosity. +In a previous study (Herrero Alonso et al. 2021), we used +68 MUSE-Wide fields to measure the LAE clustering with the +K-estimator method presented in Adelberger et al. (2005). We +computed the clustering at large scales (R > 0.6 h−1Mpc) to de- +rive the linear bias factor and the typical DMH mass of LAEs. By +splitting our main sample into subsets based on physical prop- +erties of LAEs, we also found a tentative 2σ dependence on +Lyα luminosity. Here, we extend this work with larger and more +deeply spectroscopically confirmed samples and a refined set of +analysis methods. We measured the clustering at smaller scales, +applied full HOD modeling, and studied the dependence of the +clustering properties on Lyα luminosity. +The paper is structured as follows. In Sect. 2, we describe the +data used for this work and we characterize the LAE samples. In +Sect. 3, we explain our method for measuring and analyzing the +clustering properties of our galaxy sets. We present the results of +our measurements in Sect. 4. In Sect. 5, we discuss our results +and their implications, and we investigate the clustering depen- +dence on Lyα luminosity. We give our conclusions in Sect. 6. +Throughout this paper, all distances are measured in comov- +ing coordinates and given in units of h−1Mpc (unless otherwise +stated), where h = H0/100 = 0.70 km s−1 Mpc−1. We assume +the same h to convert line fluxes to luminosities. Thus, there are +implicit h−2 +70 factors in the line luminosities. We use a ΛCDM +cosmology and adopt ΩM = 0.3, ΩΛ = 0.7, and σ8 = 0.8 (Hin- +shaw et al. 2013). All uncertainties represent 1σ (68.3%) confi- +dence intervals. +2. Data +The MUSE spectroscopic surveys are based on a wedding cake +design, namely: a first spatially wide region (bottom of the cake) +is observed with a short exposure time (1 hour), while deeper +observations (10 hours exposure) are carried out within the first +surveyed area (middle tier of the cake). Contained in the last ob- +served region, an even deeper survey (140 h) is then built up (at +the top of the cake). These three surveys are known as: MUSE- +Wide (Herenz et al. 2017; Urrutia et al. 2019), MUSE-Deep (Ba- +con et al. 2017; Inami et al. 2017; Bacon et al. 2022), and MUSE +Extremely Deep Field (MXDF; Bacon et al. 2022). Each of them +can be seen as a different layer of a wedding cake, where higher +layers become spatially smaller and correspond to deeper obser- +vations. In what follows, we give further details on survey and +galaxy sample construction. +2.1. MUSE-Wide +The spectroscopic MUSE-Wide survey (Herenz et al. 2017; Ur- +rutia et al. 2019) comprises 100 MUSE fields distributed in the +CANDELS/GOODS-S, CANDELS/COSMOS and the Hubble +Ultra Deep Field (HUDF) parallel field regions. Each MUSE +field covers 1 arcmin2. While 91 fields were observed with an ex- +posure time of one hour, nine correspond to shallow (1.6 hours) +reduced subsets of the MUSE-Deep data (see next section; as +well as Bacon et al. 2017), located within the HUDF in the +CANDELS/GOODS-S region. However, we do not include the +objects from this region since they overlap with the MUSE-Deep +sample (see next section and gap in the left panel of Fig. 1). +The slight overlap between adjacent fields leads to a total spatial +coverage of 83.52 arcmin2. The red circles in Fig. 1 display the +spatial distribution of the LAEs from the MUSE-Wide survey. +We refer to Urrutia et al. (2019) for further details on the survey +build up, reduction and flux calibration of the MUSE data cubes. +In this paper, we extend (x2 spatially, 50% more LAEs) the +sample used in Herrero Alonso et al. (2021) and include all +the 1 h exposure fields from the MUSE-Wide survey. Despite +the somewhat worse seeing (generally) in the COSMOS region +(right panel of Fig.1), we demonstrate in Appendix A that adding +these fields does not significantly impact our clustering results +but helps in minimizing the effects of cosmic sample variance. +We also expanded the redshift range of the sample. While +MUSE spectra cover 4750–9350 Å, implying a Lyα redshift in- +terval of 2.9 <∼ z <∼ 6.7, we limited the redshift range to 3 < z < 6 +(differing from the more conservative range of Herrero Alonso +et al. 2021; 3.3 < z < 6) as the details of the selection function +near the extremes are still being investigated. Section 2 of Her- +rero Alonso et al. (2021) describes the aspects relevant to our +analysis on the construction of a sample of LAEs, as well as the +strategy to measure line fluxes and redshifts. The redshift distri- +bution of the sample is shown in red in the top panel of Fig. 2. +Systematic uncertainties introduced in the redshift-derived 3D +positions of the LAEs have negligible consequences for our clus- +tering approach (see Sect. 2.2 in Herrero Alonso et al. 2021). +Within 83.52 arcmin2 and in the selected redshift interval, +we detected a total of 1030 LAEs. This implies a LAE density +of more than 13 objects per arcmin2 or n ≈ 1·10−3 h3Mpc−3 (for +3 < z < 6). At the median redshift of the sample ⟨z⟩ = 4.0, the +transverse extent of the footprint is ≈ 43 h−1Mpc. The range of +Lyα luminosities is 40.92 < log(LLyα/[erg s−1]) < 43.35 (see red +circles in Fig. 2), with a median value of log⟨LLyα/[erg s−1]⟩ = +42.34 (or ≈ L⋆ in terms of characteristic luminosity L⋆; Herenz +et al. 2019), which makes this sample the highest luminosity data +set of our three considered surveys. The Lyα luminosity distri- +bution is shown in red in the right panel of Fig. 2. The main +properties of the MUSE-Wide LAEs are summarized in Table 1. +2.2. MUSE-Deep +MUSE-Deep (10 hour MOSAIC; Bacon et al. 2017; Inami et +al. 2017; Bacon et al. 2022) encompasses nine fields located in +the CANDELS/GOODS-S region of the HUDF, each spanning +1 arcmin2 and observed with a 10 h exposure time. The total +Article number, page 2 of 17 + +Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +Fig. 1: Spatial distribution of the LAEs from the MUSE-Wide survey (red circles), MUSE-Deep (green squares) and MXDF (blue +stars). The overlapping objects between the MXDF and MUSE-Deep samples have been removed from the MUSE-Deep LAE +set, while those LAEs overlapping in MUSE-Deep and MUSE-Wide have been removed from the MUSE-Wide LAE sample. The +MUSE-Wide survey covers part of the CANDELS/GOODS-S region and the HUDF parallel fields (left panel) as well as part of +the CANDELS/COSMOS region (right panel). See Figure 1 in Urrutia et al. (2019) for the layout of the MUSE-Wide survey +without individual objects, Figure 1 in Bacon et al. (2017) for that of MUSE-Deep, and Figure 2 in Bacon et al. (2022) for that of +MUSE-Deep (MOSAIC) and MXDF together. +Table 1: Properties of the LAE samples. +Area/[arcmin2] +Number LAEs +⟨z⟩ +n/[h3Mpc−3] +log(LLyα/[erg s−1]) range +log⟨LLyα/[erg s−1]⟩ +MUSE-Wide +83.52 +1030 +4.0 +1 · 10−3 +40.92 – 43.35 +42.34 (≈ L⋆) +MUSE-Deep +9.92 +679 +4.1 +8 · 10−3 +40.84 – 43.12 +41.64 (≈ 0.2L⋆) +MXDF +1.47 +367 +4.2 +3 · 10−2 +40.15 – 43.09 +41.22 (≈ 0.08L⋆) +Notes: Properties marked with ⟨⟩ represent median values for the galaxies in the samples. +Table 2: Properties of the LAE subsamples. +Number LAEs +⟨z⟩ +log⟨LLyα/[erg s−1]⟩ +MUSE-Wide low L +(log LLyα < 42.34) +515 +3.7 +42.06 (≈ 0.5L⋆) +MUSE-Wide high L +(log LLyα > 42.34) +515 +4.1 +42.53 (≈ 1.5L⋆) +MUSE-Deep low L +(log LLyα < 41.64) +340 +3.7 +41.46 (≈ 0.1L⋆) +MUSE-Deep high L +(log LLyα > 41.64) +339 +4.5 +41.89 (≈ 0.3L⋆) +MXDF low L +(log LLyα < 41.22) +183 +4.0 +40.97 (≈ 0.04L⋆) +MXDF high L +(log LLyα > 41.22) +184 +4.5 +41.54 (≈ 0.2L⋆) +Notes: Properties marked with ⟨⟩ represent median values for the galaxies in the subsamples. +spatial coverage is 9.92 arcmin2. We represent the spatial distri- +bution of the survey in green in Fig. 1. We did, however, remove +the MUSE-Deep objects that are selected in the deepest survey, +described in the next section. We refer to Bacon et al. (2017, +2022) for a detailed description on survey construction and data +reduction. +The sources in MUSE-Deep were blindly detected and ex- +tracted using ORIGIN (Mary et al. 2020), based on a matched +filtering approach and developed to detect faint emission lines +in MUSE datacubes. While the redshift measurements and line +classifications were carried out with pyMarZ, a python version +of the redshift fitting software MarZ (Hinton et al. 2016), the +line flux determination was conducted with pyPlatefit, which is +a python module optimized to fit emission lines of high-redshift +spectra. The redshift distribution of the sample is shown in green +in the top panel of Fig. 2, also within 3 < z < 6. +The LAE density of the MUSE-Deep sample is 8 · +10−3 h3Mpc−3 (68 LAE per arcmin2 in the whole redshift range). +The survey spans ≈ 8.7 h−1Mpc transversely. The range of Lyα +luminosities is 40.84 < log(LLyα/[erg s−1]) < 43.12, represented +with green squares in Fig. 2, together with its distribution (right +panel). MUSE-Deep is our intermediate luminous dataset, with +a median luminosity of log⟨LLyα/[erg s−1]⟩ = 41.64. The sample +properties are recorded in Table 1. +2.3. MUSE Extremely Deep +The MUSE Extremely Deep Field (Bacon et al. 2022) is situated +in the CANDELS/GOODS-S region and overlaps with MUSE- +Deep and MUSE-Wide. It is composed of a single quasi circu- +lar field with inner and outer radii of 31” and 41”, respectively. +While a 140 hour exposure was employed to observe the totality +of the field, the inner field is 135 hours deep, decreasing to 10 +Article number, page 3 of 17 + +MUSE-Wide +MUSE-Deep +-27.70 +MXDF +27.75 +Dec +-27.80 +-27.85 +53.30 +53.25 +53.20 +53.15 +53.10 +53.05 +RA2.34 +2.32 +2.30 +2.28 +e +2.26 +2.24 +2.22 +2.20 +150.20 +150.18 +150.16 +150.14 +150.12 +150.10 +150.08 +RAA&A proofs: manuscript no. aanda +hours depth at the outer radius. This makes MXDF the deepest +spectroscopic survey to date. For further details see Bacon et al. +(2022) and the blue data points in Fig.1, where the MXDF field +is overplotted on the previous surveys. +The survey assembly and data reduction is described in Ba- +con et al. (2022) and is similar to the one applied to MUSE-Deep +(Bacon et al. 2017). The source extraction in MXDF and the red- +shift and flux measurements are conducted following the same +procedure as was done for MUSE-Deep. The redshift distribu- +tion of the sample is shown in blue in the top panel of Fig. 2. +Contained within ≈1.47 arcmin2 and over the same redshift +range as for the previous catalogues, we detected 367 LAEs, cor- +responding to a LAE density of n ≈ 3·10−2 h3Mpc−3 (432 LAEs +per arcmin2 at 3 < z < 6). With a median redshift of ⟨z⟩ = 4.2, +the footprint covers ≈ 2.8 h−1Mpc (transversely). The Lyα lumi- +nosities span 40.15 < log(LLyα/[erg s−1]) < 43.09 (see blue stars +in Fig. 2 and its distribution in the right panel). The median Lyα +luminosity is log⟨LLyα/[erg s−1]⟩ = 41.22 (or ≈ 0.08L⋆), more +than one order of magnitude fainter than for MUSE-Wide. This +makes MXDF the faintest ever observed sample of non-lensed +LAEs. The main properties are listed in Table 1. +2.4. LAE subsamples +We bisected the main samples into disjoint subsets based on their +median Lyα luminosity to investigate the clustering dependence +on this quantity. We did not merge the main LAE datasets be- +cause their distinct Lyα luminosities, together with their slightly +different location on the sky, might introduce systematics in the +clustering measurements. The subsample properties are summa- +rized in Table 2 and described in the following. +We split the MUSE-Wide sample at the median Lyα lu- +minosity log⟨LLyα/[erg s−1]⟩ = 42.34. The two subsamples +consist of 515 LAEs each. The low-luminosity subset has +a median redshift and Lyα luminosity of ⟨zlow⟩ = 3.7 and +log⟨LLyαlow/[erg s−1]⟩ = 42.06, while the high-luminosity sub- +sample has ⟨zhigh⟩ = 4.1 and log⟨LLyαhigh/[erg s−1]⟩ = 42.53. +The median redshift of the number of galaxy pairs for the low- +luminosity subset is zpair ≈ 3.4, and that for the high-luminosity +one is zpair ≈ 4.1. +We next bisected the MUSE-Deep set at the median Lyα lu- +minosity log⟨LLyα/[erg s−1]⟩ = 41.64. The low-luminosity sub- +sample has 340 LAEs and presents a median redshift and Lyα lu- +minosity of ⟨zlow⟩ = 3.7 and log⟨LLyαlow/[erg s−1]⟩ = 41.46. The +high-luminosity subset is formed by 339 LAEs with ⟨zhigh⟩ = +4.5 and log⟨LLyαhigh/[erg s−1]⟩ = 41.89. While for the low- +luminosity subsample zpair ≈ 3.5, for the high-luminosity one +zpair ≈ 4.4. +We also divide the sample with the largest dynamic +range of Lyα luminosities (MXDF) at the median Lyα lu- +minosity log⟨LLyα/[erg s−1]⟩ += +41.22. While the lower lu- +minosity subset contains 183 LAEs with ⟨zlow⟩ = 4.0 and +log⟨LLyαlow/[erg s−1]⟩ += +40.97, the higher luminosity sub- +sample consists of 184 LAEs with ⟨zhigh⟩ += +4.5 and +log⟨LLyαhigh/[erg s−1]⟩ = 41.54. For the low-luminosity subset, +we have zpair ≈ 3.9, and for the high-luminosity one, we have +zpair ≈ 4.8. +The redshift distribution of each subsample is shown in +Fig. 3. The corresponding median redshifts are represented with +a vertical dashed line. Despite the similar median redshifts be- +tween the subsample pairs, the redshift distributions are signifi- +cantly different, with a higher amount of spike-trough contrasts +in the high-luminosity subsets. +Fig. 2: Lyα luminosity-redshift for the LAEs in MUSE-Wide +(red circles), MUSE-Deep (green squares) and MXDF (blue +stars). The dashed colored lines correspond to the median +log LLyα values of the corresponding samples. The redshift and +LLyα distributions are shown in the top and right panel, respec- +tively. +3. Methods +3.1. K-estimator +Galaxy clustering is commonly measured by two-point corre- +lation function (2pcf) statistics. Samples investigated by this +method typically span several square degrees on the sky. With +MUSE, we encounter the opposite scenario. By design, MUSE +surveys cover small spatial extensions on the sky and provide a +broad redshift range. Although the MUSE-Wide survey is the +largest footprint of all MUSE samples, its nature is still that +of a pencil-beam survey. Its transverse scales are of the order +of 40 h−1Mpc, while in redshift space it reaches almost 1500 +h−1Mpc. If we consider the deeper samples, the difference is +even more prominent: 8.7 vs 1500 h−1Mpc for MUSE-Deep and +2.8 versus 1500 h−1Mpc for MXDF. It is thus paramount to ex- +ploit the radial scales and utilize alternative methods to the tra- +ditional 2pcf. +In Herrero Alonso et al. (2021) we applied the so-called K- +estimator, introduced by Adelberger et al. (2005), to a subset +of our current sample. Here, we build on our previous work by +extending the dataset and measuring the small-scale clustering +required to perform full HOD modeling. The details of the K- +estimator are given in Sect. 3.1 of Herrero Alonso et al. (2021). +In the following, we provide a brief description of the method. +The K-estimator measures the radial clustering along line-of- +sight distances, Zi j, by counting galaxy pairs (formed by galaxy +i and galaxy j) in redshift space at fixed transverse separations, +Ri j. Although the K-estimator does not need a random sample +to carry out the clustering measurements, its nature is very sim- +ilar to that of the projected two-point correlation function. We +bin by Ri j, shown with distinct radii in the cylinders of Fig. 4, +and count the number of pairs within individual transverse bins, +for two different ranges of Zi j, represented in red and blue in +Fig. 4. The K-estimator as a function of Ri j is then defined as +the ratio of galaxy pairs within the first Zi j interval (blue cylin- +der) and the total Zi j range (red and blue cylinder), quantifying +the excess of galaxy pairs in the first Zi j bin with respect to the +total one. We optimize the choice of the Zi j ranges, and thus the +Article number, page 4 of 17 + +0.5 +0.0 +43.0 +. +42.5 +S +42.0 +41.5 +MUSE-Wide +MUSE-Deep +41.0 +MXDF +6.0 0 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +ZYohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +Fig. 3: Redshift distribution of the subsamples bisected at the median Lyα luminosity of MUSE-Wide, MUSE-Deep and MXDF +(panels from left to right). Blue (red) colors show the low- (high-) luminosity subsets. The vertical dashed lines represent the median +redshift of the corresponding subsample. +K-estimator, by seeking out the estimator that delivers the best +sensitivity for the clustering signal (i.e., the highest signal-to- +noise ratio, S/N; see Sect. 3.1.2 in Herrero Alonso et al. 2021). +Although slightly different than in Herrero Alonso et al. (2021), +we find nearly identical K-estimators for each of the current sam- +ples (K0,7 +7,45 for MUSE-Wide, K0,7 +7,45 for MUSE-Deep, and K0,7 +7,40 for +MXDF), whose clustering signals only differ in their S/N. We +chose the same K-estimator for the three data sets, K0,7 +7,45. +The K-estimator is directly related to the average underlying +correlation function (see Eq. 2 in Herrero Alonso et al. 2021). +In fact, its definition is proportional to a combination of pro- +jected two-point correlation functions corresponding to the blue +and red cylinders of Fig. 4. While the traditional 2pcf method +integrates the correlation function ξ(Rij, Zij) over line-of-sight +separations up to a maximum line-of-sight distance πmax, the K- +estimator integrates up to a2 and a3. The correlation function +ξ(Ri j, Zi j) can be approximated with a power-law following Lim- +ber (1953) equations as we did in Herrero Alonso et al. (2021), or +modeled with a halo occupation distribution (HOD) model (see +Sect. 3.3). For reference, randomly distributed galaxies in space +(ξ(Ri j, Zi j) = 0) provide K0,7 +7,45(Rij) values equal to 7/45 (see Eq. 2 +in Herrero Alonso et al. 2021). Samples with data points signifi- +cantly above 7/45 dispense clustering signals. +3.2. Error estimation +3.2.1. Error estimation for the MUSE-Wide survey +Applying clustering statistics delivers correlated data points. +One single galaxy might be part of more than one galaxy pair +and can therefore contribute to several Rij bins, especially if +they are adjacent. In order to quantify the actual correlation be- +tween data points, we applied the jackknife resampling tech- +nique, followed by the computation of the covariance matrix (see +e.g., Krumpe et al. 2010; Miyaji et al. 2011). For the MUSE- +Wide sample, we employed ten logarithmic bins in the range +0.16 < Ri j < 27.5 h−1Mpc, discarding lower Rij scales since +they host very few galaxy pairs. +We then found a compromise between the number of inde- +pendent regions (jackknife zones) and the size of the jackknife +zones and divide the sky coverage into Njack = 10 regions, each +of which extends ≈ 4 h−1Mpc in both RA and Dec directions +(see Appendix B for a visual representation of the sky division). +The limited spatial extent of the survey does not allow for a +higher number of jackknife zones. We then constructed Njack +jackknife subsamples, excluding one jackknife zone at a time, +and computed the K-estimator for each of the subsets. The K- +estimator measurements are then used to derive the covariance +Fig. 4: Sketch of the K-estimator, representing the relative ge- +ometry that probe the one- and two-halo term scales. The empty +blue and filled red cylinders, delimited by |a2| = 7 h−1Mpc and +|a3| = 45 h−1Mpc respectively, illustrate the line-of-sight dis- +tance Zi j intervals within which we count galaxy pairs at fixed +transverse separations Ri j, represented by nested cylinders. Pairs +of LAEs connected with green lines within the same DMH (filled +gray circle) contribute to the one-halo term (small Ri j scales), +while pairs belonging to two different DMHs (yellow lines) +probe the two-halo term (larger Ri j separations). +matrix Mi j, which quantifies the correlation between bins i and +j. The matrix is expressed as +Mi j = Njack − 1 +Njack +�������� +Njack +� +k=1 +� +Kk(Ri) − ⟨K(Ri)⟩ +� +× +� +Kk(Rj) − ⟨K(Rj)⟩ +�� +, +(1) +where Kk(Ri), Kk(Rj) are the K-estimators from the k-th jack- +knife samples and ⟨K(Ri)⟩, ⟨K(Rj)⟩ are the averages over all +jackknife samples in the i, j bins, respectively. The error bar for +the K-estimator at the ith bin comes from the square root of the +diagonal element ( √Mii) of the covariance matrix, our so-called +"jackknife uncertainty." This approach could not be followed in +Herrero Alonso et al. (2021) because of the smaller sky cover- +age. Instead, we used a galaxy bootstrapping approach. In Ap- +pendix C, we compare the two techniques and show that boot- +strapping uncertainties are ≈ 50% larger than the jackknife error +bars, in agreement with Norberg et al. (2009), who found that +boostrapping overestimates the uncertainties. +Article number, page 5 of 17 + +MUSE-Wide +High 'L (log(LLyα/[erg s-1)] > 42.34) +20 +Low L (log(Llyα/[erg s-1)] < 42.34) +15 +LI +A +10 +5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +ZMUSE-Deep +Highi L (log(LLyα/[erg s-1)l > 41.64) +20 +Low L (log(LLyα/[erg s-1)] < 4l.64) +15 +L +A +之 +10 +5 +0 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +ZMXDF +High L (log(LLyα/[erg s-1)] > 41.22) +20 +Low L (log(LLyα/[erg s-1)] < 41.22) +15 +AE +10 +5 +0 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +Zα3 = 45 +α²= 7 +αi = 0 +-α2 = 7 +—α3 = —45 +Zij +Riji1 +Rij2 +Rij4 +Rij3A&A proofs: manuscript no. aanda +We next search for the best-fit parameters by minimizing the +correlated χ2 values according to +χ2 = +Nbins +� +i=1 +Nbins +� +j=1 +� +K(Ri) − K(Ri)HOD +� +× M−1 +ij +� +K(Rj) − K(Rj)HOD +� +, +(2) +where Nbins = 10 is the number of Rij bins, K(Ri), K(Rj) are +the measured K-estimators and K(Ri)HOD, K(Rj)HOD are the K- +estimators predicted by the HOD model for each i, j bin, respec- +tively. +Regardless of the larger sample considered in this work, we +are still limited by the spatial size of the survey, which only per- +mits a small number of jackknife zones. The insufficient statis- +tics naturally lead to a higher noise contribution in the covari- +ance matrix, which cause the χ2 minimization to mathematically +fail (i.e., cases of χ2 < 0) when the full covariance matrix is in- +cluded. Hence, we only incorporated the main diagonal of the +matrix and its two contiguous diagonals. In Appendix B, we dis- +cuss the high level of noise in the matrix elements corresponding +to bins that are significantly apart from each other. We also ver- +ify the robustness of our approach and show that our clustering +results are not altered (within 1σ) by this choice. +3.2.2. Error estimation for the deeper surveys +The small sky coverage of the deeper surveys does not allow us +to follow the same error estimation approach as for the MUSE- +Wide survey. In Appendix C, we not only compare the boot- +strapping technique applied in Herrero Alonso et al. (2021) +to the jackknife approach performed in MUSE-Wide, but we +also consider the Poisson uncertainties. We demonstrate that +Poisson and jackknife errors are comparable in our sample. In +fact, we show that while bootstrapping uncertainties are ≈ 50% +larger than jackknife errors, Poisson uncertainties are only ≈ 7% +higher. Thus, and similarly to Adelberger et al. (2005); Diener +et al. (2017); Khostovan et al. (2018), we stick to Poisson un- +certainties for the MUSE-Deep and MXDF samples. For these +datasets, we measure the K-estimator in eight and six loga- +rithmic bins in the ranges 0.09 < Rij/[h−1Mpc] < 4.75 and +0.09 < Ri j/[h−1Mpc] < 1.45, respectively, constrained by the +spatial extent of the surveys. +We then perform a standard χ2 minimization to find the best +fitting parameters to the K-estimator measurements. Namely, +χ2 = +Nbins +� +i=1 +� K(Ri) − K(Ri)HOD +σi +�2 +, +(3) +where K(Ri), K(Ri)HOD, and σi denote the measured K- +estimator, the HOD modeled K-estimator and the Poisson un- +certainty in the ith bin, respectively. +We note that the standard χ2 minimization does not account +for the correlation between bins. Although in Appendix B we +show that only contiguous bins are moderately correlated, we +should take the resulting fit uncertainties with caution. +3.3. Halo occupation distribution modeling +The clustering statistics can be approximated with a power- +law or modeled with state-of-the-art HOD modeling. Traditional +clustering studies make use of power laws to derive the corre- +lation length and slope, from which they infer large-scale bias +factors and typical DMH masses. This simple approach deviates +from the actual shape of the clustering statistic curve, even in the +linear regime, and its inferred DMH masses suffer from system- +atic errors (e.g., Jenkins et al. 1998 and references therein). To +overcome these concerns, physically motivated HOD models do +not treat the linear and non-linear regime alike but differentiate +between the clustering contribution from galaxy pairs that reside +in the same DMH and pairs that occupy different DMHs. +In Herrero Alonso et al. (2021) we only modeled the two- +halo term of the K-estimator with HOD modeling, which only +delivered the large-scale bias factor and the typical DMH mass +of the sample. We now extend into the non-linear regime (i.e., +Ri j < 0.6 h−1Mpc) of the one-halo term. We can then model +the clustering measured by the K-estimator with a full HOD +model, combining the separate contributions from the one- (1h, +i.e., galaxy pairs residing in the same DMH) and the two-halo +(2h, i.e., galaxy pairs residing in different DMHs) terms: +ξ = ξ1h + ξ2h, +(4) +where ξ is the correlation function. +The HOD model we used is the same as in Herrero Alonso +et al. (2021), an improved version of that described by Miyaji et +al. (2011); Krumpe et al. (2012, 2015, 2018). We assumed that +LAEs are associated with DMHs, linked by the bias-halo mass +relation from Tinker et al. (2005). From Tinker et al. (2005), +we also included the effects of halo-halo collisions and scale- +dependent bias. The mass function of DMHs, which is denoted +by φ(Mh)dMh, is based on Sheth et al. (2001), and the DMH +profile is taken from Navarro, Frenk & White (1997). We use +the concentration parameter from Zheng et al. (2007), and the +weakly redshift-dependent collapse overdensity from Navarro, +Frenk & White (1997); van den Bosch et al. (2013). We fur- +ther incorporated redshift space distortions (RSDs) in the two- +halo term using linear theory (Kaiser infall; Kaiser 1987 and +van den Bosch et al. 2013). We did not model RSDs in the one- +halo term because the peculiar velocity has negligible effects to +our K-estimator as demonstrated in the following. The veloc- +ity dispersion (σv) of satellites in a Mh halo can be estimated +by σ2 +v ≈ GMh/(2Rvir), where Rvir is the virial radius (Tinker +2007). Its effect on the line-of-sight physical distance estimate +is then σv/H(z). For 1011−12 h−1M⊙ DMH masses, which are +typical for our sample, with virial radii of ≈ 0.02 − 0.05 (physi- +cal) h−1Mpc, the line-of-sight distance estimation is deviated by +≈ 0.15 − 0.30 h−1Mpc, corresponding to a peculiar velocity dis- +persion of σv ≈ 80 − 170 km s−1. This is significantly small +compared to our a2 = 7 h−1Mpc. We thus assume that the one- +halo term contributes only to the Zi j = 0 − 7 h−1Mpc bin. We +evaluated the HOD model at the median redshift of N(z)2, where +N(z) is the redshift distribution of the sampled galaxy pairs. For +our three main datasets, zpair ≈ 3.8. +The mean halo occupation function is a simplified version +of the five parameter model by Zheng et al. (2007). We fixed +the halo mass at which the satellite occupation becomes zero to +M0 = 0 and the smoothing scale of the central halo occupation +lower mass cutoff to σlog M = 0, due to sample size limitations. +We define the mean occupation distribution of the central galaxy +⟨Nc(Mh)⟩ as +⟨Nc(Mh)⟩ = +� 1 +(Mh ≥ Mmin) +0 +(Mh < Mmin) +(5) +and that of satellite galaxies ⟨Ns(Mh)⟩ as +⟨Ns(Mh)⟩ = ⟨Nc(Mh)⟩ · +� Mh +M1 +�α +, +(6) +Article number, page 6 of 17 + +Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +where Mmin is the minimum halo mass required to host a central +galaxy, M1 is the halo mass threshold to host (on average) one +satellite galaxy, and α is the high-mass power-law slope of the +satellite galaxy mean occupation function. The total halo occu- +pation is given by the sum of central and satellite galaxy halo +occupations, N(Mh) = Nc(Mh) + Ns(Mh). +The dependencies of the HOD parameters on the shape of the +K-estimator are detailed in Appendix D. In short, for the HOD +parameters there selected, the clustering amplitude of the two- +halo term is ascertained by the hosting DMHs and is thus very +sensitive to their mass, Mmin, and to the fraction of galaxies in +massive halos with respect to lower-mass halos, linked to α. The +clustering in the one-halo term regime, however, is affected by +the three parameters in a complex manner; roughly Mmin and α +vary the amplitude, and α as well as (moderately) M1 modify the +slope. +To find the best-fit HOD model, we construct a 3D parame- +ter grid for Mmin, M1, and α. We vary log(Mmin/[h−1M⊙]) in the +range 9.5 − 11.2, log(M1/Mmin) from 0.5 to 2.5, and α within +0.2 − 4.3, all in steps of 0.1. For each parameter combination, +we computed ξ (Eq. 4), converted it to the K-estimator using +Eq. 2 in Herrero Alonso et al. (2021), and computed a χ2 value +(Eqs. 2 or 3). We then used the resulting 3D χ2 grid to estimate +the confidence intervals for the HOD parameters. For each point +on a 2D plane, we search for the minimum χ2 for the contour- +ing along the remaining parameter. The contours we plot are at +∆χ2 = 3.53 and 8.02, which correspond to Gaussian 68% (1σ) +and 95% (2σ) confidence levels, respectively, applying the χ2 +distribution for three degrees of freedom. The projections of the +68% probability contours on the three interesting parameters are +then used to compute the uncertainty of each HOD parameter. +For each point in the three parameter grid, we also computed +the large-scale galaxy bias factor, b, and the fraction of satellite +galaxies per halo, fsat, as follows: +b = +� +⟨N(Mh)⟩ bh(Mh) φ(Mh) dMh +� +⟨N(Mh)⟩ φ(Mh) dMh +, +(7) +fsat = +� +⟨Ns (Mh)⟩ φ(Mh) dMh +� +⟨N(Mh)⟩ φ(Mh) dMh +, +(8) +where bh(Mh) denotes the large-scale halo bias. The typical +DMH mass is determined by the large-scale galaxy bias factor. +We ultimately compute the bias and fsat distributions from the +HOD models that fall within the 68% confidence (for the three- +parameter space) contours. These distributions are then used to +assess the uncertainties in the bias and fsat. +4. Results from HOD modeling +4.1. Fit results from the MUSE-Wide survey +Using the K-estimator K0,7 +7,45, we compute the clustering of +our LAE sample in ten logarithmic bins in the range 0.16 < +Ri j/[h−1Mpc] < 27.5, with error bars calculated following the +jackknife resampling technique described in Sect. 3.2.1. In the +left top panel of Figure 5, we show the measured clustering sig- +nal, with all MUSE-Wide data points significantly above the 7/45 +baseline, which represents the expected clustering of an unclus- +tered population. +Following the procedure laid out in Sect. 3.3, we obtain con- +straints on the HOD parameters. From the grid search and the +χ2 minimization, we find the best HOD fit to the K-estimator, +colored in black in the same figure and dissected into the one- +and two-halo term contributions. It can be seen from the residu- +als (bottom) that the model is in remarkable agreement with the +measurements. +A somewhat intriguing feature, at least at first sight, is the +kink in the two-halo term profile at 0.2 < Ri j/[h−1Mpc] < 0.4. +This reflects the effect of the halo-halo collision introduced in the +HOD model formalism by Tinker et al. (2005), where the galaxy +pairs within the same DMH cannot contribute to the two-halo +term. +Our fitting allows us to find the best-fit HOD from Eqs. 5 and +6. In the right top panel of Fig. 5, we represent the best HODs +for the central, satellite, and total LAEs from the MUSE-Wide +survey. While the halo mass needed to host one (central) LAE +is log(Mh/[h−1M⊙]) > 10.6, satellite galaxies are only present +if the DMHs are at least one order of magnitude more massive +(log(Mh/[h−1M⊙]) > 11.6). +As described in Sect. 3.3, we also compute the confidence +regions for the HOD parameters. We show the probability con- +tours (red) in Fig. 6. The wobbliness of the curves, especially +those involving α, is caused by making use of a discrete grid. +For our sample, the contours are constrained to have α > 1, +log(M1/Mmin) > 1, and log(Mmin/[h−1M⊙]) > 10.4. +We list the best-fit HOD parameters in Table 3. While +the minimum DMH mass required to host a central galaxy is +log(Mmin/[h−1M⊙]) = 10.7+0.2 +−0.3, that needed to host one central +and (on average) one satellite is log(M1/[h−1M⊙]) = 12.4+0.4 +−0.6 +(i.e., log(M1/Mmin) = 1.7+0.4 +−0.6). The power-law slope of the num- +ber of satellites is found to be α = 2.8+0.9 +−0.7. The inferred typical +DMH mass is log(Mh/[h−1M⊙]) = 11.09+0.10 +−0.09, corresponding to +a large-scale bias factor of b = 2.65+0.13 +−0.11. The high values of +log M1 and α, considering the typical DMH mass of LAEs, sug- +gest a low number of satellite galaxies detected in our sample. +Seeking robust information about the number of satellite +galaxies, we compute the satellite fraction fsat (Eq. 8) for each +parameter combination. We find fsat ≲ 0.10 at the 3σ confidence +level, being fsat = 0.012+0.018 +−0.009. That is, ≈ 3% (1σ upper limit) +of the LAEs in the MUSE-Wide survey are satellites. In other +words, at most ≈ 2 out of ≈ 65 DMHs in our sample host one +satellite LAE. +4.2. Fit results from MUSE-Deep +We measure the clustering of the MUSE-Deep LAE sample with +the same K-estimator in eight logarithmic bins within 0.09 < +Ri j/[h−1Mpc] < 4.75. We compute Poisson uncertainties as laid +out in Sect. 3.2.2 and display the result in the middle left panel +of Fig. 5. Overplotted on the clustering signal, we show the best +HOD fit, split into the one- and two-halo term contributions. The +good quality of the fit is quantified with the residuals in the bot- +tom panel of the figure. +Following the procedure described in Sect. 3.2.2, we com- +pute the confidence intervals for the HOD parameters and list +them in Table 3. We plot the probability contours (green) in +Fig. 6, which overlap significantly with those from the MUSE- +Wide sample. Central LAEs can occupy DMHs if these are at +least as massive as log(Mmin/[h−1Mpc]) = 10.5+0.2 +−0.1, whereas, +in order to host satellite LAEs, the halos must have masses +log(M1/[h−1Mpc]) = 12.4+0.3 +−0.2 (log(M1/Mmin) = 1.9+0.3 +−0.2). These +values correspond to a large-scale bias and typical DMH mass +b = 2.42+0.10 +−0.09 and log(Mh/[h−1M⊙]) = 10.89+0.09 +−0.09, which are +similar to those found in the MUSE-Wide survey. The derived +Article number, page 7 of 17 + +A&A proofs: manuscript no. aanda +Fig. 5: Best-fit HOD models to the LAE clustering measurements (blue data points) from MUSE samples. Top left: Blue dashed, +red dotted, and black continuous curves show the one-halo, two-halo, and total clustering terms from the MUSE-Wide sample, +respectively. The black straight line shows the expected K value of an unclustered sample. The residuals are shown below. The +uncertainties are computed with the jackknife technique described in Sect. 3.2.1. Top right: Best-fit HODs for central (red dot- +ted), satellite (blue dashed), and total LAEs (black continuous) from the MUSE-Wide survey. Shaded regions correspond to 1σ +confidence space. Middle: Same but for MUSE-Deep and using Poisson error bars. Bottom: Same but for MXDF and Poisson +uncertainties. +satellite fraction is fsat = 0.004+0.009 +−0.002, consistent with that from +the MUSE-Wide LAE sample. +We then compute the best-fit HOD for central, satellite and +total LAEs (middle right panel of Fig. 5). In line with the best- +fit HOD parameters and somewhat lower than the values found +for the MUSE-Wide survey, the smallest DMH that can host a +central LAE has a mass of log(Mh/[h−1M⊙]) > 10.4, more than +one order of magnitude lower than that required to host one ad- +ditional LAE (satellite). +4.3. Fit results from the MUSE Extremely Deep Field +We make use of six logarithmic bins in the range 0.09 < +Ri j/[h−1Mpc] < 1.45 and Poisson errors (see Sect. 3.2.2) to +quantify the clustering of the sample of LAEs from MXDF. We +show the K-estimator measurements in the bottom left panel of +Fig. 5, along with the corresponding best HOD fit. +The probability contours are plotted in blue in Fig. 6, sig- +nificantly apart from those of MUSE-Wide and MUSE-Deep. +While the minimum DMH mass to host a central LAE is +log(Mmin/[h−1Mpc]) = 10.3+0.2 +−0.3, that to host one central and one +Article number, page 8 of 17 + +Total (1h+2h) +MUSE-Deep +2-halo term +0.4 +1-halo term +5 +0.3 +0.2 +Data/model +1.25 +1.00 +0.1 +1.0 +10.0 +Rij [h-1Mpc]total +centrals +satellites +1.0 +N(Mh) +0.1 +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +logM [h-1M]Total (1h+2h) +MXDF +2-halo term +0.4 +1-halo term +5 +0.3 +0.2 +Data/model +1.25 +1.00 +0.1 +1.0 +10.0 +Rij [h-1Mpc]total +centrals +satellites +1.0 +N(Mh) +0.1 +12.5 +10.0 +10.5 +11.0 +11.5 +12.0 +13.0 +logM [h-1M]Total (1h+2h) +MUSE-Wide +2-halo term +0.4 +1-halo term +5 +0.3 +0.2 +Data/model +1.25 +1.00 +0.1 +1.0 +10.0 +Rij [h-1Mpc]total +centrals +satellites +1.0 +N(Mh) +0.1 +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +13.0 +logM [h-1M]Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +Table 3: Best-fit HOD parameters for the main samples of LAEs. +⟨z⟩ +log(Mmin/[h−1M⊙]) +log(M1/Mmin) +α +fsat +b +log(Mh/[h−1M⊙]) +MUSE-Wide +4.0 +10.7+0.2 +−0.3 +1.7+0.4 +−0.6 +2.8+0.9 +−0.7 +0.012+0.018 +−0.009 +2.65+0.13 +−0.11 +11.09+0.10 +−0.09 +MUSE-Deep +4.1 +10.5+0.2 +−0.1 +1.9+0.3 +−0.2 +3.0+0.4 +−0.5 +0.004+0.009 +−0.002 +2.42+0.10 +−0.09 +10.89+0.09 +−0.09 +MXDF +4.2 +10.3+0.2 +−0.3 +1.4+0.3 +−0.2 +1.5+0.5 +−0.5 +0.08+0.02 +−0.05 +2.43+0.15 +−0.15 +10.77+0.13 +−0.15 +Notes: ⟨z⟩ is the median redshift of the sample. Mmin, M1 are the threshold DMH masses to host a central and a satellite +LAE, respectively. α is the high-mass power-law slope of the number of satellite galaxies, fsat is the satellite fraction, b is the +large-scale bias factor and Mh is the typical DMH mass of the galaxy sample. +Fig. 6: Confidence contours in the three HOD parameter space. Red corresponds to MUSE-Wide, green to MUSE-Deep, and blue +to MXDF. The thick (dashed) contours represent the 68.3% (95.5%) confidence, at ∆χ2 = 3.53 (8.02) level. The crosses stand for +best-fit (χ2 +min), searched along the remaining parameter for each 2D parameter plane. +satellite LAE is log(M1/[h−1Mpc]) = 11.7+0.3 +−0.2 (log(M1/Mmin) = +1.4+0.3 +−0.2). These values are somewhat lower than those found for +the MUSE-Wide survey and correspond to a bias factor and +typical halo mass of b = 2.43+0.15 +−0.15 and log(Mh/[h−1M⊙]) = +10.77+0.13 +−0.15, respectively. The inferred satellite fraction is fsat = +0.08+0.02 +−0.05 ( fsat ≲ 0.2 at the 3σ confidence level), tentatively +higher than that found in the MUSE-Wide survey. +From the best-fit HOD parameters, we calculate the HODs +for central, satellite and total LAEs and show them in the bottom +right panel of Fig. 5. Significantly lower than in the MUSE-Wide +survey, central LAEs reside in DMHs if these are more massive +than log(Mh/[h−1M⊙]) > 10.2. For the satellite case, and simi- +larly to the previous LAE samples, they only exist if the halos +are around one order of magnitude more massive. +Article number, page 9 of 17 + +2.50 +2.25 +2.00 +1.75 +1.50 +MUSE-Wide +1.25 +MUSE-Deep +MXDF +1.00 +68.3% +0.75 +95.5% +0.50 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +10.0 +10.5 +1.0 +11.0 0.5 +1.5 +2.0 +2.5 +logM1/Mmin +log(Mmin/h-1M)A&A proofs: manuscript no. aanda +It is worth pointing out that the three HOD parameters have +some degree of degeneracy, printed out in the diagonally elon- +gated probability contours in log M1/Mmin – α space in the bot- +tom right panel of Fig. 6. This can be understood as follows: +a higher α in the models causes an increase of satellites at +high mass halos, but this can be compensated by producing less +satellites by increasing log M1/Mmin. While this correlation is +clearly visible for the MUSE-Wide and MUSE-Deep samples, +the MXDF dataset only seems to be affected in the 95% con- +fidence contour. We did not observe clear correlations between +other parameters with any of our samples. Appendix A shows +how our K-estimator varies with the parameters. The causes of +parameter degeneracies are also noticeable in Fig. D.1. We note +however that while the correlation between the HOD parame- +ters leads to the perturbed shape of the probability contours, the +lowest (MXDF) and highest luminosity (MUSE-Wide) sample +contours are detached from each other. Thus, for the purposes of +this study, simultaneously fitting the three HOD parameters and +showing their correlations is preferable over, for instance, fixing +α to a dubious value. +5. Discussion +5.1. Clustering dependence on Lyα luminosity +The complex radiative transfer processes that the Lyα photons +are subject to make the search for correlations between Lyα lu- +minosity and other physical properties a difficult task. Despite +this complication, Yajima et al. (2018) predicted a correlation +between simulated LLyα and halo mass based on halo merger +trees and Lyα radiative transfer calculations. Khostovan et al. +(2019) is, however, the only study so far that has reported a clear +(5σ) relation between these quantities using observational data. +Motivated by these results, we exploited the large dynamic range +of Lyα luminosities that we cover to investigate the relation be- +tween Lyα luminosity and DMH mass. As a first step, we com- +pare the K-estimator measurements in the MUSE-Wide survey +(highest luminosity LAE sample: ⟨LLyα⟩ ≈ 1042.34 erg s−1, but +still fainter than those in Khostovan et al. (2019)) and in MXDF +(faintest LAE sample; ⟨LLyα⟩ ≈ 1041.22 erg s−1) and show the +outcome of this comparison in the left panel of Fig. 7. +The relatively luminous LAEs from the MUSE-Wide survey +cluster slightly more strongly (bWide = 2.65+0.13 +−0.11) than the low- +luminosity LAEs from MXDF (bMXDF = 2.43+0.15 +−0.15). The cluster- +ing measurements and bias factor (b = 2.42+0.10 +−0.09) in MUSE-Deep +(log(LLyα/[erg s−1]) = 41.64) fall between those from MUSE- +Wide and MXDF. We convert the bias factors from the three +main samples of this study into typical DMH masses and plot +them as a function of their median Lyα luminosity with colored +symbols in Fig. 8. +Although the three main datasets sample the same region of +the sky, their transverse coverage is limited and somewhat dif- +fers. Therefore, our results are affected by cosmic sample vari- +ance. Ideally, this uncertainty is estimated from the variance of +clustering measurements from simulated mocks in different lines +of sight. Inferring cosmic variance from a large set of mocks +that are able to reproduce the observed clustering of our LAEs is +however beyond the scope of this paper. +We further investigate the possible dependence on LLyα by +splitting the main LAE samples into disjoint subsets (see Ta- +ble 2). We compute the K-estimator in each LLyα subsample, +find the best HOD fit and list the large-scale bias factors and the +typical DMH masses in Table 4. We also plot the typical DMH +masses in Fig. 8 (empty symbols) as a function of the median +Table 4: Best HOD fit large-scale bias factor and typical DMH +mass for the LAE subsamples. +Subsample +⟨z⟩ +b +log(Mh/[h−1M⊙]) +MUSE-Wide high L +4.1 +3.13+0.08 +−0.15 +11.43+0.04 +−0.10 +MUSE-Wide low L +3.7 +2.45+0.10 +−0.12 +10.92+0.09 +−0.11 +MUSE-Deep high L +4.5 +2.41+0.12 +−0.10 +10.40+0.12 +−0.10 +MUSE-Deep low L +3.7 +2.20+0.09 +−0.11 +10.68+0.09 +−0.13 +MXDF high L +4.5 +3.10+0.24 +−0.22 +10.96+0.15 +−0.15 +MXDF low L +4.0 +1.79+0.08 +−0.06 +10.00+0.12 +−0.09 +Notes. ⟨z⟩ is the median redshift of the subsample. The uncertainties do +not include cosmic sample variance. +LLyα of the subsamples. We find that typical halo mass increases +from 1010.00 to 1011.43M⊙ between 1040.97 and 1042.53 erg s−1 in +line luminosity. +For each subsample pair, the high-luminosity subset always +clusters more strongly than the low-luminosity one and, in this +case, cosmic sample variance effects can be completely ne- +glected because subset pairs span the exact same area on the +sky. The most pronounced difference is found when splitting the +MXDF sample, the dataset with the largest dynamic range of +Lyα luminosity. The best HOD fits deliver blow = 1.79+0.08 +−0.06 and +bhigh = 3.10+0.24 +−0.22 (3.9σ significant). +Despite its higher luminosity, we infer a less massive DMH +for the MUSE-Deep high-luminosity subsample than for the +main dataset. This is due to the higher zpair of the subset (see +Sect. 2.4 and 3.3). Because we evaluate the HOD model at +zpair, a higher redshift corresponds to HOD models in which the +halo mass function presents a lower number density of mas- +sive halos and, thus, deliver less massive typical DMHs. The +same reasoning applies when comparing the high-luminosity +MXDF and low-luminosity MUSE-Deep subsamples and the +high-luminosity MUSE-Deep and low-luminosity MUSE-Wide +subsets. While each subsample pair presents similar median lu- +minosities, the former also has similar zpair, unlike the latter one +(see Sect. 2.4). This translates into similar DMH masses for the +first pair but significantly distinct masses for the second. +We last consider the most extreme cases, the low-luminosity +subset from MXDF and the high-luminosity one from the +MUSE-Wide survey. We show the measured clustering in +the two subsamples in the right panel of Fig. 7. The high- +luminosity LAEs cluster 8σ more strongly than the low- +luminosity LAEs, without accounting for cosmic variance. We +find that LAEs with log(LLyα/[erg s−1]) ≈ 42.53 reside in DMHs +of log(Mh/[h−1M⊙]) = 11.43+0.04 +−0.10 and that lower luminosity +LAEs (log(LLyα/[erg s−1]) ≈ 40.97) are hosted by DMHs of +masses ranging log(Mh/[h−1M⊙]) = 10.00+0.12 +−0.09. These results fit +well within the assumed framework in which star-forming galax- +ies that reside in more massive halos present higher star forma- +tion rates and thus show more luminous nebular emission lines +(Kusakabe et al. 2018). This dependence can then be weakened +by low Lyα escape fractions in high mass halos. +Following Sect. 5.4.1 of Herrero Alonso et al. (2021), we +matched the redshift distributions of the three main samples and +of each subsample pair to verify that the difference in cluster- +ing amplitude is not driven by the different redshift distribu- +tion of the datasets. For each main sample, we compare indi- +vidual bins between their corresponding z-distributions and se- +lect the one that contains a higher number of objects. We then +Article number, page 10 of 17 + +Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +Fig. 7: Clustering dependence on Lyα luminosity. Left: K-estimator measurements in the MUSE-Wide survey (red; ⟨LLyα⟩ ≈ 1042.34 +erg s−1) and MXDF (blue; ⟨log LLyα⟩ ≈ 1041.22 erg s−1). The dotted curves represent the best HOD fits. The black straight line shows +the expected K-estimator of an unclustered sample. Right: Same for the high LLyα subset (red) from the MUSE-Wide survey and +the low LLyα subsample (blue) from MXDF. +randomly remove LAEs until we match the number counts of +the non-selected samples in that bin. Once all bins have been +inspected, we obtain "matched" z-distributions (i.e., equivalent), +but with still different Lyα luminosity distributions. We ran the +K-estimator in the three "matched" datasets and find consistent +results with the original ones. We follow the same approach for +the subsamples such that the low- and high-luminosity subsets +have exactly the same z-distribution. We find that the cluster- +ing difference between the "matched" and original subsamples +varies within 1σ. Besides, as we did for LLyα, we also searched +for a possible clustering dependence on redshift and found no +trend. Thus, we discarded the possibility of a possible clustering +dependence on Lyα luminosity driven by z. +Our results are not driven by AGN or low-redshift emission +line contamination either. The Lyα-emitting AGN fraction for +LLyα < 1043 erg s−1 is close to zero (Spinoso et al. 2020 and ref- +erences therein) and the four known X-ray detected AGNs (Luo +et al. 2017), which only affect MUSE-Wide and MUSE-Deep, +were not included in our datasets. Besides, Urrutia et al. (2019) +performed a stacking experiment of X-ray images centered on +MUSE-Wide LAEs, yielding no signal. The presence of low- +redshift interlopers in our spectroscopic samples is also unlikely. +[O ii] emitters are the typical contaminants of high-redshift LAE +samples but the high resolution of the MUSE instrument allows +to distinguish the [O ii] emission line doublet with high confi- +dence. +These results are in line with the tentative trends seen in +Ouchi et al. (2003); Kusakabe et al. (2018); Herrero Alonso et +al. (2021) and the clear dependence found in Khostovan et al. +(2019). While Ouchi et al. (2003) noted a slight difference in the +correlation amplitude of two LLyα subsamples (30 and 57 LAEs +in each subset at z = 4.86 with log(LLyα/[erg s−1]) > 42.2 and +log(LLyα/[erg s−1]) < 42.2, respectively), Kusakabe et al. (2018) +observed a tendency (< 2σ) of larger bias factors corresponding +to higher luminosity LAEs. They used four deep survey fields +at z = 2 with limiting Lyα luminosities within the range of +41.3 < log(LLyα/[erg s−1]) < 42 computed from NB387 mag- +nitudes. +More significant is the dependence found in Khostovan et +al. (2019) and Herrero Alonso et al. (2021). While the lat- +ter measured a 2σ difference in bias factors or DMH masses +between two subsets of 349 and 346 LAEs at z ≈ 4 with +Fig. 8: Typical dark matter halo mass against observed me- +dian Lyα luminosity. Filled and unfilled symbols correspond to +the values derived from the samples and subsamples described +in Sect. 2, respectively. Red circles, green triangles and blue +squares belong to MUSE-Wide, MUSE-Deep and MXDF, re- +spectively. Gray crosses represent the results from Khostovan et +al. (2019) in the Lyα luminosity interval relevant for this study. +log(LLyα/[erg s−1]) ≈ 42.14 and log(LLyα/[erg s−1]) ≈ 42.57, +the former used various surveys with discrete redshift slices be- +tween 2.5 < z < 6 and 42.0 < log(LLyα/[erg s−1]) < 43.6 to +find that halo mass clearly (5σ) increases with increasing line +luminosity. For a direct comparison, we plot in Fig. 8 (gray +crosses) the DMH masses computed by Khostovan et al. (2019) +from samples with similar redshifts (z ≈ 3) and Lyα luminosi- +ties (log(LLyα/[erg s−1]) ≈ 42) to our current LAE samples. Our +results are in good agreement and extend to much fainter Lyα +luminosities. +Our results, along with those from the literature, demonstrate +that having a broad dynamic range of LLyα (nearly extending two +orders of magnitude) and a large number of LAEs in the samples +is crucial to detect the clustering dependence on LLyα. +Article number, page 11 of 17 + +0.7 +MXDF (log(LLyα/[erg s-1]) ～ 41.22) +MUSE-Wide (log(LLyα/[erg s-1]) ~ 42.34) +0.6 +0.5 +4 +0.4 +0.3 +0.2 +0.1 +1.0 +10.0 +Rij [h-1Mpc]0.7 +Low L (MXDF, log(LLyα/[erg s-1]) 40.97) +High L (MUSE-Wide, log(LLyα/[erg s-1j) ～ 42.53) +0.6 +0.5 +5 +0.4 +0.3 +0.2 +0.1 +1.0 +10.0 +Rii [h-1Mpc]MUSE-Wide +MUSE-Deep +12.0 +MXDF +log(Mn / [h-1Mol) +Khostovan et al. 2019 +11.5 +11.0 +10.5 +中 +10.0 +41.0 +41.2 +41.4 +41.6 +41.8 +42.0 +42.2 +42.4 +42.6 +log(LLyα/erg s-1)A&A proofs: manuscript no. aanda +5.2. Comparison to Herrero Alonso et al. (2021) +In this section we compare our results with the findings of our +previous study (Herrero Alonso et al. 2021, hereafter HA21), +where we measured the clustering of a subset (68 fields of the +MUSE-Wide survey) of our current sample (91 fields of the +MUSE-Wide survey) and fitted the corresponding signal with +a two-halo term only HOD modeling. In order to envisage the +methodological and statistical improvement of our new investi- +gation, we applied our K0,7 +7,45 estimator to the sample considered +in HA21 (695 LAEs at 3.3 < z < 6). We compare the outcome +to our current clustering measurement in Fig. 9. +The two datasets show good agreement within the uncer- +tainties, with smaller errors for the current sample. Besides the +higher number of LAEs and larger spatial coverage, the error es- +timation was carried out following different procedures. While +the spatial coverage of the full MUSE-Wide survey allows us +to compute the covariance matrix from the jackknife resampling +technique, the smaller transverse extent covered by the 68 fields +did not allow the split of the surveyed area into a significant num- +ber of jackknife zones. Thus, in HA21, we chose bootstrapping +error bars as our next most conservative and realistic approach. +The slightly puzzling hump seen in Sect. 4 of HA21 at +4 ≲ Ri j/[h−1Mpc] ≲ 7 is no longer visible in our new dataset. +This confirms the judgement in HA21 that the feature was con- +sistent with a statistical fluctuation resulting from the correlation +between datapoints. +In HA21, we limited the range of transverse separations to +Ri j > 0.6 h−1Mpc, excluding the smallest scales of the one-halo +term. Thus, we fitted the signal with a two-halo term only HOD +model (red dotted curve in Fig. 9) in contrast to the full HOD +modeling performed in this work (blue dotted curve). While the +former only constrained the large-scale bias factor and the typi- +cal DMH mass of LAEs, the latter further determines the num- +ber of central and satellite galaxies, as well as the required DMH +mass to host each type of galaxy. Despite these dissimilarities, +the two fits are in good agreement: the bias factor (b = 2.80+0.38 +−0.38) +and the typical DMH mass of LAEs (log(MDMH / [h−1M⊙]) = +11.34+0.23 +−0.27) from HA21 are consistent with those derived in this +work (b = 2.65+0.13 +−0.11 and log(MDMH / [h−1M⊙]) = 11.09+0.10 +−0.09). The +higher accuracy of our current measurements originates from the +larger sample, the availability of more realistic error bars, and +constraints from the one-halo term. +5.3. Comparison to the literature +A common way to infer the host DMH masses of LAEs is to +quantify the galaxy clustering of the detected population through +clustering statistics, which is then traditionally approximated +with power-laws or fit with physically motivated HOD models. +Following the traditional approach, Gawiser et al. (2007), +Ouchi et al. (2010) and Bielby et al. (2016) focused on the clus- +tering of a few hundred LAEs at z = 3.1 − 6.6 to obtain typi- +cal DMH masses in the range 1010 − 1011 M⊙. Similar masses +were found by Khostovan et al. (2018) in a much larger sample +(≈ 5000 LAEs) in discrete redshift slices within 2.5 < z < 6, +adopting the same procedure. A major improvement in terms +of methodology was presented in Lee et al. (2006); Durkalec +et al. (2014); Ouchi et al. (2018); Durkalec et al. (2018), who +considered samples of high-z galaxies (2000-3000 mainly LAEs +and Lyman-break galaxies, LBGs) and quantified the clustering +with HOD modeling. While Ouchi et al. (2018) found that their +LAEs at z = 5.7 (6.6) are hosted by DMHs with typical masses +Fig. 9: Clustering of the full MUSE-Wide sample (blue; this +work) compared to the subset considered in HA21 (red). The for- +mer measurements show jackknife uncertainties (see Sect. 3.2.1) +and the latter bootstrapping errors (see Sect. 3.1.3 in HA21). The +blue dotted curve represents our best-fit from full HOD model- +ing. The red dotted curve displays the two-halo term only best +HOD fit found in Sect. 4.3 of HA21. The black straight line +shows the expected K value of an unclustered sample. +of log(Mh/M⊙) = 11.1+0.2 +−0.4 (10.8+0.3 +−0.5), Lee et al. (2006) and +Durkalec et al. (2014, 2018) computed log(Mh/h−1M⊙) ≈ 11.7 +for their sample of galaxies at z = 4 − 5 and z = 3, respectively. +Considering that we have performed a full HOD modeling at +the median redshift of our number of galaxy pairs (zpair = 3.8) +and that the DMH masses are predicted to evolve with cos- +mic time, our derived typical DMH masses log(Mh/h−1M⊙) ≈ +10.77 − 11.09 are in good agreement with the literature. +Besides the computation of typical DMH masses, modeling +the one-halo term of the clustering statistics with HOD mod- +els delivers the minimum DMH mass required to host a central +galaxy, Mmin, that is needed for a satellite galaxy, M1, and the +power-law slope of number of satellites, α. These three parame- +ters constrain the satellite fraction, fsat. Ouchi et al. (2018) par- +tially exploited the power of HOD models in a sample of ≈ 2000 +LAEs to obtain log(Mmin/M⊙) = 9.5+0.5 +−1.2 (9.1+0.7 +−1.9) at z = 5.7 +(6.6). Our derived minimum masses to host a central galaxy at +zpair = 3.8 are considerably larger (log(Mmin/M⊙) ≈ 10.3−10.7), +which can be explained by the different Lyα luminosities cov- +ered in the two studies, and by the fact that several HOD pa- +rameters were fixed in Ouchi et al. (2018), namely, σlog M = 0.2, +log M0 = 0.76M1+2.3, log M1 = 1.18 log Mmin−1.28, and α = 1, +which are not compatible with ours. This was the only previous +study that performed HOD modeling in a sample of LAEs. +Lee et al. (2006) and Durkalec et al. (2014) made use of the +full potential of HOD models to reproduce the clustering of their +LBG population at z = 4 − 5 and 2.9 < z < 5, respectively. +Although it is still under debate whether LBGs and LAEs are +the same galaxy population (Garel et al. 2015 and references +therein), Lee et al. (2006) computed a minimum DMH mass to +host a central LBG of log(Mmin/M⊙) ≈ 10.8, to host a satel- +lite LBG of log(M1/M⊙) ≈ 12.0, and a power-law slope α for +the number of satellites of α ≈ 0.7, with considerable uncer- +tainties. Similarly, Durkalec et al. (2014) found log(Mmin/M⊙) = +11.18+0.56 +−0.70, log(M1/M⊙) = 12.55+0.85 +−0.88, and α = 0.73+0.23 +−0.30. While +their halo masses are in agreement with our findings, their slope +is somewhat shallower. This is partially expected given the dis- +Article number, page 12 of 17 + +Herrero Alonso et al. 2021 +0.40 +This work +0.35 +0.30 +5 +R +0.25 +0.20 +0.15 +0.1 +1.0 +10.0 +Rij [h-1Mpc]Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +similarities in the galaxy populations (i.e., disparate observa- +tional selection techniques detect distinct galaxy populations). +5.4. Satellite fraction +In the above discussions on HOD modeling, we limit ourselves +to the HOD model form expressed by Eqs. 5 and 6, which is +rather restrictive. The underlying assumption of the model is that +the center of the halo with mass Mh > Mmin is always occupied +by one galaxy in the sample (or at least at a Mh-independent +constant probability). This form may be appropriate for instance, +for luminosity or stellar mass thresholding samples, but there is +no reason that this has to be the case for samples selected by +other criteria. +We note that the inferred value of fsat is sensitive to the form +of the parameterized model of the central and satellite HODs. +In this work and in the literature, a power-law form of the satel- +lite HOD is customarily assumed. In this case, a lower α would +increase the model ⟨Ns(Mh)⟩ at the lower Mh end, near Mmin, +and yield fewer satellites in higher mass halos. Since the halo +mass function drops with increasing mass, fsat is mainly deter- +mined by the HOD behavior around Mh ∼ Mmin ∼ 1010.5 h−1M⊙, +where the halo mass function is large and the virial radius is +rvir ≈ 0.08 h−1Mpc at z ∼ 3.8 (Zheng et al. 2007). These +scales are too small to be well constrained by our observations. +Our observed one-halo term mainly constrains the satellite frac- +tion at larger mass halos (Mh ∼ Mmin ∼ 1013 h−1M⊙, where +rvir ≈ 0.5 h−1Mpc at the same redshift). Thus, the fsat values +from the HOD modeling should be viewed with caution and may +well reflect the artefacts of the assumed form of the model. On +the other hand, the sheer presence of a significant one-halo term +indicates the existence of some satellites at higher halo masses. +The extent of the one-halo term up to Rij ≈ 0.5 h−1Mpc shows +that there are indeed satellites up to Mh ∼ 1013 h−1M⊙. +In spite of the above caveats, the small satellite fraction of the +LAEs is likely to be robust. The small fsat values for the assumed +HOD model indicate that not only central-satellite pairs are rare, +but also satellite-satellite pairs are as well, suggesting that only +a small fraction of halos contain multiple LAEs. The small Mmin +values themselves are also an indication that a large majority of +the halos (at the low mass end) that contain a LAE are indeed +dominated by one galaxy and in this case, the LAE is probably +the central galaxy. +5.5. Implications +The clustering results of this study do not only have implications +on the baryonic-DM relation, but also on evolving Lyα lumi- +nosity functions, signatures of incomplete reionization, and halo +mass-dependent Lyα escape fractions. We address these aspects +in the following. +The relation between halo mass (or clustering strength) and +Lyα luminosity (Table 4 and Fig. 8) demonstrates that high- +luminosity LAEs tend to reside in higher density environments +than lower luminosity ones. As a result, overdense regions con- +tain a larger fraction of high-luminosity sources (and a lower +fraction of less luminous ones) than environments of lower den- +sity. These inferences affect the Lyα LF measurements at 3 < +z < 6. While we expect a shallower faint-end slope of the Lyα +LF in overdense regions, the slope should steepen in average or +low density environments. As a consequence, surveys for rela- +tively high-luminosity (LLyα ≈ 1042 erg s−1) LAEs are implicitly +biased against the lowest density regions and thus gives a biased +shape for the LF, which should not be extrapolated towards lower +Lyα luminosities. +Assuming that our LLyα − Mh relation still holds at higher +redshifts, the Lyα LF at z ≥ 6 would be even more affected, +not only because of the above discussion but also because higher +redshift bins are mainly populated by high-luminosity sources, +contrary to lower redshift bins (typical case for telescopes with +higher sensitivity at bluer wavelengths). Thus, it is important to +be careful when interpreting Lyα LFs, especially near the epoch +of reionization (EoR), where a shallow to steep variation in the +slope of the LF from higher (z ≈ 7) to lower redshifts (z ≈ 5.7) +is commonly interpreted as a sign of incomplete reionization +(Konno et al. 2014; Matthee et al. 2015; Santos et al. 2016). +Simulations at those higher redshifts also tend to find that +high-luminosity LAEs are more likely to be observed than low- +luminosity ones because they are able to ionize their surround- +ings and form H ii regions around them (i.e., ionized bubbles; +Matthee et al. 2015; Hutter et al. 2015; Yoshioka et al. 2022). +These allow Lyα photons to redshift out of the resonance wave- +length and escape the region. Lower luminosity LAEs are then +observed if they reside within the ionized bubbles of higher lumi- +nosity LAEs or if they are able to transmit enough flux through +the IGM (Matthee et al. 2015). If our LLyα − Mh relation is still +valid at these redshifts, our results would support this simulation +paradigm since high-luminosity LAEs (situated in overdense re- +gions) could form large ionized bubbles more efficiently than +low-luminosity sources which tend to be located in lower den- +sity environments (Tilvi et al. 2020). +Theoretical studies (e.g., Furlanetto et al. 2006; McQuinn +et al. 2007) have modeled the size distribution of these H ii re- +gions and predicted an increase in the apparent clustering sig- +nal of LAEs towards the epoch of reionization (i.e., towards a +more neutral IGM). Large ionized bubbles become rarer as the +ionizing fraction declines. This patchy distribution of H ii re- +gions, which mostly surrounds large galaxy overdensities, boosts +the apparent clustering of LAEs. This is commonly interpreted +as another sign of incomplete reionization (e.g., Matthee et al. +2015; Hutter et al. 2015). Comparisons between observed intrin- +sic LAE clustering and model predictions have therefore been +used to infer the fraction of neutral hydrogen at the EoR (e.g., +Ouchi et al. 2018). Nevertheless, if the clustering dependence on +Lyα luminosity continues to z ≈ 6, this comparison should be +performed with caution. Because the observed high redshift bins +(z ≥ 6) mainly contain high-luminosity LAEs, a strong cluster- +ing signal at z ≈ 6 may be wrongly interpreted as incomplete +reionization when, in fact, it may only reflect the natural relation +between Lyα luminosity and clustering strength. +We speculate that our results also play a role in the amount of +escaping Lyα photons (Lyα fesc). Durkalec et al. (2018) observed +a dependence between halo mass and absolute UV magnitude +(MUV). The interpretation of their relation goes as follows: MUV +traces star formation rate (SFR; e.g., Walter et al. 2012), which, +in turn, tracks stellar mass (M∗; e.g., Salmon et al. 2015), which +correlates with halo mass (e.g., Moster et al. 2010). Because we +observe a similar relation of Mh with LLyα, LLyα is presumably +also a tracer of star formation. If this is correct, the object-to- +object variations in Lyα escape fraction cannot be so large that +they obscure the trend of SFR – M∗ – Mh. Given the typical Lyα +luminosities of our sample, this is in agreement with the model +suggestions of Schaerer et al. (2011a); Garel et al. (2015), where +the Lyα fesc is of the order of unity for sources with SFR ≈ +1 M⊙ yr−1. The Lyα luminosity would then be a good tracer of +the SFR for less luminous LAEs. +Article number, page 13 of 17 + +A&A proofs: manuscript no. aanda +6. Conclusions +We report a strong clustering dependence on Lyα luminosity +from the clustering measurements of three MUSE Lyα emitting +galaxy (LAE) samples at 3 < z < 6. Following the pencil-beam +design of MUSE surveys from spatially large and shallow ob- +servation to spatially small and deep observation, we use 1030 +LAEs from the full MUSE-Wide survey (1 h exposure time), +679 LAEs from MUSE-Deep (10 h), and 367 LAEs from MXDF +(140 h). We thus connect the clustering properties of L⋆ LAEs +with those of much fainter ones in the MXDF. We applied an +optimized version of the K-estimator as the clustering statistic, +coupled to state-of-the-art halo occupation distribution (HOD) +modeling. +From our full HOD analysis, we derive constraints on +the HOD of high-luminosity (log(LLyα/erg s−1) ≈ 42.34), in- +termediate (log(LLyα/erg s−1) +≈ +41.64) and low-luminosity +(log(LLyα/erg s−1) ≈ 41.22) LAEs. We modeled the LAE HOD +with three parameters: the threshold dark matter halo (DMH) +mass for hosting a central LAE (Mmin), for hosting (on average) +one satellite LAE (M1), and the power-law slope of the num- +ber of satellites per halo (α) as a function of halo mass. For +the high-luminosity sample we derived a typical DMH mass of +log(Mh/[h−1M⊙]) = 11.09+0.10 +−0.09, corresponding to a bias factor +of b = 2.65+0.13 +−0.11. These findings, although more accurate, are in +agreement with the results based on the two-halo term only HOD +modeling performed in Herrero Alonso et al. (2021) for a subset +of our MUSE-Wide sample. For the lower luminosity samples +we found lower DMH masses. While for the log(LLyα/erg s−1) ≈ +41.64 dataset we inferred log(Mh/[h−1M⊙]) = 10.89+0.09 +−0.09 (b = +2.42+0.10 +−0.09), for the low-luminosity LAE sample we computed +log(Mh/[h−1M⊙]) = 10.77+0.13 +−0.15 (b = 2.43+0.15 +−0.15). +We also derived threshold DMH masses for centrals and +satellites for each sample. We found that the minimum +DMH mass to host a central LAE is log(Mmin/[h−1M⊙]) = +10.3+0.2 +−0.3, +10.5+0.2 +−0.1, +10.7+0.2 +−0.3 for low-, intermediate-, and +high-luminosity LAEs, respectively. The threshold halo mass +for satellites and the power-law slope of the number of +satellite LAEs also increase with Lyα luminosity, from +log(M1/[h−1M⊙]) += +11.7+0.3 +−0.2 and α += +1.5 ± 0.5 to +log(M1/[h−1M⊙]) += +12.4+0.3 +−0.2 and α += +3.0+0.4 +−0.5 and to +log(M1/[h−1M⊙]) = 12.4+0.4 +−0.6 and α = 2.8+0.9 +−0.7. These HOD con- +straints imply a decreasing number of detected satellite LAEs +with luminosity. Indeed we infer satellite fractions of fsat ≲ +10, 20% (at 3σ confidence level) for high- and low-luminosity +LAEs, respectively. This suggests that the most common sce- +nario for current MUSE surveys is that in which DMHs mainly +host a single detected LAE. +Motivated by these results, we aimed to further explore the +clustering dependence on Lyα luminosity. Exploiting the large +dynamic range of LLyα from MXDF, we split the main LAE +sample at its median LLyα. We found a 3.9σ difference be- +tween the clustering of the low-luminosity (log(LLyα/erg s−1) ≈ +40.97, blow += +1.79+0.08 +−0.06) and the high-luminosity subset +(log(LLyα/erg s−1) ≈ 41.54, bhigh = 3.10+0.24 +−0.22). We then selected +the highest luminosity LAE subset from the MUSE-Wide survey +(log(LLyα/erg s−1) ≈ 42.53) and the lowest luminosity LAE sub- +sample from MXDF (log(LLyα/erg s−1) ≈ 40.97), resulting in +a clear clustering dependence where the high-luminosity LAEs +from MUSE-Wide cluster more strongly (bhigh = 3.13+0.08 +−0.15 or +log(Mh/[h−1M⊙]) = 11.43+0.04 +−0.10) than the low-luminosity ones +from MXDF (blow = 1.79+0.08 +−0.06 or log(Mh/[h−1M⊙]) = 10.00+0.12 +−0.09) +at 8σ significance, excluding cosmic variance effects. The on- +going Hobby-Eberly Telescope Dark Energy Experiment (HET- +DEX; Gebhardt et al. 2021) survey will complement these re- +sults at the high-luminosity end and at somewhat lower redshifts +(1.9 < z < 3.5). +The implications of this framework are however not only +relevant for LAE clustering studies, but also for reported mea- +surements of evolving Lyα luminosity functions, detections of +incomplete reionization at z ≈ 6, and the relation between Lyα +escape fraction and halo mass. Our results are also crucial for +the much debated relevance of unresolved satellite LAEs (fainter +than those in MXDF) for the measured Lyα surface brightness +profiles. +Acknowledgements. The authors give thanks to the staff at ESO for extensive +support during the visitor-mode campaigns at Paranal Observatory. We thank the +eScience group at AIP for help with the functionality of the MUSE-Wide data +release webpage. T.M. and H.A. thank for financial support by CONACyT Grant +Científica Básica #252531 and by UNAM-DGAPA (PASPA, PAPIIT IN111319 +and IN114423). L.W. and T.U. by the Deutsche Forschungsgemeinschaft through +grant Wi 1369/32-1. M.K. acknowledges support by DLR grant 50OR1904 and +DFG grant KR 3338/4-1.The data were obtained with the European Southern +Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A- +0791. This research made use of Astropy, a community-developed core Python +package for Astronomy (Astropy Collaboration et al. 2013). +References +Adelberger, K. L., Steidel, C. C., Pettini, M., Shapley, A. E., Reddy, N. A., & +Erb, D. K. 2005, ApJ, 619, 697-713 +Adelberger, Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, +A&A, 558, A33 +Bacon, R., Conseil, D., Mary, D., et al. 2017, A&A, 608, A1 +Bacon, +R., +Brinchmann, +J., +Conseil, +S., +et +al. +2022, +arXiv +e-prints, +arXiv:2211.08493 +Bielby, R. M., Tummuangpak, P., Shanks, T., et al. 2016, MNRAS, 456, 4061 +Davis, M. & Peebles, P. J. E. 1983, ApJ, 267, 465 +Diener, C., Wisotzki, L., Schmidt, K. B., et al. 2017, MNRAS, 471, 3186-3192 +Durkalec, A., Le Fèvre, O., Pollo, A., et al. 2014, A&A, 583, A128 +Durkalec, A., Le Fèvre, O., Pollo, A., et al. 2018, A&A, 612, A42 +Furlanetto, S. R., Zaldarriaga, M., Hernquist, L., et al. 2006, MNRAS, 365, 1012 +Garel, T., Blaizot, J., Guiderdoni, B., et al. 2015, MNRAS, 450, 1279 +Gawiser, E., Francke, H., Lai, K., et al. 2007, ApJ, 671, 278 +Gebhardt, K., Cooper, E. M., Ciardullo, R., et al. 2021, ApJ, 923, 217 +Hatfield, P. W., Bowler, R. A. A., Jarvis, M. J., et al. 2018, MNRAS, 477, 3760 +Herenz, E. C., Urrutia, T., Wisotzki, L., et al. 2017, A&A, 606, A12 +Herenz, E. C., Wisotzki, L., Saust, R., et al. 2019, A&A, 621, A107 +Harikane, Y., Ouchi, M., Ono, Y., et al. 2018, Publications of the Astronomical +Society of Japan, 70, S11 +Herrero Alonso, Y., Krumpe, M., Wisotzki, L., et al. 2021, A&A, 653, A136 +Hinshaw, G., Larson, D., Komatsu, E., et al. 2013, AJSS, 208, 19 +Hinton, S. R., Davis, T. M., Lidman, C., et al. 2016, Astronomy and Computing, +15, 61 +Hu, E. M., Cowie, L. L., McMahon, & R. G. 1998, ApJ, 502, L99 +Hutter, A., Dayal, P., & Müller, V. 2015, mnras, 450, 4025 +Inami, H., Bacon, R., Brinchmann, J., et al. 2017, A&A, 608, A2 +Jenkins, A., Frenk, C. S., Pearce, F. R., et al. 1998, ApJ, 499, 20 +Kaiser, N. 1987, MNRAS, 227, 1-21 +Khostovan, A. A., Sobral, D., Mobasher, B., et al. 2018, MNRAS, 478, 2999– +3015 +Khostovan, A. A., Sobral, D., Mobasher, B., et al. 2019, MNRAS, 489, 555-573 +Konno, A., Ouchi, M., Ono, Y., et al. 2014, ApJ, 797, 16 +Krumpe, M., Miyaji, T. & Coil, A. L. 2010, ApJ, 713, 558 +Krumpe, M., Miyaji, T., Coil, A. L. & Aceves H. 2012, ApJ, 746, 1 +Krumpe, M., Miyaji, T. Husemann, B., et al. 2015, ApJ, 815, 21 +Krumpe, M., Miyaji, T. Coil, A. L. & Aceves, H. 2018, MNRAS, 474, 1773 +Kusakabe, H., Shimasaku, K., Ouchi, M., et al. 2018, Publications of the Astro- +nomical Society of Japan, 70, 4 +Lee, K.-S., Giavalisco, M., Gnedin, O., et al. 2006, ApJ, 642, 63 +Limber, D. N. 1953, ApJ, 117, 134 +Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2017, ApJSS, 228, 2 +Madau, M. A., Cohen, D. P, Maruyama, M., et al. 2017, ApJ, 850, 5 +Mary, D., Bacon, R., Conseil, S., et al. 2020, A&A, 635,A194 +Matthee, J., Sobral, D., Santos, S., et al. 2015, MNRAS, 451, 400 +McQuinn, M., Hernquist, L., Zaldarriaga, M., et al. 2007, MNRAS, 381, 75 +Article number, page 14 of 17 + +Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +Miyaji, T., Krumpe, M., Coil, A. & Aceves, H. 2011, ApJ, 726, 83 +Moster, B. P., Somerville, R. S., Maulbetsch, C., et al. 2010, ApJ, 710, 903 +Navarro, J. F., Frenk, C. S. & White, S.D.M 1997, ApJ, 490, 493 +Norberg, P., Baugh, C. M., Gaztañaga, E., & Croton, D. J. 2009, MNRAS, 396, +19 +Ouchi, M., Shimasaku, K., Furusawa, H., et al. 2003, ApJ, 582, 60 +Ouchi, M., Shimasaku, K., Furusawa, H., et al. 2010, ApJ, 723, 869 +Ouchi, M., Harikane, Y., Shibuya, T., et al. 2018, PASJ, 70, S13 +Salmon, B., Papovich, C., Finkelstein, S. L, et al. 2015, ApJ, 799, 183 +Santos, S., Sobral, D. & Matthee, J. 2016, MNRAS, 463, 1678 +Schaerer, D., Hayes, M., Verhamme, A., et al. 2011a, A&A, 531, A12 +Sheth, R., Mo, H. J. & Tormen, G. 2001, 323, 1-12 +Spinoso, D., Orsi, A., López-Sanjuan, C. 2020, A&A, 643, A149 +Steidel, C. C., Giavalisco, M., Pettini, M. 1996, ApJ, 462, L17 +Tilvi, V., Malhotra, S., Rhoads, J. E., et al. 2020, ApJL, 891, L10 +Tinker, J. L., Weinberg, D. H. & and Zheng, Z. 2005, MNRAS, 368, 85 +Tinker, J. L. 2007, MNRAS, 374, 477 +Urrutia, T., Wisotzki, L., Kerutt, J., et al. 2019, A&A, 624, 24 +Van Den Bosch, F. C., More, S., Cacciato, M., et al. 2013, MNRAS, 430, 725 +Walter, F., Decarli, R., Carilli, C., et al. 2012, ApJ, 752, 93 +Wechsler, R. H. & Tinker, J. L. 2018, Annual Review of Astronomy and Astro- +physics, 56, 435 +Yoshioka, T., Kashikawa, N., Inoue, A., et al. 2022, ApJ, 927, 32 +Yajima, H., Sugimura, K., & Hasegawa, K. 2018, MNRAS, 477, 5406 +Zheng, Z., Coil, A. & Zehavi, I. 2007, ApJ, 667, 760-779 +Appendices +A. Effect of different fields on the clustering +measurements +In this work, we have analyzed the clustering of LAEs in the +full MUSE-Wide sample, including the CANDELS/COSMOS +fields and the HUDF parallel fields. Here, we explore the pos- +sible effects on the MUSE-Wide clustering results when includ- +ing or excluding various sets of fields. In appendix A of Her- +rero Alonso et al. (2021), we showed that the HUDF parallel +fields did not alter the clustering results, their exclusion or inclu- +sion mainly affected the clustering uncertainties. We therefore +explore the effect of including the CANDELS/COSMOS region +by comparing the clustering of the full MUSE-Wide survey with +that present in a subsample without the CANDELS/COSMOS +fields. The number of LAEs in the CANDELS/COSMOS region +is 250. +It is clear from Fig. A.1 that the clustering in both samples +is in good agreement. The large-scales bias factors derived from +the two curves are indistinguishable (within 1σ). The uncertain- +ties corresponding to the smaller sample are (on average) 20% +larger than in the full MUSE-Wide sample. We conclude that the +inclusion of these fields has no notable effect on our clustering +results but helps in reducing cosmic sample variance uncertain- +ties. +B. Covariance matrix +A common approach to quantify the correlation of the clustering +data points is to resample the set of galaxies with the jackknife +technique, followed by the calculation of the covariance matrix. +To apply the jackknife method, we find a compromise between +the number and the size of the jackknife zones. Thus, we split +the sky area into ten independent regions (see Fig. B.1) with a +spatial extent of ≈ 4 h−1Mpc in both RA and Dec directions. We +then construct ten different subsamples, each of them excluding +one jackknife zone, and compute the K-estimator in each subset. +These measurements are then used to build up the covariance +matrix using Eq. 1 (see Sect. 3.2.1). +Fig. A.1: Clustering of the LAEs in the full MUSE-Wide sample +(blue, see Fig. 1) and without the CANDELS/COSMOS fields +(red, see right panel of Fig. 1). The black baseline represents the +expected clustering of an unclustered sample. The error bars are +Poissonian. The red measurements have been shifted along the +x-axis for visual purposes. +Considering that the probability of one galaxy pair to con- +tribute to various adjacent bins is higher than that to contribute +to several distant bins, one would naively expect a higher corre- +lation in the former case. This is indeed what the (normalized) +covariance matrix reflects in the left panel of Fig. B.2. In fact, +the noise in the matrix elements corresponding to notably sepa- +rate bins is substantial. In the right panel of Fig. B.2, we plot the +normalized matrix elements as a function of bin i for each bin j +to better illustrate the high level of noise in the matrix, especially +for bins i > 6, where most curves become negative. This is likely +due to the limited spatial size of the survey, which does not allow +neither for a higher number of jackknife zones nor for spatially +larger zones. +As a result of the considerable noise in the matrix on account +of barely correlated bins significantly apart from each other, the +minimization of the χ2 values (Eq. 2) including the full covari- +ance fails (i.e., various χ2 values become negative). We there- +fore limit the use of the covariance matrix to its main diagonal +and two adjacent diagonals (see red section in the left panel of +Fig. B.2; our so-called reduced covariance matrix). This means +we set the negative part of the curves in the right panel of Fig. B.2 +to zero (i.e., no correlation between those bins), in an attempt to +smooth out the noise. +While incorporating more diagonals re- +sults mathematically problematic for the χ2 minimization, we +have verified that the number of adjacent diagonals (one or two) +slightly modifies the χ2 values but the probability contours rep- +resented in Fig. 6 remain unaltered. Thus, so do the best-fit HOD +parameters. +Article number, page 15 of 17 + +0.45 +Full sample +Without COSMOS fields +0.40 +0.35 +5 +0.30 ++ +0.25 +0.20 +0.15 +1.0 +10.0 +Rij [h-1Mpc]A&A proofs: manuscript no. aanda +Fig. B.1: Ten Jackknife zones in the spatial coverage of the full MUSE-Wide survey (83.52 arcmin2). Each Jackknife zone has a +spatial extent of ≈ 4 h−1Mpc in both RA and Dec directions. +Fig. B.2: Covariance matrix computed from ten independent K-estimator measurements from the jackknife resampling technique. +Left: Normalized covariance matrix for bins i and j. The red region defines the main diagonal and the two adjacent diagonals used +for our reduced covariance matrix. Right: Normalized covariance matrix elements as a function of bin i for each bin j (colored). +Fig. C.1: Error estimation method comparison for the sample of +LAEs in the MUSE-Wide survey. Uncertainties from the covari- +ance matrix and the jackknife resampling technique described in +Sect. 3.2.1 are colored in blue, those from the bootstrapping ap- +proach used in Herrero Alonso et al. (2021) in red, and Poisson +uncertainties in green. +Despite current limitations, jackknife is still the most robust +method to compute the K-estimator uncertainties. While galaxy +bootstrapping or Poisson error bars do not account for bin to +bin correlations, our reduced covariance matrix only neglects the +correlation between bins remarkably separated (expected to be +minimal), but accounts for the correlation between nearby bins. +C. Error estimation comparison +In order to quantify the correlation between the K-estimator +bins, the covariance matrix must be computed. By splitting the +sky area into independent regions, following the jackknife re- +sampling technique, we create as many subsamples from the +MUSE-Wide sample as jackknife zones (see Sect. 3.2.1). The K- +estimator is then computed in each subset and the measurements +are used to quantify the covariance matrix, whose diagonal pro- +vides the variance of each clustering data point. The square root +of the diagonal represents the 1σ uncertainties and are repre- +sented in blue in Fig. C.1 (same along the main paper). +The jackknife resampling method requires a division of the +sky area into several independent regions, each of which should +ideally be large enough to cover the full range of scales under +consideration. Out of the three samples examined in this study, +this can only be partially achieved in the MUSE-Wide dataset. +Article number, page 16 of 17 + +6.0 +-27.70 +5.5 +zone5 +zone3 +zone7 +5.0 +zone9 +-27.75 +e +4.5 +N +D +-27.80 +zone4 +4.0 +-27.85 +zone6 +zone8 +3.5 +zonel0 +-27.90 +3.0 +53.30 +53.25 +53.20 +53.15 +53.10 +53.05 +RA2.34 +6.0 +2.32 +5.5 +2.30 +5.0 +2.28 +zone1 +zone2 +e +4.5 +¥N +2.26 +D +2.24 +4.0 +2.22 +3.5 +2.20 +3.0 +150.18 150.16 150.14 150.12 150.10 150.08 +RA1.0 +0.8 +8 +0.6 +6 +0.4 +"w"W +J +M +0.2 +4 +0.0 +2 +-0.2 +-0.4 +0 +2 +4 +6 +81.0 +j=0 +0.8 +j=1 +0.6 +j=2 +j=3 +0.4 +j=4 +/MiMji +M +j=5 +0.2 +j=6 +0.0 +j=7 +j=8 +-0.2 +j=9 +-0.4 +-0.6 +0 +2 +4 +6 +8Jackknife +0.45 +Poisson +Bootstrapping +0.40 +0.35 +0.30 +0.25 +0.20 +II +0.15 +1.0 +10.0 +Rij [h-1Mpc]Yohana Herrero Alonso et al.: Strong clustering dependence on Lyα luminosity at 3 < z < 6 +Fig. D.1: Effect of HOD parameters on the shape of the K-estimator. Left: Dependence on log(Mmin) for fixed log(M1/Mmin) = 1.2 +and α = 2.4. Middle: Dependence on log(M1/Mmin) for fixed log(Mmin/[h−1M⊙]) = 10.9 and α = 2.4. Right: Dependence on α for +fixed log(Mmin/[h−1M⊙]) = 10.9 and log(M1/Mmin) = 1.2. +MUSE-Deep and MXDF do not allow for a spatial split into in- +dependent zones. We are thus left with two options for the deeper +samples: the bootstrapping technique applied in Herrero Alonso +et al. (2021), shown in red in Fig. C.1, and Poisson uncertainties, +shown in green. +We find that Poisson (bootstrapping) errors are, on average, +7% (46%) larger than those computed with the jackknife tech- +nique. These findings corroborate the results from Norberg et +al. (2009), who found that the bootstrapping approach overesti- +mates the uncertainties. +Similarly as for the MUSE-Wide survey, we find that boot- +strapping uncertainties are ≈ 40% (on average) larger than Pois- +son in both MUSE-Deep and MXDF. We thus decide to use Pois- +son errors for the deeper samples in an attempt to least overvalue +the uncertainties. +We verified that the error estimation method does not sig- +nificantly affect our clustering results. The best-fit parameters +from MUSE-Deep and MXDF using bootstrapping error bars +and the χ2 minimization described in Sect. 3.1.3 of Herrero +Alonso et al. (2021) are consistent with those delivered from +Poisson statistics. Although in agreement, bootstrapping deliv- +ers ≈ 45% larger uncertainties than Poisson for the best-fit HOD +parameters. +We last perform the same experiment in MUSE-Deep and +MXDF but using scaled Poisson error bars. We decreased the +Poisson in 7% (excess found in MUSE-Wide) and find that the +best-fit parameters are ≈ 10% less uncertain than if Poisson er- +rors are directly applied. +D. Dependence of HOD parameters on the shape of +the K-estimator +Here we visualize and qualitatively describe the effect of the +HOD parameters on the K-estimator. Figure D.1 shows the K- +estimator for numerous HOD models. Each panel represents the +result of varying one HOD parameter with the other two parame- +ters fixed. Before detailing the major effects, it should be pointed +out that the exact change in the shape of the K-estimator does +not only depend on the varied parameter but also on the specific +choice of the other two. Hence, these panels should merely be +seen as illustrative examples. +The left panel of Fig. D.1 shows the dependence of the K- +estimator on Mmin. Higher values of log Mmin (i.e., more massive +halos) raise the expected K-estimator at all Rij scales (one- and +two- halo terms). At large scales, this occurs because more mas- +sive halos present larger bias factors, whereas at small scales, +this is due to the decline in the contribution from less massive +DMHs. +The middle panel of Fig. D.1 shows the dependence of the K- +estimator on M1/Mmin. Larger log(M1/Mmin) values (i.e., more +massive halos) reduce the one-halo term clustering amplitude +because of the decrease in the contribution from less massive +DMHs. The clustering in the two-halo term does not depend on +M1. +The right panel of Fig. D.1 shows the dependence of the K- +estimator on α. Higher values of α increase the fraction of galax- +ies in massive DMHs with respect to smaller mass DMHs. Given +that more massive halos are more strongly biased, the ampli- +tude of the two-halo term increases. The change observed in the +one-halo term is explained because galaxies hosted by massive +DMHs can contribute to the one-halo term on its largest scales, +while galaxies residing in less massive halos can only contribute +to the one-halo term at smaller Ri j scales. Since α modifies the +fraction of galaxies in massive DMHs to less mass DMHs, the +corresponding fraction of the clustering contribution also varies. +This alters the slope of the one-halo term. +Article number, page 17 of 17 + +1.0 +log(Mmin) = 11.l +log(Mmin) = 11.0 +0.8 +log(Mmin) = 10.9 +5 +log(Mmin) = 10.8 +0.6 +4 +log(Mmin) = 10.7 +0.4 +0.2 +0.1 +1.0 +10.0 +Rij [h-1Mpc]1.0 +log(Mi/Mmin) = 1.6 +log(M1/Mmin) = 1.4 +0.8 +log(Mi/Mmin) = 1.2 +log(Mi/Mmin) = 1.0 +5 +.4 +0.6 +log(M1/Mmin) = 0.8 +K +0.4 +0.2 +0.1 +1.0 +10.0 +Rij [h-1Mpc]1.0 +α=3.0 +α= 2.7 +0.8 +α=2.4 +α= 2.1 +5 +74 +0.6 +α= 1.8 +07 +K +0.4 +0.2 +0.1 +1.0 +10.0 +Rii [h-1Mpc] \ No newline at end of file diff --git a/n9E2T4oBgHgl3EQfzwgf/content/tmp_files/load_file.txt b/n9E2T4oBgHgl3EQfzwgf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31c9be65a671db400380d47e6f4f357cbe6280c4 --- /dev/null +++ b/n9E2T4oBgHgl3EQfzwgf/content/tmp_files/load_file.txt @@ -0,0 +1,2136 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf,len=2135 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda ©ESO 2023 January 11, 2023 Clustering dependence on Lyα luminosity from MUSE surveys at 3 < z < 6 Yohana Herrero Alonso,1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Miyaji,2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Wisotzki,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Krumpe,1 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Matthee,4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Schaye,3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Aceves,2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Kusakabe,5 and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Urrutia1 1 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany e-mail: yherreroalonso@aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='de 2 Universidad Nacional Autónoma de México, Instituto de Astronomía (IA-UNAM-E), AP 106, Ensenada 22860, BC, México 3 Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Box 9513, 2300 RA, Leiden, The Netherlands 4 Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland 5 Observatoire de Gèneve, Université de Gèneve, 51 Chemin de Pégase, 1290 Versoix, Switzerland Received xxx/Accepted xxx ABSTRACT We investigate the dependence of Lyα emitter (LAE) clustering on Lyα luminosity and connect the clustering properties of ≈ L⋆ LAEs with those of much fainter ones, namely, ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04L⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We use 1030 LAEs from the MUSE-Wide survey, 679 LAEs from MUSE-Deep, and 367 LAEs from the to-date deepest ever spectroscopic survey, the MUSE Extremely Deep Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' All objects have spectroscopic redshifts of 3 < z < 6 and cover a large dynamic range of Lyα luminosities: 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 < log(LLyα/erg s−1) < 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We apply the Adelberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' K-estimator as the clustering statistic and fit the measurements with state-of-the-art halo occu- pation distribution (HOD) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We find that the large-scale bias factor increases weakly with an increasing line luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For the low-luminosity (log⟨LLyα/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) and intermediate-luminosity (log⟨LLyα/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) LAEs, we com- pute consistent bias factors blow = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 and binterm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09, whereas for the high-luminosity (log⟨LLyα/[erg s−1]⟩ = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34) LAEs we calculated bhigh = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Consequently, high-luminosity LAEs occupy dark matter halos (DMHs) with typical masses of log(Mh/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09, while low-luminosity LAEs reside in halos of log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The minimum masses to host one central LAE, Mmin, and (on average) one satellite LAE, M1, also vary with Lyα luminosity, growing from log(Mmin/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 and log(M1/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 to log(Mmin/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 and log(M1/[h−1M⊙]) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 from low- to high-luminosity samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The satellite fractions are ≲ 10% (≲ 20%) at 1σ (3σ) confidence level, sup- porting a scenario in which DMHs typically host one single LAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We next bisected the three main samples into disjoint subsets to thoroughly explore the dependence of the clustering properties on LLyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We report a strong (8σ) clustering dependence on Lyα lumi- nosity, not accounting for cosmic variance effects, where the highest luminosity LAE subsample (log(LLyα/erg s−1) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53) clusters more strongly (bhighest = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15) and resides in more massive DMHs (log(Mh/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10) than the lowest luminosity one (log(LLyα/erg s−1) ≈ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97), which presents a bias of blowest = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06 and occupies log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We discuss the implications of these results for evolving Lyα luminosity functions, halo mass dependent Lyα escape fractions, and incomplete reionization signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' large-scale structure – high-redshift galaxies – HOD models – dark matter halo – satellite galaxies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Introduction Dark matter halos (DMHs) serve as sites of galaxy formation but their co-evolution is still a matter of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Observations deliver snapshots of the luminosities of galaxies at given red- shifts, while numerical analyses succeed at simulating the evolu- tion and copiousness of DMHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Linking these two constituents is not straightforward but, because the spatial distribution of bary- onic matter is biased against that of dark matter (DM), the former indirectly traces the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The evolutionary stage of the two dis- tributions depends on both the epoch of galaxy formation and the physical properties of galaxies (see Wechsler & Tinker 2018 for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, studying the dependence of the baryonic-DM relation on galaxy properties is essential for better understand- ing the evolution of the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Exploring the spatial distribution of high-redshift (z > 2) galaxies and its dependence on physical properties provides an insight into the early formation and evolution of the galaxies we observe today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Clustering statistics yield observational con- straints on the relationship between galaxies and DMHs, as well as on their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Traditional studies of high-z galaxies (Stei- del et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019) model the large-scale (R ≳ 1 − 2 h−1cMpc) clustering statistics with a two parameter power-law correlation function that takes the form ξ = (r/r0)−γ (Davis & Peebles 1983) to derive the large-scale lin- ear galaxy bias and the associated typical DMH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' To make full use of the clustering measurements, the smaller separations of the nonlinear regime (R ≲ 1 − 2 h−1cMpc) are modeled by relating galaxies to DMHs within the nonlinear framework of halo occupation distribution (HOD) modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In this context, the mean number of galaxies in the DMH is modeled as a func- tion of DMH mass, further assessing whether these galaxies oc- cupy the centers of the DMHs or whether they are satellite galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 1 of 17 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04133v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='GA] 10 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda Although clustering studies of high-redshift galaxies are plentiful, HOD modeling has been rarely used to interpret the re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While several works have focused on Lyman-break galaxy (LBG) surveys, only one study fit a sample of Lyman-α emitters (LAEs) with HOD models (Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Malkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Hatfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Harikane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) applied the full HOD framework to sets of LBGs to put constraints on the central and satellite galaxy populations, while Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) partially exploited the power of HOD models in a sample of LAEs to infer the threshold DMH mass for central galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The number of studies that have investigated the correlations between clustering strength and physical properties of high- redshift galaxies is slightly higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In [O ii] and [O iii] emission- line-selected galaxy samples, Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) found a strong halo mass dependence on the line luminosity and stel- lar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) also observed a correlation with stellar mass, together with a further dependence on UV lumi- nosity, in a sample of LBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' However, these correlations be- come somewhat unclear near the epoch of reionization (z ≈ 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Based on LAEs surveys, Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Bielby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) revealed tentative trends (≈ 1σ) between luminosity (both UV and Lyα) and clustering strength, while only Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) reported a clear (5σ) correlation between inferred DMH mass and Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In a previous study (Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021), we used 68 MUSE-Wide fields to measure the LAE clustering with the K-estimator method presented in Adelberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We computed the clustering at large scales (R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 h−1Mpc) to de- rive the linear bias factor and the typical DMH mass of LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' By splitting our main sample into subsets based on physical prop- erties of LAEs, we also found a tentative 2σ dependence on Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Here, we extend this work with larger and more deeply spectroscopically confirmed samples and a refined set of analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We measured the clustering at smaller scales, applied full HOD modeling, and studied the dependence of the clustering properties on Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2, we describe the data used for this work and we characterize the LAE samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3, we explain our method for measuring and analyzing the clustering properties of our galaxy sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We present the results of our measurements in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5, we discuss our results and their implications, and we investigate the clustering depen- dence on Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We give our conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Throughout this paper, all distances are measured in comov- ing coordinates and given in units of h−1Mpc (unless otherwise stated), where h = H0/100 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We assume the same h to convert line fluxes to luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, there are implicit h−2 70 factors in the line luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We use a ΛCDM cosmology and adopt ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7, and σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 (Hin- shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' All uncertainties represent 1σ (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3%) confi- dence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Data The MUSE spectroscopic surveys are based on a wedding cake design, namely: a first spatially wide region (bottom of the cake) is observed with a short exposure time (1 hour), while deeper observations (10 hours exposure) are carried out within the first surveyed area (middle tier of the cake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Contained in the last ob- served region, an even deeper survey (140 h) is then built up (at the top of the cake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These three surveys are known as: MUSE- Wide (Herenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Urrutia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019), MUSE-Deep (Ba- con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022), and MUSE Extremely Deep Field (MXDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Each of them can be seen as a different layer of a wedding cake, where higher layers become spatially smaller and correspond to deeper obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In what follows, we give further details on survey and galaxy sample construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' MUSE-Wide The spectroscopic MUSE-Wide survey (Herenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Ur- rutia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019) comprises 100 MUSE fields distributed in the CANDELS/GOODS-S, CANDELS/COSMOS and the Hubble Ultra Deep Field (HUDF) parallel field regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Each MUSE field covers 1 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While 91 fields were observed with an ex- posure time of one hour, nine correspond to shallow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 hours) reduced subsets of the MUSE-Deep data (see next section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' as well as Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017), located within the HUDF in the CANDELS/GOODS-S region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' However, we do not include the objects from this region since they overlap with the MUSE-Deep sample (see next section and gap in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The slight overlap between adjacent fields leads to a total spatial coverage of 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='52 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1 display the spatial distribution of the LAEs from the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We refer to Urrutia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) for further details on the survey build up, reduction and flux calibration of the MUSE data cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In this paper, we extend (x2 spatially, 50% more LAEs) the sample used in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) and include all the 1 h exposure fields from the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Despite the somewhat worse seeing (generally) in the COSMOS region (right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1), we demonstrate in Appendix A that adding these fields does not significantly impact our clustering results but helps in minimizing the effects of cosmic sample variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We also expanded the redshift range of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While MUSE spectra cover 4750–9350 Å, implying a Lyα redshift in- terval of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 <∼ z <∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7, we limited the redshift range to 3 < z < 6 (differing from the more conservative range of Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 < z < 6) as the details of the selection function near the extremes are still being investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Section 2 of Her- rero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) describes the aspects relevant to our analysis on the construction of a sample of LAEs, as well as the strategy to measure line fluxes and redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The redshift distri- bution of the sample is shown in red in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Systematic uncertainties introduced in the redshift-derived 3D positions of the LAEs have negligible consequences for our clus- tering approach (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Within 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='52 arcmin2 and in the selected redshift interval, we detected a total of 1030 LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This implies a LAE density of more than 13 objects per arcmin2 or n ≈ 1·10−3 h3Mpc−3 (for 3 < z < 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' At the median redshift of the sample ⟨z⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0, the transverse extent of the footprint is ≈ 43 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The range of Lyα luminosities is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='92 < log(LLyα/[erg s−1]) < 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='35 (see red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2), with a median value of log⟨LLyα/[erg s−1]⟩ = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34 (or ≈ L⋆ in terms of characteristic luminosity L⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Herenz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019), which makes this sample the highest luminosity data set of our three considered surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The Lyα luminosity distri- bution is shown in red in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The main properties of the MUSE-Wide LAEs are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' MUSE-Deep MUSE-Deep (10 hour MOSAIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022) encompasses nine fields located in the CANDELS/GOODS-S region of the HUDF, each spanning 1 arcmin2 and observed with a 10 h exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The total Article number, page 2 of 17 Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1: Spatial distribution of the LAEs from the MUSE-Wide survey (red circles), MUSE-Deep (green squares) and MXDF (blue stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The overlapping objects between the MXDF and MUSE-Deep samples have been removed from the MUSE-Deep LAE set, while those LAEs overlapping in MUSE-Deep and MUSE-Wide have been removed from the MUSE-Wide LAE sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The MUSE-Wide survey covers part of the CANDELS/GOODS-S region and the HUDF parallel fields (left panel) as well as part of the CANDELS/COSMOS region (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' See Figure 1 in Urrutia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) for the layout of the MUSE-Wide survey without individual objects, Figure 1 in Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2017) for that of MUSE-Deep, and Figure 2 in Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2022) for that of MUSE-Deep (MOSAIC) and MXDF together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Table 1: Properties of the LAE samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Area/[arcmin2] Number LAEs ⟨z⟩ n/[h3Mpc−3] log(LLyα/[erg s−1]) range log⟨LLyα/[erg s−1]⟩ MUSE-Wide 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='52 1030 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 1 · 10−3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='92 – 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='35 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34 (≈ L⋆) MUSE-Deep 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='92 679 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 8 · 10−3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='84 – 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2L⋆) MXDF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='47 367 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 3 · 10−2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 – 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08L⋆) Notes: Properties marked with ⟨⟩ represent median values for the galaxies in the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Table 2: Properties of the LAE subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Number LAEs ⟨z⟩ log⟨LLyα/[erg s−1]⟩ MUSE-Wide low L (log LLyα < 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34) 515 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5L⋆) MUSE-Wide high L (log LLyα > 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34) 515 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53 (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5L⋆) MUSE-Deep low L (log LLyα < 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) 340 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='46 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1L⋆) MUSE-Deep high L (log LLyα > 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) 339 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='89 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3L⋆) MXDF low L (log LLyα < 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) 183 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04L⋆) MXDF high L (log LLyα > 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) 184 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='54 (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2L⋆) Notes: Properties marked with ⟨⟩ represent median values for the galaxies in the subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' spatial coverage is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='92 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We represent the spatial distri- bution of the survey in green in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We did, however, remove the MUSE-Deep objects that are selected in the deepest survey, described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We refer to Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2017, 2022) for a detailed description on survey construction and data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The sources in MUSE-Deep were blindly detected and ex- tracted using ORIGIN (Mary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2020), based on a matched filtering approach and developed to detect faint emission lines in MUSE datacubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the redshift measurements and line classifications were carried out with pyMarZ, a python version of the redshift fitting software MarZ (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2016), the line flux determination was conducted with pyPlatefit, which is a python module optimized to fit emission lines of high-redshift spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The redshift distribution of the sample is shown in green in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2, also within 3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The LAE density of the MUSE-Deep sample is 8 · 10−3 h3Mpc−3 (68 LAE per arcmin2 in the whole redshift range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The survey spans ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 h−1Mpc transversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The range of Lyα luminosities is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='84 < log(LLyα/[erg s−1]) < 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12, represented with green squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2, together with its distribution (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' MUSE-Deep is our intermediate luminous dataset, with a median luminosity of log⟨LLyα/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The sample properties are recorded in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' MUSE Extremely Deep The MUSE Extremely Deep Field (Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022) is situated in the CANDELS/GOODS-S region and overlaps with MUSE- Deep and MUSE-Wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' It is composed of a single quasi circu- lar field with inner and outer radii of 31” and 41”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While a 140 hour exposure was employed to observe the totality of the field, the inner field is 135 hours deep, decreasing to 10 Article number, page 3 of 17 MUSE-Wide MUSE-Deep 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='70 MXDF 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='75 Dec 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='80 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='85 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='05 RA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='28 e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='18 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='16 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='14 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 RAA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda hours depth at the outer radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This makes MXDF the deepest spectroscopic survey to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For further details see Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2022) and the blue data points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1, where the MXDF field is overplotted on the previous surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The survey assembly and data reduction is described in Ba- con et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2022) and is similar to the one applied to MUSE-Deep (Bacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The source extraction in MXDF and the red- shift and flux measurements are conducted following the same procedure as was done for MUSE-Deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The redshift distribu- tion of the sample is shown in blue in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Contained within ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='47 arcmin2 and over the same redshift range as for the previous catalogues, we detected 367 LAEs, cor- responding to a LAE density of n ≈ 3·10−2 h3Mpc−3 (432 LAEs per arcmin2 at 3 < z < 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' With a median redshift of ⟨z⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2, the footprint covers ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 h−1Mpc (transversely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The Lyα lumi- nosities span 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 < log(LLyα/[erg s−1]) < 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 (see blue stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2 and its distribution in the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The median Lyα luminosity is log⟨LLyα/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 (or ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08L⋆), more than one order of magnitude fainter than for MUSE-Wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This makes MXDF the faintest ever observed sample of non-lensed LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The main properties are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' LAE subsamples We bisected the main samples into disjoint subsets based on their median Lyα luminosity to investigate the clustering dependence on this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We did not merge the main LAE datasets be- cause their distinct Lyα luminosities, together with their slightly different location on the sky, might introduce systematics in the clustering measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The subsample properties are summa- rized in Table 2 and described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We split the MUSE-Wide sample at the median Lyα lu- minosity log⟨LLyα/[erg s−1]⟩ = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The two subsamples consist of 515 LAEs each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The low-luminosity subset has a median redshift and Lyα luminosity of ⟨zlow⟩ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 and log⟨LLyαlow/[erg s−1]⟩ = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06, while the high-luminosity sub- sample has ⟨zhigh⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 and log⟨LLyαhigh/[erg s−1]⟩ = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The median redshift of the number of galaxy pairs for the low- luminosity subset is zpair ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4, and that for the high-luminosity one is zpair ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We next bisected the MUSE-Deep set at the median Lyα lu- minosity log⟨LLyα/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The low-luminosity sub- sample has 340 LAEs and presents a median redshift and Lyα lu- minosity of ⟨zlow⟩ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 and log⟨LLyαlow/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The high-luminosity subset is formed by 339 LAEs with ⟨zhigh⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 and log⟨LLyαhigh/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While for the low- luminosity subsample zpair ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5, for the high-luminosity one zpair ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We also divide the sample with the largest dynamic range of Lyα luminosities (MXDF) at the median Lyα lu- minosity log⟨LLyα/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the lower lu- minosity subset contains 183 LAEs with ⟨zlow⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 and log⟨LLyαlow/[erg s−1]⟩ = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97, the higher luminosity sub- sample consists of 184 LAEs with ⟨zhigh⟩ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 and log⟨LLyαhigh/[erg s−1]⟩ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For the low-luminosity subset, we have zpair ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9, and for the high-luminosity one, we have zpair ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The redshift distribution of each subsample is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The corresponding median redshifts are represented with a vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Despite the similar median redshifts be- tween the subsample pairs, the redshift distributions are signifi- cantly different, with a higher amount of spike-trough contrasts in the high-luminosity subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2: Lyα luminosity-redshift for the LAEs in MUSE-Wide (red circles), MUSE-Deep (green squares) and MXDF (blue stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The dashed colored lines correspond to the median log LLyα values of the corresponding samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The redshift and LLyα distributions are shown in the top and right panel, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' K-estimator Galaxy clustering is commonly measured by two-point corre- lation function (2pcf) statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Samples investigated by this method typically span several square degrees on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' With MUSE, we encounter the opposite scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' By design, MUSE surveys cover small spatial extensions on the sky and provide a broad redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although the MUSE-Wide survey is the largest footprint of all MUSE samples, its nature is still that of a pencil-beam survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Its transverse scales are of the order of 40 h−1Mpc, while in redshift space it reaches almost 1500 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' If we consider the deeper samples, the difference is even more prominent: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 vs 1500 h−1Mpc for MUSE-Deep and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 versus 1500 h−1Mpc for MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' It is thus paramount to ex- ploit the radial scales and utilize alternative methods to the tra- ditional 2pcf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) we applied the so-called K- estimator, introduced by Adelberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2005), to a subset of our current sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Here, we build on our previous work by extending the dataset and measuring the small-scale clustering required to perform full HOD modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The details of the K- estimator are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 of Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In the following, we provide a brief description of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The K-estimator measures the radial clustering along line-of- sight distances, Zi j, by counting galaxy pairs (formed by galaxy i and galaxy j) in redshift space at fixed transverse separations, Ri j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although the K-estimator does not need a random sample to carry out the clustering measurements, its nature is very sim- ilar to that of the projected two-point correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We bin by Ri j, shown with distinct radii in the cylinders of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4, and count the number of pairs within individual transverse bins, for two different ranges of Zi j, represented in red and blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The K-estimator as a function of Ri j is then defined as the ratio of galaxy pairs within the first Zi j interval (blue cylin- der) and the total Zi j range (red and blue cylinder), quantifying the excess of galaxy pairs in the first Zi j bin with respect to the total one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We optimize the choice of the Zi j ranges, and thus the Article number, page 4 of 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 S 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 MUSE-Wide MUSE-Deep 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 MXDF 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 ZYohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3: Redshift distribution of the subsamples bisected at the median Lyα luminosity of MUSE-Wide, MUSE-Deep and MXDF (panels from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Blue (red) colors show the low- (high-) luminosity subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The vertical dashed lines represent the median redshift of the corresponding subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' K-estimator, by seeking out the estimator that delivers the best sensitivity for the clustering signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', the highest signal-to- noise ratio, S/N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although slightly different than in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), we find nearly identical K-estimators for each of the current sam- ples (K0,7 7,45 for MUSE-Wide, K0,7 7,45 for MUSE-Deep, and K0,7 7,40 for MXDF), whose clustering signals only differ in their S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We chose the same K-estimator for the three data sets, K0,7 7,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The K-estimator is directly related to the average underlying correlation function (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2 in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In fact, its definition is proportional to a combination of pro- jected two-point correlation functions corresponding to the blue and red cylinders of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the traditional 2pcf method integrates the correlation function ξ(Rij, Zij) over line-of-sight separations up to a maximum line-of-sight distance πmax, the K- estimator integrates up to a2 and a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The correlation function ξ(Ri j, Zi j) can be approximated with a power-law following Lim- ber (1953) equations as we did in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), or modeled with a halo occupation distribution (HOD) model (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For reference, randomly distributed galaxies in space (ξ(Ri j, Zi j) = 0) provide K0,7 7,45(Rij) values equal to 7/45 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2 in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Samples with data points signifi- cantly above 7/45 dispense clustering signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Error estimation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Error estimation for the MUSE-Wide survey Applying clustering statistics delivers correlated data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' One single galaxy might be part of more than one galaxy pair and can therefore contribute to several Rij bins, especially if they are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In order to quantify the actual correlation be- tween data points, we applied the jackknife resampling tech- nique, followed by the computation of the covariance matrix (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Krumpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For the MUSE- Wide sample, we employed ten logarithmic bins in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='16 < Ri j < 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 h−1Mpc, discarding lower Rij scales since they host very few galaxy pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then found a compromise between the number of inde- pendent regions (jackknife zones) and the size of the jackknife zones and divide the sky coverage into Njack = 10 regions, each of which extends ≈ 4 h−1Mpc in both RA and Dec directions (see Appendix B for a visual representation of the sky division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The limited spatial extent of the survey does not allow for a higher number of jackknife zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then constructed Njack jackknife subsamples, excluding one jackknife zone at a time, and computed the K-estimator for each of the subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The K- estimator measurements are then used to derive the covariance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4: Sketch of the K-estimator, representing the relative ge- ometry that probe the one- and two-halo term scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The empty blue and filled red cylinders, delimited by |a2| = 7 h−1Mpc and |a3| = 45 h−1Mpc respectively, illustrate the line-of-sight dis- tance Zi j intervals within which we count galaxy pairs at fixed transverse separations Ri j, represented by nested cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Pairs of LAEs connected with green lines within the same DMH (filled gray circle) contribute to the one-halo term (small Ri j scales), while pairs belonging to two different DMHs (yellow lines) probe the two-halo term (larger Ri j separations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' matrix Mi j, which quantifies the correlation between bins i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The matrix is expressed as Mi j = Njack − 1 Njack �������� Njack � k=1 � Kk(Ri) − ⟨K(Ri)⟩ � × � Kk(Rj) − ⟨K(Rj)⟩ �� , (1) where Kk(Ri), Kk(Rj) are the K-estimators from the k-th jack- knife samples and ⟨K(Ri)⟩, ⟨K(Rj)⟩ are the averages over all jackknife samples in the i, j bins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The error bar for the K-estimator at the ith bin comes from the square root of the diagonal element ( √Mii) of the covariance matrix, our so-called "jackknife uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='" This approach could not be followed in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) because of the smaller sky cover- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Instead, we used a galaxy bootstrapping approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Ap- pendix C, we compare the two techniques and show that boot- strapping uncertainties are ≈ 50% larger than the jackknife error bars, in agreement with Norberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2009), who found that boostrapping overestimates the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=" Article number, page 5 of 17 MUSE-Wide High 'L (log(LLyα/[erg s-1)] > 42." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34) 20 Low L (log(Llyα/[erg s-1)] < 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34) 15 LI A 10 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 ZMUSE-Deep Highi L (log(LLyα/[erg s-1)l > 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) 20 Low L (log(LLyα/[erg s-1)] < 4l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) 15 L A 之 10 5 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 ZMXDF High L (log(LLyα/[erg s-1)] > 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) 20 Low L (log(LLyα/[erg s-1)] < 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) 15 AE 10 5 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Zα3 = 45 α²= 7 αi = 0 α2 = 7 —α3 = —45 Zij Riji1 Rij2 Rij4 Rij3A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda We next search for the best-fit parameters by minimizing the correlated χ2 values according to χ2 = Nbins � i=1 Nbins � j=1 � K(Ri) − K(Ri)HOD � × M−1 ij � K(Rj) − K(Rj)HOD � , (2) where Nbins = 10 is the number of Rij bins, K(Ri), K(Rj) are the measured K-estimators and K(Ri)HOD, K(Rj)HOD are the K- estimators predicted by the HOD model for each i, j bin, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Regardless of the larger sample considered in this work, we are still limited by the spatial size of the survey, which only per- mits a small number of jackknife zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The insufficient statis- tics naturally lead to a higher noise contribution in the covari- ance matrix, which cause the χ2 minimization to mathematically fail (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', cases of χ2 < 0) when the full covariance matrix is in- cluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Hence, we only incorporated the main diagonal of the matrix and its two contiguous diagonals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Appendix B, we dis- cuss the high level of noise in the matrix elements corresponding to bins that are significantly apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We also ver- ify the robustness of our approach and show that our clustering results are not altered (within 1σ) by this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Error estimation for the deeper surveys The small sky coverage of the deeper surveys does not allow us to follow the same error estimation approach as for the MUSE- Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Appendix C, we not only compare the boot- strapping technique applied in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) to the jackknife approach performed in MUSE-Wide, but we also consider the Poisson uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We demonstrate that Poisson and jackknife errors are comparable in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In fact, we show that while bootstrapping uncertainties are ≈ 50% larger than jackknife errors, Poisson uncertainties are only ≈ 7% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, and similarly to Adelberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Diener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018), we stick to Poisson un- certainties for the MUSE-Deep and MXDF samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For these datasets, we measure the K-estimator in eight and six loga- rithmic bins in the ranges 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 < Rij/[h−1Mpc] < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='75 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 < Ri j/[h−1Mpc] < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='45, respectively, constrained by the spatial extent of the surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then perform a standard χ2 minimization to find the best fitting parameters to the K-estimator measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Namely, χ2 = Nbins � i=1 � K(Ri) − K(Ri)HOD σi �2 , (3) where K(Ri), K(Ri)HOD, and σi denote the measured K- estimator, the HOD modeled K-estimator and the Poisson un- certainty in the ith bin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We note that the standard χ2 minimization does not account for the correlation between bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although in Appendix B we show that only contiguous bins are moderately correlated, we should take the resulting fit uncertainties with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Halo occupation distribution modeling The clustering statistics can be approximated with a power- law or modeled with state-of-the-art HOD modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Traditional clustering studies make use of power laws to derive the corre- lation length and slope, from which they infer large-scale bias factors and typical DMH masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This simple approach deviates from the actual shape of the clustering statistic curve, even in the linear regime, and its inferred DMH masses suffer from system- atic errors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1998 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' To overcome these concerns, physically motivated HOD models do not treat the linear and non-linear regime alike but differentiate between the clustering contribution from galaxy pairs that reside in the same DMH and pairs that occupy different DMHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) we only modeled the two- halo term of the K-estimator with HOD modeling, which only delivered the large-scale bias factor and the typical DMH mass of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We now extend into the non-linear regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ri j < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 h−1Mpc) of the one-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We can then model the clustering measured by the K-estimator with a full HOD model, combining the separate contributions from the one- (1h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', galaxy pairs residing in the same DMH) and the two-halo (2h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', galaxy pairs residing in different DMHs) terms: ξ = ξ1h + ξ2h, (4) where ξ is the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The HOD model we used is the same as in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), an improved version of that described by Miyaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Krumpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2012, 2015, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We assumed that LAEs are associated with DMHs, linked by the bias-halo mass relation from Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' From Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2005), we also included the effects of halo-halo collisions and scale- dependent bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The mass function of DMHs, which is denoted by φ(Mh)dMh, is based on Sheth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2001), and the DMH profile is taken from Navarro, Frenk & White (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We use the concentration parameter from Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2007), and the weakly redshift-dependent collapse overdensity from Navarro, Frenk & White (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We fur- ther incorporated redshift space distortions (RSDs) in the two- halo term using linear theory (Kaiser infall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Kaiser 1987 and van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We did not model RSDs in the one- halo term because the peculiar velocity has negligible effects to our K-estimator as demonstrated in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The veloc- ity dispersion (σv) of satellites in a Mh halo can be estimated by σ2 v ≈ GMh/(2Rvir), where Rvir is the virial radius (Tinker 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Its effect on the line-of-sight physical distance estimate is then σv/H(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For 1011−12 h−1M⊙ DMH masses, which are typical for our sample, with virial radii of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='02 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='05 (physi- cal) h−1Mpc, the line-of-sight distance estimation is deviated by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30 h−1Mpc, corresponding to a peculiar velocity dis- persion of σv ≈ 80 − 170 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This is significantly small compared to our a2 = 7 h−1Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We thus assume that the one- halo term contributes only to the Zi j = 0 − 7 h−1Mpc bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We evaluated the HOD model at the median redshift of N(z)2, where N(z) is the redshift distribution of the sampled galaxy pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For our three main datasets, zpair ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The mean halo occupation function is a simplified version of the five parameter model by Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We fixed the halo mass at which the satellite occupation becomes zero to M0 = 0 and the smoothing scale of the central halo occupation lower mass cutoff to σlog M = 0, due to sample size limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We define the mean occupation distribution of the central galaxy ⟨Nc(Mh)⟩ as ⟨Nc(Mh)⟩ = � 1 (Mh ≥ Mmin) 0 (Mh < Mmin) (5) and that of satellite galaxies ⟨Ns(Mh)⟩ as ⟨Ns(Mh)⟩ = ⟨Nc(Mh)⟩ · � Mh M1 �α , (6) Article number, page 6 of 17 Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 where Mmin is the minimum halo mass required to host a central galaxy, M1 is the halo mass threshold to host (on average) one satellite galaxy, and α is the high-mass power-law slope of the satellite galaxy mean occupation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The total halo occu- pation is given by the sum of central and satellite galaxy halo occupations, N(Mh) = Nc(Mh) + Ns(Mh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The dependencies of the HOD parameters on the shape of the K-estimator are detailed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In short, for the HOD parameters there selected, the clustering amplitude of the two- halo term is ascertained by the hosting DMHs and is thus very sensitive to their mass, Mmin, and to the fraction of galaxies in massive halos with respect to lower-mass halos, linked to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The clustering in the one-halo term regime, however, is affected by the three parameters in a complex manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' roughly Mmin and α vary the amplitude, and α as well as (moderately) M1 modify the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' To find the best-fit HOD model, we construct a 3D parame- ter grid for Mmin, M1, and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We vary log(Mmin/[h−1M⊙]) in the range 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2, log(M1/Mmin) from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5, and α within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, all in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For each parameter combination, we computed ξ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4), converted it to the K-estimator using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2 in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), and computed a χ2 value (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2 or 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then used the resulting 3D χ2 grid to estimate the confidence intervals for the HOD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For each point on a 2D plane, we search for the minimum χ2 for the contour- ing along the remaining parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The contours we plot are at ∆χ2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='02, which correspond to Gaussian 68% (1σ) and 95% (2σ) confidence levels, respectively, applying the χ2 distribution for three degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The projections of the 68% probability contours on the three interesting parameters are then used to compute the uncertainty of each HOD parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For each point in the three parameter grid, we also computed the large-scale galaxy bias factor, b, and the fraction of satellite galaxies per halo, fsat, as follows: b = � ⟨N(Mh)⟩ bh(Mh) φ(Mh) dMh � ⟨N(Mh)⟩ φ(Mh) dMh , (7) fsat = � ⟨Ns (Mh)⟩ φ(Mh) dMh � ⟨N(Mh)⟩ φ(Mh) dMh , (8) where bh(Mh) denotes the large-scale halo bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The typical DMH mass is determined by the large-scale galaxy bias factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We ultimately compute the bias and fsat distributions from the HOD models that fall within the 68% confidence (for the three- parameter space) contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These distributions are then used to assess the uncertainties in the bias and fsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Results from HOD modeling 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fit results from the MUSE-Wide survey Using the K-estimator K0,7 7,45, we compute the clustering of our LAE sample in ten logarithmic bins in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='16 < Ri j/[h−1Mpc] < 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5, with error bars calculated following the jackknife resampling technique described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In the left top panel of Figure 5, we show the measured clustering sig- nal, with all MUSE-Wide data points significantly above the 7/45 baseline, which represents the expected clustering of an unclus- tered population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Following the procedure laid out in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, we obtain con- straints on the HOD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' From the grid search and the χ2 minimization, we find the best HOD fit to the K-estimator, colored in black in the same figure and dissected into the one- and two-halo term contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' It can be seen from the residu- als (bottom) that the model is in remarkable agreement with the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A somewhat intriguing feature, at least at first sight, is the kink in the two-halo term profile at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 < Ri j/[h−1Mpc] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This reflects the effect of the halo-halo collision introduced in the HOD model formalism by Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2005), where the galaxy pairs within the same DMH cannot contribute to the two-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our fitting allows us to find the best-fit HOD from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In the right top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5, we represent the best HODs for the central, satellite, and total LAEs from the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the halo mass needed to host one (central) LAE is log(Mh/[h−1M⊙]) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6, satellite galaxies are only present if the DMHs are at least one order of magnitude more massive (log(Mh/[h−1M⊙]) > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' As described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, we also compute the confidence regions for the HOD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We show the probability con- tours (red) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The wobbliness of the curves, especially those involving α, is caused by making use of a discrete grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For our sample, the contours are constrained to have α > 1, log(M1/Mmin) > 1, and log(Mmin/[h−1M⊙]) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We list the best-fit HOD parameters in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the minimum DMH mass required to host a central galaxy is log(Mmin/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, that needed to host one central and (on average) one satellite is log(M1/[h−1M⊙]) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', log(M1/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The power-law slope of the num- ber of satellites is found to be α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The inferred typical DMH mass is log(Mh/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09, corresponding to a large-scale bias factor of b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The high values of log M1 and α, considering the typical DMH mass of LAEs, sug- gest a low number of satellite galaxies detected in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Seeking robust information about the number of satellite galaxies, we compute the satellite fraction fsat (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 8) for each parameter combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We find fsat ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 at the 3σ confidence level, being fsat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='012+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='018 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' That is, ≈ 3% (1σ upper limit) of the LAEs in the MUSE-Wide survey are satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In other words, at most ≈ 2 out of ≈ 65 DMHs in our sample host one satellite LAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fit results from MUSE-Deep We measure the clustering of the MUSE-Deep LAE sample with the same K-estimator in eight logarithmic bins within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 < Ri j/[h−1Mpc] < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We compute Poisson uncertainties as laid out in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 and display the result in the middle left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Overplotted on the clustering signal, we show the best HOD fit, split into the one- and two-halo term contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The good quality of the fit is quantified with the residuals in the bot- tom panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Following the procedure described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2, we com- pute the confidence intervals for the HOD parameters and list them in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We plot the probability contours (green) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6, which overlap significantly with those from the MUSE- Wide sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Central LAEs can occupy DMHs if these are at least as massive as log(Mmin/[h−1Mpc]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1, whereas, in order to host satellite LAEs, the halos must have masses log(M1/[h−1Mpc]) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 (log(M1/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These values correspond to a large-scale bias and typical DMH mass b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 and log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09, which are similar to those found in the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The derived Article number, page 7 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5: Best-fit HOD models to the LAE clustering measurements (blue data points) from MUSE samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Top left: Blue dashed, red dotted, and black continuous curves show the one-halo, two-halo, and total clustering terms from the MUSE-Wide sample, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The black straight line shows the expected K value of an unclustered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The residuals are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The uncertainties are computed with the jackknife technique described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Top right: Best-fit HODs for central (red dot- ted), satellite (blue dashed), and total LAEs (black continuous) from the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Shaded regions correspond to 1σ confidence space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Middle: Same but for MUSE-Deep and using Poisson error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Bottom: Same but for MXDF and Poisson uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' satellite fraction is fsat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='004+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='009 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='002, consistent with that from the MUSE-Wide LAE sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then compute the best-fit HOD for central, satellite and total LAEs (middle right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In line with the best- fit HOD parameters and somewhat lower than the values found for the MUSE-Wide survey, the smallest DMH that can host a central LAE has a mass of log(Mh/[h−1M⊙]) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4, more than one order of magnitude lower than that required to host one ad- ditional LAE (satellite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fit results from the MUSE Extremely Deep Field We make use of six logarithmic bins in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 < Ri j/[h−1Mpc] < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='45 and Poisson errors (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2) to quantify the clustering of the sample of LAEs from MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We show the K-estimator measurements in the bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5, along with the corresponding best HOD fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The probability contours are plotted in blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6, sig- nificantly apart from those of MUSE-Wide and MUSE-Deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the minimum DMH mass to host a central LAE is log(Mmin/[h−1Mpc]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, that to host one central and one Article number, page 8 of 17 Total (1h+2h) MUSE-Deep 2-halo term 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 1-halo term 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 0.' 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+page_content='0 N(Mh) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 logM [h-1M]Total (1h+2h) MXDF 2-halo term 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 1-halo term 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 Data/model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 logM [h-1M]Total (1h+2h) MUSE-Wide 2-halo term 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 1-halo term 5 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 N(Mh) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 logM [h-1M]Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 Table 3: Best-fit HOD parameters for the main samples of LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' ⟨z⟩ log(Mmin/[h−1M⊙]) log(M1/Mmin) α fsat b log(Mh/[h−1M⊙]) MUSE-Wide 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 MUSE-Deep 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 MXDF 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 Notes: ⟨z⟩ is the median redshift of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Mmin, M1 are the threshold DMH masses to host a central and a satellite LAE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' α is the high-mass power-law slope of the number of satellite galaxies, fsat is the satellite fraction, b is the large-scale bias factor and Mh is the typical DMH mass of the galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6: Confidence contours in the three HOD parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Red corresponds to MUSE-Wide, green to MUSE-Deep, and blue to MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The thick (dashed) contours represent the 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3% (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5%) confidence, at ∆χ2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='02) level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The crosses stand for best-fit (χ2 min), searched along the remaining parameter for each 2D parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' satellite LAE is log(M1/[h−1Mpc]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 (log(M1/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These values are somewhat lower than those found for the MUSE-Wide survey and correspond to a bias factor and typical halo mass of b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 and log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The inferred satellite fraction is fsat = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='05 ( fsat ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 at the 3σ confidence level), tentatively higher than that found in the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' From the best-fit HOD parameters, we calculate the HODs for central, satellite and total LAEs and show them in the bottom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Significantly lower than in the MUSE-Wide survey, central LAEs reside in DMHs if these are more massive than log(Mh/[h−1M⊙]) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For the satellite case, and simi- larly to the previous LAE samples, they only exist if the halos are around one order of magnitude more massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 9 of 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='50 MUSE-Wide 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 MUSE-Deep MXDF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='75 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 logM1/Mmin log(Mmin/h-1M)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda It is worth pointing out that the three HOD parameters have some degree of degeneracy, printed out in the diagonally elon- gated probability contours in log M1/Mmin – α space in the bot- tom right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This can be understood as follows: a higher α in the models causes an increase of satellites at high mass halos, but this can be compensated by producing less satellites by increasing log M1/Mmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While this correlation is clearly visible for the MUSE-Wide and MUSE-Deep samples, the MXDF dataset only seems to be affected in the 95% con- fidence contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We did not observe clear correlations between other parameters with any of our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Appendix A shows how our K-estimator varies with the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The causes of parameter degeneracies are also noticeable in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We note however that while the correlation between the HOD parame- ters leads to the perturbed shape of the probability contours, the lowest (MXDF) and highest luminosity (MUSE-Wide) sample contours are detached from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, for the purposes of this study, simultaneously fitting the three HOD parameters and showing their correlations is preferable over, for instance, fixing α to a dubious value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Clustering dependence on Lyα luminosity The complex radiative transfer processes that the Lyα photons are subject to make the search for correlations between Lyα lu- minosity and other physical properties a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Despite this complication, Yajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) predicted a correlation between simulated LLyα and halo mass based on halo merger trees and Lyα radiative transfer calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) is, however, the only study so far that has reported a clear (5σ) relation between these quantities using observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Motivated by these results, we exploited the large dynamic range of Lyα luminosities that we cover to investigate the relation be- tween Lyα luminosity and DMH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' As a first step, we com- pare the K-estimator measurements in the MUSE-Wide survey (highest luminosity LAE sample: ⟨LLyα⟩ ≈ 1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34 erg s−1, but still fainter than those in Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019)) and in MXDF (faintest LAE sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' ⟨LLyα⟩ ≈ 1041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 erg s−1) and show the outcome of this comparison in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The relatively luminous LAEs from the MUSE-Wide survey cluster slightly more strongly (bWide = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11) than the low- luminosity LAEs from MXDF (bMXDF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The cluster- ing measurements and bias factor (b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09) in MUSE-Deep (log(LLyα/[erg s−1]) = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) fall between those from MUSE- Wide and MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We convert the bias factors from the three main samples of this study into typical DMH masses and plot them as a function of their median Lyα luminosity with colored symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although the three main datasets sample the same region of the sky, their transverse coverage is limited and somewhat dif- fers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Therefore, our results are affected by cosmic sample vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Ideally, this uncertainty is estimated from the variance of clustering measurements from simulated mocks in different lines of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Inferring cosmic variance from a large set of mocks that are able to reproduce the observed clustering of our LAEs is however beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We further investigate the possible dependence on LLyα by splitting the main LAE samples into disjoint subsets (see Ta- ble 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We compute the K-estimator in each LLyα subsample, find the best HOD fit and list the large-scale bias factors and the typical DMH masses in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We also plot the typical DMH masses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 8 (empty symbols) as a function of the median Table 4: Best HOD fit large-scale bias factor and typical DMH mass for the LAE subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Subsample ⟨z⟩ b log(Mh/[h−1M⊙]) MUSE-Wide high L 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 MUSE-Wide low L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='92+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11 MUSE-Deep high L 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 MUSE-Deep low L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='68+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 MXDF high L 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='96+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 MXDF low L 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' ⟨z⟩ is the median redshift of the subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The uncertainties do not include cosmic sample variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' LLyα of the subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We find that typical halo mass increases from 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00 to 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43M⊙ between 1040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97 and 1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53 erg s−1 in line luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For each subsample pair, the high-luminosity subset always clusters more strongly than the low-luminosity one and, in this case, cosmic sample variance effects can be completely ne- glected because subset pairs span the exact same area on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The most pronounced difference is found when splitting the MXDF sample, the dataset with the largest dynamic range of Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The best HOD fits deliver blow = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06 and bhigh = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9σ significant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Despite its higher luminosity, we infer a less massive DMH for the MUSE-Deep high-luminosity subsample than for the main dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This is due to the higher zpair of the subset (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Because we evaluate the HOD model at zpair, a higher redshift corresponds to HOD models in which the halo mass function presents a lower number density of mas- sive halos and, thus, deliver less massive typical DMHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The same reasoning applies when comparing the high-luminosity MXDF and low-luminosity MUSE-Deep subsamples and the high-luminosity MUSE-Deep and low-luminosity MUSE-Wide subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While each subsample pair presents similar median lu- minosities, the former also has similar zpair, unlike the latter one (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This translates into similar DMH masses for the first pair but significantly distinct masses for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We last consider the most extreme cases, the low-luminosity subset from MXDF and the high-luminosity one from the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We show the measured clustering in the two subsamples in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The high- luminosity LAEs cluster 8σ more strongly than the low- luminosity LAEs, without accounting for cosmic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We find that LAEs with log(LLyα/[erg s−1]) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53 reside in DMHs of log(Mh/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 and that lower luminosity LAEs (log(LLyα/[erg s−1]) ≈ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97) are hosted by DMHs of masses ranging log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These results fit well within the assumed framework in which star-forming galax- ies that reside in more massive halos present higher star forma- tion rates and thus show more luminous nebular emission lines (Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This dependence can then be weakened by low Lyα escape fractions in high mass halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Following Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 of Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), we matched the redshift distributions of the three main samples and of each subsample pair to verify that the difference in cluster- ing amplitude is not driven by the different redshift distribu- tion of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For each main sample, we compare indi- vidual bins between their corresponding z-distributions and se- lect the one that contains a higher number of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then Article number, page 10 of 17 Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 7: Clustering dependence on Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Left: K-estimator measurements in the MUSE-Wide survey (red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' ⟨LLyα⟩ ≈ 1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34 erg s−1) and MXDF (blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' ⟨log LLyα⟩ ≈ 1041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The dotted curves represent the best HOD fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The black straight line shows the expected K-estimator of an unclustered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Right: Same for the high LLyα subset (red) from the MUSE-Wide survey and the low LLyα subsample (blue) from MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' randomly remove LAEs until we match the number counts of the non-selected samples in that bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Once all bins have been inspected, we obtain "matched" z-distributions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', equivalent), but with still different Lyα luminosity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We ran the K-estimator in the three "matched" datasets and find consistent results with the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We follow the same approach for the subsamples such that the low- and high-luminosity subsets have exactly the same z-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We find that the cluster- ing difference between the "matched" and original subsamples varies within 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Besides, as we did for LLyα, we also searched for a possible clustering dependence on redshift and found no trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, we discarded the possibility of a possible clustering dependence on Lyα luminosity driven by z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our results are not driven by AGN or low-redshift emission line contamination either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The Lyα-emitting AGN fraction for LLyα < 1043 erg s−1 is close to zero (Spinoso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2020 and ref- erences therein) and the four known X-ray detected AGNs (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017), which only affect MUSE-Wide and MUSE-Deep, were not included in our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Besides, Urrutia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) performed a stacking experiment of X-ray images centered on MUSE-Wide LAEs, yielding no signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The presence of low- redshift interlopers in our spectroscopic samples is also unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' [O ii] emitters are the typical contaminants of high-redshift LAE samples but the high resolution of the MUSE instrument allows to distinguish the [O ii] emission line doublet with high confi- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These results are in line with the tentative trends seen in Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) and the clear dependence found in Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2003) noted a slight difference in the correlation amplitude of two LLyα subsamples (30 and 57 LAEs in each subset at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='86 with log(LLyα/[erg s−1]) > 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 and log(LLyα/[erg s−1]) < 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2, respectively), Kusakabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) observed a tendency (< 2σ) of larger bias factors corresponding to higher luminosity LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' They used four deep survey fields at z = 2 with limiting Lyα luminosities within the range of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 < log(LLyα/[erg s−1]) < 42 computed from NB387 mag- nitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' More significant is the dependence found in Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) and Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the lat- ter measured a 2σ difference in bias factors or DMH masses between two subsets of 349 and 346 LAEs at z ≈ 4 with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 8: Typical dark matter halo mass against observed me- dian Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Filled and unfilled symbols correspond to the values derived from the samples and subsamples described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Red circles, green triangles and blue squares belong to MUSE-Wide, MUSE-Deep and MXDF, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Gray crosses represent the results from Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) in the Lyα luminosity interval relevant for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' log(LLyα/[erg s−1]) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='14 and log(LLyα/[erg s−1]) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='57, the former used various surveys with discrete redshift slices be- tween 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 < z < 6 and 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 < log(LLyα/[erg s−1]) < 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 to find that halo mass clearly (5σ) increases with increasing line luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For a direct comparison, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 8 (gray crosses) the DMH masses computed by Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2019) from samples with similar redshifts (z ≈ 3) and Lyα luminosi- ties (log(LLyα/[erg s−1]) ≈ 42) to our current LAE samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our results are in good agreement and extend to much fainter Lyα luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our results, along with those from the literature, demonstrate that having a broad dynamic range of LLyα (nearly extending two orders of magnitude) and a large number of LAEs in the samples is crucial to detect the clustering dependence on LLyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 11 of 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 MXDF (log(LLyα/[erg s-1]) ～ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) MUSE-Wide (log(LLyα/[erg s-1]) ~ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rij [h-1Mpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 Low L (MXDF, log(LLyα/[erg s-1]) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97) High L (MUSE-Wide, log(LLyα/[erg s-1j) ～ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rii [h-1Mpc]MUSE-Wide MUSE-Deep 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 MXDF log(Mn / [h-1Mol) Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 中 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 log(LLyα/erg s-1)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Comparison to Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) In this section we compare our results with the findings of our previous study (Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021, hereafter HA21), where we measured the clustering of a subset (68 fields of the MUSE-Wide survey) of our current sample (91 fields of the MUSE-Wide survey) and fitted the corresponding signal with a two-halo term only HOD modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In order to envisage the methodological and statistical improvement of our new investi- gation, we applied our K0,7 7,45 estimator to the sample considered in HA21 (695 LAEs at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 < z < 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We compare the outcome to our current clustering measurement in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The two datasets show good agreement within the uncer- tainties, with smaller errors for the current sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Besides the higher number of LAEs and larger spatial coverage, the error es- timation was carried out following different procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the spatial coverage of the full MUSE-Wide survey allows us to compute the covariance matrix from the jackknife resampling technique, the smaller transverse extent covered by the 68 fields did not allow the split of the surveyed area into a significant num- ber of jackknife zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, in HA21, we chose bootstrapping error bars as our next most conservative and realistic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The slightly puzzling hump seen in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4 of HA21 at 4 ≲ Ri j/[h−1Mpc] ≲ 7 is no longer visible in our new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This confirms the judgement in HA21 that the feature was con- sistent with a statistical fluctuation resulting from the correlation between datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In HA21, we limited the range of transverse separations to Ri j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 h−1Mpc, excluding the smallest scales of the one-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, we fitted the signal with a two-halo term only HOD model (red dotted curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 9) in contrast to the full HOD modeling performed in this work (blue dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While the former only constrained the large-scale bias factor and the typi- cal DMH mass of LAEs, the latter further determines the num- ber of central and satellite galaxies, as well as the required DMH mass to host each type of galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Despite these dissimilarities, the two fits are in good agreement: the bias factor (b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='38) and the typical DMH mass of LAEs (log(MDMH / [h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='27) from HA21 are consistent with those derived in this work (b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11 and log(MDMH / [h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The higher accuracy of our current measurements originates from the larger sample, the availability of more realistic error bars, and constraints from the one-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Comparison to the literature A common way to infer the host DMH masses of LAEs is to quantify the galaxy clustering of the detected population through clustering statistics, which is then traditionally approximated with power-laws or fit with physically motivated HOD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Following the traditional approach, Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2007), Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2010) and Bielby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2016) focused on the clus- tering of a few hundred LAEs at z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 to obtain typi- cal DMH masses in the range 1010 − 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Similar masses were found by Khostovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) in a much larger sample (≈ 5000 LAEs) in discrete redshift slices within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 < z < 6, adopting the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A major improvement in terms of methodology was presented in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018), who considered samples of high-z galaxies (2000-3000 mainly LAEs and Lyman-break galaxies, LBGs) and quantified the clustering with HOD modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) found that their LAEs at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6) are hosted by DMHs with typical masses Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 9: Clustering of the full MUSE-Wide sample (blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' this work) compared to the subset considered in HA21 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The for- mer measurements show jackknife uncertainties (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1) and the latter bootstrapping errors (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 in HA21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The blue dotted curve represents our best-fit from full HOD model- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The red dotted curve displays the two-halo term only best HOD fit found in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 of HA21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The black straight line shows the expected K value of an unclustered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' of log(Mh/M⊙) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2006) and Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2014, 2018) computed log(Mh/h−1M⊙) ≈ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 for their sample of galaxies at z = 4 − 5 and z = 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Considering that we have performed a full HOD modeling at the median redshift of our number of galaxy pairs (zpair = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8) and that the DMH masses are predicted to evolve with cos- mic time, our derived typical DMH masses log(Mh/h−1M⊙) ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='77 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 are in good agreement with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Besides the computation of typical DMH masses, modeling the one-halo term of the clustering statistics with HOD mod- els delivers the minimum DMH mass required to host a central galaxy, Mmin, that is needed for a satellite galaxy, M1, and the power-law slope of number of satellites, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These three parame- ters constrain the satellite fraction, fsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) par- tially exploited the power of HOD models in a sample of ≈ 2000 LAEs to obtain log(Mmin/M⊙) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9) at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our derived minimum masses to host a central galaxy at zpair = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 are considerably larger (log(Mmin/M⊙) ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7), which can be explained by the different Lyα luminosities cov- ered in the two studies, and by the fact that several HOD pa- rameters were fixed in Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018), namely, σlog M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2, log M0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='76M1+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, log M1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='18 log Mmin−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='28, and α = 1, which are not compatible with ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This was the only previous study that performed HOD modeling in a sample of LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2006) and Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2014) made use of the full potential of HOD models to reproduce the clustering of their LBG population at z = 4 − 5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 < z < 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although it is still under debate whether LBGs and LAEs are the same galaxy population (Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015 and references therein), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2006) computed a minimum DMH mass to host a central LBG of log(Mmin/M⊙) ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8, to host a satel- lite LBG of log(M1/M⊙) ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0, and a power-law slope α for the number of satellites of α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7, with considerable uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Similarly, Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2014) found log(Mmin/M⊙) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='70, log(M1/M⊙) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='88, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While their halo masses are in agreement with our findings, their slope is somewhat shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This is partially expected given the dis- Article number, page 12 of 17 Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='40 This work 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30 5 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rij [h-1Mpc]Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 similarities in the galaxy populations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', disparate observa- tional selection techniques detect distinct galaxy populations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Satellite fraction In the above discussions on HOD modeling, we limit ourselves to the HOD model form expressed by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5 and 6, which is rather restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The underlying assumption of the model is that the center of the halo with mass Mh > Mmin is always occupied by one galaxy in the sample (or at least at a Mh-independent constant probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This form may be appropriate for instance, for luminosity or stellar mass thresholding samples, but there is no reason that this has to be the case for samples selected by other criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We note that the inferred value of fsat is sensitive to the form of the parameterized model of the central and satellite HODs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In this work and in the literature, a power-law form of the satel- lite HOD is customarily assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In this case, a lower α would increase the model ⟨Ns(Mh)⟩ at the lower Mh end, near Mmin, and yield fewer satellites in higher mass halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Since the halo mass function drops with increasing mass, fsat is mainly deter- mined by the HOD behavior around Mh ∼ Mmin ∼ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 h−1M⊙, where the halo mass function is large and the virial radius is rvir ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 h−1Mpc at z ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These scales are too small to be well constrained by our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our observed one-halo term mainly constrains the satellite frac- tion at larger mass halos (Mh ∼ Mmin ∼ 1013 h−1M⊙, where rvir ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 h−1Mpc at the same redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, the fsat values from the HOD modeling should be viewed with caution and may well reflect the artefacts of the assumed form of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' On the other hand, the sheer presence of a significant one-halo term indicates the existence of some satellites at higher halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The extent of the one-halo term up to Rij ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 h−1Mpc shows that there are indeed satellites up to Mh ∼ 1013 h−1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In spite of the above caveats, the small satellite fraction of the LAEs is likely to be robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The small fsat values for the assumed HOD model indicate that not only central-satellite pairs are rare, but also satellite-satellite pairs are as well, suggesting that only a small fraction of halos contain multiple LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The small Mmin values themselves are also an indication that a large majority of the halos (at the low mass end) that contain a LAE are indeed dominated by one galaxy and in this case, the LAE is probably the central galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Implications The clustering results of this study do not only have implications on the baryonic-DM relation, but also on evolving Lyα lumi- nosity functions, signatures of incomplete reionization, and halo mass-dependent Lyα escape fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We address these aspects in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The relation between halo mass (or clustering strength) and Lyα luminosity (Table 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 8) demonstrates that high- luminosity LAEs tend to reside in higher density environments than lower luminosity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' As a result, overdense regions con- tain a larger fraction of high-luminosity sources (and a lower fraction of less luminous ones) than environments of lower den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These inferences affect the Lyα LF measurements at 3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While we expect a shallower faint-end slope of the Lyα LF in overdense regions, the slope should steepen in average or low density environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' As a consequence, surveys for rela- tively high-luminosity (LLyα ≈ 1042 erg s−1) LAEs are implicitly biased against the lowest density regions and thus gives a biased shape for the LF, which should not be extrapolated towards lower Lyα luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Assuming that our LLyα − Mh relation still holds at higher redshifts, the Lyα LF at z ≥ 6 would be even more affected, not only because of the above discussion but also because higher redshift bins are mainly populated by high-luminosity sources, contrary to lower redshift bins (typical case for telescopes with higher sensitivity at bluer wavelengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, it is important to be careful when interpreting Lyα LFs, especially near the epoch of reionization (EoR), where a shallow to steep variation in the slope of the LF from higher (z ≈ 7) to lower redshifts (z ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7) is commonly interpreted as a sign of incomplete reionization (Konno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Simulations at those higher redshifts also tend to find that high-luminosity LAEs are more likely to be observed than low- luminosity ones because they are able to ionize their surround- ings and form H ii regions around them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', ionized bubbles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Hutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Yoshioka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These allow Lyα photons to redshift out of the resonance wave- length and escape the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Lower luminosity LAEs are then observed if they reside within the ionized bubbles of higher lumi- nosity LAEs or if they are able to transmit enough flux through the IGM (Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' If our LLyα − Mh relation is still valid at these redshifts, our results would support this simulation paradigm since high-luminosity LAEs (situated in overdense re- gions) could form large ionized bubbles more efficiently than low-luminosity sources which tend to be located in lower den- sity environments (Tilvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Theoretical studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Furlanetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007) have modeled the size distribution of these H ii re- gions and predicted an increase in the apparent clustering sig- nal of LAEs towards the epoch of reionization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', towards a more neutral IGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Large ionized bubbles become rarer as the ionizing fraction declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This patchy distribution of H ii re- gions, which mostly surrounds large galaxy overdensities, boosts the apparent clustering of LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This is commonly interpreted as another sign of incomplete reionization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Hutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Comparisons between observed intrin- sic LAE clustering and model predictions have therefore been used to infer the fraction of neutral hydrogen at the EoR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ouchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Nevertheless, if the clustering dependence on Lyα luminosity continues to z ≈ 6, this comparison should be performed with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Because the observed high redshift bins (z ≥ 6) mainly contain high-luminosity LAEs, a strong cluster- ing signal at z ≈ 6 may be wrongly interpreted as incomplete reionization when, in fact, it may only reflect the natural relation between Lyα luminosity and clustering strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We speculate that our results also play a role in the amount of escaping Lyα photons (Lyα fesc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Durkalec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2018) observed a dependence between halo mass and absolute UV magnitude (MUV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The interpretation of their relation goes as follows: MUV traces star formation rate (SFR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Walter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2012), which, in turn, tracks stellar mass (M∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Salmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015), which correlates with halo mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Because we observe a similar relation of Mh with LLyα, LLyα is presumably also a tracer of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' If this is correct, the object-to- object variations in Lyα escape fraction cannot be so large that they obscure the trend of SFR – M∗ – Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Given the typical Lyα luminosities of our sample, this is in agreement with the model suggestions of Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2011a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2015), where the Lyα fesc is of the order of unity for sources with SFR ≈ 1 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The Lyα luminosity would then be a good tracer of the SFR for less luminous LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 13 of 17 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Conclusions We report a strong clustering dependence on Lyα luminosity from the clustering measurements of three MUSE Lyα emitting galaxy (LAE) samples at 3 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Following the pencil-beam design of MUSE surveys from spatially large and shallow ob- servation to spatially small and deep observation, we use 1030 LAEs from the full MUSE-Wide survey (1 h exposure time), 679 LAEs from MUSE-Deep (10 h), and 367 LAEs from MXDF (140 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We thus connect the clustering properties of L⋆ LAEs with those of much fainter ones in the MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We applied an optimized version of the K-estimator as the clustering statistic, coupled to state-of-the-art halo occupation distribution (HOD) modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' From our full HOD analysis, we derive constraints on the HOD of high-luminosity (log(LLyα/erg s−1) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='34), in- termediate (log(LLyα/erg s−1) ≈ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64) and low-luminosity (log(LLyα/erg s−1) ≈ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22) LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We modeled the LAE HOD with three parameters: the threshold dark matter halo (DMH) mass for hosting a central LAE (Mmin), for hosting (on average) one satellite LAE (M1), and the power-law slope of the num- ber of satellites per halo (α) as a function of halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For the high-luminosity sample we derived a typical DMH mass of log(Mh/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09, corresponding to a bias factor of b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These findings, although more accurate, are in agreement with the results based on the two-halo term only HOD modeling performed in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) for a subset of our MUSE-Wide sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' For the lower luminosity samples we found lower DMH masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While for the log(LLyα/erg s−1) ≈ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='64 dataset we inferred log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09 (b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09), for the low-luminosity LAE sample we computed log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 (b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We also derived threshold DMH masses for centrals and satellites for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We found that the minimum DMH mass to host a central LAE is log(Mmin/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 for low-, intermediate-, and high-luminosity LAEs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The threshold halo mass for satellites and the power-law slope of the number of satellite LAEs also increase with Lyα luminosity, from log(M1/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 to log(M1/[h−1M⊙]) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 and α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5 and to log(M1/[h−1M⊙]) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These HOD con- straints imply a decreasing number of detected satellite LAEs with luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Indeed we infer satellite fractions of fsat ≲ 10, 20% (at 3σ confidence level) for high- and low-luminosity LAEs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This suggests that the most common sce- nario for current MUSE surveys is that in which DMHs mainly host a single detected LAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Motivated by these results, we aimed to further explore the clustering dependence on Lyα luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Exploiting the large dynamic range of LLyα from MXDF, we split the main LAE sample at its median LLyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We found a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9σ difference be- tween the clustering of the low-luminosity (log(LLyα/erg s−1) ≈ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97, blow = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06) and the high-luminosity subset (log(LLyα/erg s−1) ≈ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='54, bhigh = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then selected the highest luminosity LAE subset from the MUSE-Wide survey (log(LLyα/erg s−1) ≈ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='53) and the lowest luminosity LAE sub- sample from MXDF (log(LLyα/erg s−1) ≈ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='97), resulting in a clear clustering dependence where the high-luminosity LAEs from MUSE-Wide cluster more strongly (bhigh = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 or log(Mh/[h−1M⊙]) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='10) than the low-luminosity ones from MXDF (blow = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='06 or log(Mh/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='09) at 8σ significance, excluding cosmic variance effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The on- going Hobby-Eberly Telescope Dark Energy Experiment (HET- DEX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021) survey will complement these re- sults at the high-luminosity end and at somewhat lower redshifts (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The implications of this framework are however not only relevant for LAE clustering studies, but also for reported mea- surements of evolving Lyα luminosity functions, detections of incomplete reionization at z ≈ 6, and the relation between Lyα escape fraction and halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Our results are also crucial for the much debated relevance of unresolved satellite LAEs (fainter than those in MXDF) for the measured Lyα surface brightness profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The authors give thanks to the staff at ESO for extensive support during the visitor-mode campaigns at Paranal Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We thank the eScience group at AIP for help with the functionality of the MUSE-Wide data release webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' thank for financial support by CONACyT Grant Científica Básica #252531 and by UNAM-DGAPA (PASPA, PAPIIT IN111319 and IN114423).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' by the Deutsche Forschungsgemeinschaft through grant Wi 1369/32-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' acknowledges support by DLR grant 50OR1904 and DFG grant KR 3338/4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='The data were obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='A- 0791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' References Adelberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Steidel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Pettini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Shapley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Reddy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', & Erb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2005, ApJ, 619, 697-713 Adelberger, Astropy Collaboration, Robitaille, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Tollerud, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2013, A&A, 558, A33 Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Conseil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Mary, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017, A&A, 608, A1 Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Brinchmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Conseil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='08493 Bielby, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Tummuangpak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Shanks, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2016, MNRAS, 456, 4061 Davis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Peebles, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1983, ApJ, 267, 465 Diener, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Wisotzki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Schmidt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017, MNRAS, 471, 3186-3192 Durkalec, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Le Fèvre, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Pollo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2014, A&A, 583, A128 Durkalec, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Le Fèvre, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Pollo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, A&A, 612, A42 Furlanetto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Zaldarriaga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2006, MNRAS, 365, 1012 Garel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Blaizot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Guiderdoni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015, MNRAS, 450, 1279 Gawiser, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Francke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Lai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007, ApJ, 671, 278 Gebhardt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Cooper, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ciardullo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021, ApJ, 923, 217 Hatfield, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Bowler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Jarvis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, MNRAS, 477, 3760 Herenz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Urrutia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Wisotzki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017, A&A, 606, A12 Herenz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Wisotzki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Saust, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019, A&A, 621, A107 Harikane, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ono, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, Publications of the Astronomical Society of Japan, 70, S11 Herrero Alonso, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Krumpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Wisotzki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2021, A&A, 653, A136 Hinshaw, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Larson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Komatsu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2013, AJSS, 208, 19 Hinton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Davis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Lidman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2016, Astronomy and Computing, 15, 61 Hu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', McMahon, & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1998, ApJ, 502, L99 Hutter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Dayal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', & Müller, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015, mnras, 450, 4025 Inami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Brinchmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017, A&A, 608, A2 Jenkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Pearce, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1998, ApJ, 499, 20 Kaiser, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1987, MNRAS, 227, 1-21 Khostovan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Sobral, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Mobasher, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, MNRAS, 478, 2999– 3015 Khostovan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Sobral, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Mobasher, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019, MNRAS, 489, 555-573 Konno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ono, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2014, ApJ, 797, 16 Krumpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Miyaji, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Coil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2010, ApJ, 713, 558 Krumpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Miyaji, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Coil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Aceves H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2012, ApJ, 746, 1 Krumpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Miyaji, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Husemann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015, ApJ, 815, 21 Krumpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Miyaji, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Coil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Aceves, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, MNRAS, 474, 1773 Kusakabe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Shimasaku, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, Publications of the Astro- nomical Society of Japan, 70, 4 Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Giavalisco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Gnedin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2006, ApJ, 642, 63 Limber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1953, ApJ, 117, 134 Luo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Brandt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Xue, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017, ApJSS, 228, 2 Madau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Cohen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' P, Maruyama, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2017, ApJ, 850, 5 Mary, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Bacon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Conseil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2020, A&A, 635,A194 Matthee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Sobral, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Santos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015, MNRAS, 451, 400 McQuinn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Zaldarriaga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007, MNRAS, 381, 75 Article number, page 14 of 17 Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 Miyaji, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Krumpe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Coil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Aceves, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2011, ApJ, 726, 83 Moster, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Somerville, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Maulbetsch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2010, ApJ, 710, 903 Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='M 1997, ApJ, 490, 493 Norberg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Baugh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Gaztañaga, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', & Croton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2009, MNRAS, 396, 19 Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Shimasaku, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Furusawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2003, ApJ, 582, 60 Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Shimasaku, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Furusawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2010, ApJ, 723, 869 Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Harikane, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Shibuya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, PASJ, 70, S13 Salmon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Papovich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2015, ApJ, 799, 183 Santos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Sobral, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Matthee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2016, MNRAS, 463, 1678 Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Hayes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Verhamme, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2011a, A&A, 531, A12 Sheth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Mo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Tormen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2001, 323, 1-12 Spinoso, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Orsi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', López-Sanjuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2020, A&A, 643, A149 Steidel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Giavalisco, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Pettini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1996, ApJ, 462, L17 Tilvi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Malhotra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Rhoads, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2020, ApJL, 891, L10 Tinker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Weinberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & and Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2005, MNRAS, 368, 85 Tinker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007, MNRAS, 374, 477 Urrutia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Wisotzki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Kerutt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2019, A&A, 624, 24 Van Den Bosch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', More, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Cacciato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2013, MNRAS, 430, 725 Walter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Decarli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Carilli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2012, ApJ, 752, 93 Wechsler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Tinker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, Annual Review of Astronomy and Astro- physics, 56, 435 Yoshioka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Kashikawa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Inoue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2022, ApJ, 927, 32 Yajima, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Sugimura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', & Hasegawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2018, MNRAS, 477, 5406 Zheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', Coil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' & Zehavi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2007, ApJ, 667, 760-779 Appendices A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Effect of different fields on the clustering measurements In this work, we have analyzed the clustering of LAEs in the full MUSE-Wide sample, including the CANDELS/COSMOS fields and the HUDF parallel fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Here, we explore the pos- sible effects on the MUSE-Wide clustering results when includ- ing or excluding various sets of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In appendix A of Her- rero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), we showed that the HUDF parallel fields did not alter the clustering results, their exclusion or inclu- sion mainly affected the clustering uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We therefore explore the effect of including the CANDELS/COSMOS region by comparing the clustering of the full MUSE-Wide survey with that present in a subsample without the CANDELS/COSMOS fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The number of LAEs in the CANDELS/COSMOS region is 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' It is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 that the clustering in both samples is in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The large-scales bias factors derived from the two curves are indistinguishable (within 1σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The uncertain- ties corresponding to the smaller sample are (on average) 20% larger than in the full MUSE-Wide sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We conclude that the inclusion of these fields has no notable effect on our clustering results but helps in reducing cosmic sample variance uncertain- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Covariance matrix A common approach to quantify the correlation of the clustering data points is to resample the set of galaxies with the jackknife technique, followed by the calculation of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' To apply the jackknife method, we find a compromise between the number and the size of the jackknife zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, we split the sky area into ten independent regions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1) with a spatial extent of ≈ 4 h−1Mpc in both RA and Dec directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We then construct ten different subsamples, each of them excluding one jackknife zone, and compute the K-estimator in each subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These measurements are then used to build up the covariance matrix using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1: Clustering of the LAEs in the full MUSE-Wide sample (blue, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1) and without the CANDELS/COSMOS fields (red, see right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The black baseline represents the expected clustering of an unclustered sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The error bars are Poissonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The red measurements have been shifted along the x-axis for visual purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Considering that the probability of one galaxy pair to con- tribute to various adjacent bins is higher than that to contribute to several distant bins, one would naively expect a higher corre- lation in the former case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This is indeed what the (normalized) covariance matrix reflects in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In fact, the noise in the matrix elements corresponding to notably sepa- rate bins is substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2, we plot the normalized matrix elements as a function of bin i for each bin j to better illustrate the high level of noise in the matrix, especially for bins i > 6, where most curves become negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This is likely due to the limited spatial size of the survey, which does not allow neither for a higher number of jackknife zones nor for spatially larger zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' As a result of the considerable noise in the matrix on account of barely correlated bins significantly apart from each other, the minimization of the χ2 values (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 2) including the full covari- ance fails (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', various χ2 values become negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We there- fore limit the use of the covariance matrix to its main diagonal and two adjacent diagonals (see red section in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' our so-called reduced covariance matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This means we set the negative part of the curves in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 to zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', no correlation between those bins), in an attempt to smooth out the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While incorporating more diagonals re- sults mathematically problematic for the χ2 minimization, we have verified that the number of adjacent diagonals (one or two) slightly modifies the χ2 values but the probability contours rep- resented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 6 remain unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Thus, so do the best-fit HOD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 15 of 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='45 Full sample Without COSMOS fields 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='35 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rij [h-1Mpc]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1: Ten Jackknife zones in the spatial coverage of the full MUSE-Wide survey (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='52 arcmin2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Each Jackknife zone has a spatial extent of ≈ 4 h−1Mpc in both RA and Dec directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2: Covariance matrix computed from ten independent K-estimator measurements from the jackknife resampling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Left: Normalized covariance matrix for bins i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The red region defines the main diagonal and the two adjacent diagonals used for our reduced covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Right: Normalized covariance matrix elements as a function of bin i for each bin j (colored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1: Error estimation method comparison for the sample of LAEs in the MUSE-Wide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Uncertainties from the covari- ance matrix and the jackknife resampling technique described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 are colored in blue, those from the bootstrapping ap- proach used in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) in red, and Poisson uncertainties in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Despite current limitations, jackknife is still the most robust method to compute the K-estimator uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' While galaxy bootstrapping or Poisson error bars do not account for bin to bin correlations, our reduced covariance matrix only neglects the correlation between bins remarkably separated (expected to be minimal), but accounts for the correlation between nearby bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Error estimation comparison In order to quantify the correlation between the K-estimator bins, the covariance matrix must be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' By splitting the sky area into independent regions, following the jackknife re- sampling technique, we create as many subsamples from the MUSE-Wide sample as jackknife zones (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The K- estimator is then computed in each subset and the measurements are used to quantify the covariance matrix, whose diagonal pro- vides the variance of each clustering data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The square root of the diagonal represents the 1σ uncertainties and are repre- sented in blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 (same along the main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The jackknife resampling method requires a division of the sky area into several independent regions, each of which should ideally be large enough to cover the full range of scales under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Out of the three samples examined in this study, this can only be partially achieved in the MUSE-Wide dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 16 of 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} 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Poisson Bootstrapping 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='20 II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rij [h-1Mpc]Yohana Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' : Strong clustering dependence on Lyα luminosity at 3 < z < 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1: Effect of HOD parameters on the shape of the K-estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Left: Dependence on log(Mmin) for fixed log(M1/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Middle: Dependence on log(M1/Mmin) for fixed log(Mmin/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Right: Dependence on α for fixed log(Mmin/[h−1M⊙]) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 and log(M1/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' MUSE-Deep and MXDF do not allow for a spatial split into in- dependent zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We are thus left with two options for the deeper samples: the bootstrapping technique applied in Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021), shown in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1, and Poisson uncertainties, shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We find that Poisson (bootstrapping) errors are, on average, 7% (46%) larger than those computed with the jackknife tech- nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' These findings corroborate the results from Norberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2009), who found that the bootstrapping approach overesti- mates the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Similarly as for the MUSE-Wide survey, we find that boot- strapping uncertainties are ≈ 40% (on average) larger than Pois- son in both MUSE-Deep and MXDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We thus decide to use Pois- son errors for the deeper samples in an attempt to least overvalue the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We verified that the error estimation method does not sig- nificantly affect our clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The best-fit parameters from MUSE-Deep and MXDF using bootstrapping error bars and the χ2 minimization described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='3 of Herrero Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' (2021) are consistent with those delivered from Poisson statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Although in agreement, bootstrapping deliv- ers ≈ 45% larger uncertainties than Poisson for the best-fit HOD parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We last perform the same experiment in MUSE-Deep and MXDF but using scaled Poisson error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' We decreased the Poisson in 7% (excess found in MUSE-Wide) and find that the best-fit parameters are ≈ 10% less uncertain than if Poisson er- rors are directly applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Dependence of HOD parameters on the shape of the K-estimator Here we visualize and qualitatively describe the effect of the HOD parameters on the K-estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 shows the K- estimator for numerous HOD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Each panel represents the result of varying one HOD parameter with the other two parame- ters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Before detailing the major effects, it should be pointed out that the exact change in the shape of the K-estimator does not only depend on the varied parameter but also on the specific choice of the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Hence, these panels should merely be seen as illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 shows the dependence of the K- estimator on Mmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Higher values of log Mmin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', more massive halos) raise the expected K-estimator at all Rij scales (one- and two- halo terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' At large scales, this occurs because more mas- sive halos present larger bias factors, whereas at small scales, this is due to the decline in the contribution from less massive DMHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 shows the dependence of the K- estimator on M1/Mmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Larger log(M1/Mmin) values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=', more massive halos) reduce the one-halo term clustering amplitude because of the decrease in the contribution from less massive DMHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The clustering in the two-halo term does not depend on M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 shows the dependence of the K- estimator on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Higher values of α increase the fraction of galax- ies in massive DMHs with respect to smaller mass DMHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Given that more massive halos are more strongly biased, the ampli- tude of the two-halo term increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' The change observed in the one-halo term is explained because galaxies hosted by massive DMHs can contribute to the one-halo term on its largest scales, while galaxies residing in less massive halos can only contribute to the one-halo term at smaller Ri j scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Since α modifies the fraction of galaxies in massive DMHs to less mass DMHs, the corresponding fraction of the clustering contribution also varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' This alters the slope of the one-halo term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content=' Article number, page 17 of 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 log(Mmin) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='l log(Mmin) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 log(Mmin) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='9 5 log(Mmin) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 4 log(Mmin) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rij [h-1Mpc]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 log(Mi/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 log(M1/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 log(Mi/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 log(Mi/Mmin) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 log(M1/Mmin) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rij [h-1Mpc]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 α=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 α= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 α= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 5 74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='6 α= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='8 07 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} +page_content='0 Rii [h-1Mpc]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQfzwgf/content/2301.04133v1.pdf'} diff --git a/ndE3T4oBgHgl3EQfiwq0/content/tmp_files/2301.04583v1.pdf.txt b/ndE3T4oBgHgl3EQfiwq0/content/tmp_files/2301.04583v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6a85c0e4de26702f88cd4ab0147af1b5116fd87 --- /dev/null +++ b/ndE3T4oBgHgl3EQfiwq0/content/tmp_files/2301.04583v1.pdf.txt @@ -0,0 +1,1432 @@ +Magneto optics of a doubly charged quantum dot - Observation of a negative +diamagnetic shift +Giora Peniakov,1, 2, ∗ Ayal Beck,1, ∗ Eilon Poem,3 Zu-En Su,1 Boaz Taitler,1 and David Gershoni1, † +1The Physics Department and the Solid State Institute, +Technion–Israel Institute of Technology, 3200003 Haifa, Israel +2Technische Physik, Physikalisches Institï¿œt and Wilhelm Conrad Rï¿œntgen-Center for Complex Material Systems, +Universitat Wï¿œrzburg, Am Hubland, D-97074 Wï¿œzburg, Germany +3Department of Physics of Complex Systems, Weizmann Institute of Science, 7610001 Rehovot, Israel +(Dated: January 12, 2023) +We present magneto-optical studies of a self-assembled semiconductor quantum dot, concentrating +specifically on the case in which the dot is doubly positively charged, studying this way the confined +hole - hole exchange interaction. A simple harmonic potential model, which we extend to capture +the influence of an externally applied magnetic field in Faraday configuration fully describe the +observed polarization sensitive magneto-photoluminscence spectra. We deduce the effective compo- +sition of the quantum dot from its measured electronic g-factor. Using this value we determine the +dot effective permittivity and quantitatively describe various measured excitonic transitions, their +measured Zeeman splittings and their diamagnetic shifts. In particular, the model quantitatively +accounts for an observed pronounced negative diamagnetic shift, which provides a direct measure +for the hole-hole exchange interaction and its dependence on the externally applied magnetic field +strength. +I. +INTRODUCTION +Self-assembled Quantum Dots (QDs) in semiconduc- +tors form a well-known platform for quantum technolo- +gies. +They have proven to be the best contemporary +single-photon sources [1–4], while providing an excellent +interface between anchored spin qubits and “flying” pho- +ton qubits. Much progress has been made in controlling +confined-spin qubits in QDs [5–8] and entangling them +with photons Sen[9–14] , enabling deterministic genera- +tion of long strings of entangled photons [15–17]. +In +addition, QDs still provide a convenient platform for +studying the many-body states of confined many carri- +ers complexes. Interesting properties of such complexes +include the relative interactions between the consisting +particles, the form of their spatial wavefunctions, and +their response to externally applied fields, to name a few. +In particular, an externally applied magnetic field causes +the associated optical transitions to energetically shift - +an effect known as the diamagnetic shift. Modeling those +shifts in confined systems is still an ongoing effort [18–22]. +Here we present a magneto-optical study of semicon- +ductor QDs in which we focus our attention on the optical +properties of a doubly positively charged QD and its fun- +damental excitonic transition denoted by X+2. The QD +confined X+2 exciton contains three heavy holes and an +electron. After radiative recombination of an electron- +hole pair, the QD remains with two heavy-holes. The +pairs of holes may form either three spin triplet states or +one spin singlet state. Our work was spurred by notic- +ing an anomaly in the diamagnetic shifts of the opti- +cal transitions into the singlet state, the X+2 +S0 , which we +∗ These authors contributed to this work equally. +† giora.peniakov@physik.uni-wuerzburg.de +found to be negative. In the effort to understand this +phenomenon, we found that the X+2 excitonic transi- +tions form an excellent platform for studying the hole- +hole Coulomb exchange interaction and its dependence +on externally applied magnetic field. We show, using a +simple harmonic model for the QD spatial potential, that +the measured diamagnetic shifts of the X+2 transitions +and the measured electron and hole g-factors can be ob- +tained using one free fitting parameter which describes +the effective composition of the QD. +The paper is organized as follows. First, we present +our measurements of the electron and hole g-factors by +measuring the spectral Zeeman splittings of the bright +and dark neutral excitons. From the values of the mea- +sured g-factors, we extract the composition of the QD +as captured by the parameter x, defining the ratio of In- +dium and Gallium in the QD, InxGa1−xAs. +Next, we +present full polarization-sensitive magneto-PL measure- +ments displaying the diamagnetic shifts of various opti- +cal transitions and their optical transition selection rules. +We then concentrate on the anomalous negative shift of +the doubly positively charged exciton, the X+2 +S0 , and fit +its optical transition field dependence using no additional +free parameters. Finally, we use a Hartree-Fock approx- +imation to calculate the absolute values of the X+2 dia- +magnetic shifts in terms of the diamagnetic shift of the +bright exciton, X0 +BE. +II. +EXPERIMENTAL SYSTEM +We studied a single InxGa1−xAs self-assembled QD +embedded in a planar microcavity grown along the [001] +direction [23]. We used an Attocube closed-cycle cryo- +stat to cool the sample down to 4 Kelvin. A built-in vec- +tor magnet enabled us to apply a magnetic field in any +arXiv:2301.04583v1 [cond-mat.mes-hall] 11 Jan 2023 + +2 +desired direction. The emitted photoluminescence (PL) +was collected by a ×60 objective. Its polarization was +analyzed by pairs of liquid crystal variable retarders and +polarizing beam splitters, enabling PL polarization pro- +jecting on any direction in the Poincarï¿œ sphere. The +PL was then spectrally analyzed using an 80 cm double +monochromator, providing a spectral resolution of ∼ 20 +µeV . +The QD was optically excited using an above band-gap +CW red HeNe or a blue diode laser, emitting at 633 or +445 nm, respectively. The excitation colors induce the +average charge state of the QD. While HeNe illumina- +tion results in positive charging blue excitation leads to +negative charging [24]. +We defined the symmetry axis of the QD and the op- +tical beam direction as the z-axis of the experimental +system. The x and y axes were defined along the polar- +ization eigenstates of the QD’s bright exciton, X0 +BE. The +X0 +BE is an electron-hole pair which can be expressed in +the spin basis {|+z⟩ = |⇑↓⟩ , |−z⟩ = |⇓↑⟩} with ⇑\⇓ and +↑\↓ denoting the spin projections of the heavy-hole and +electron onto the z-axis. Since a heavy-hole and an elec- +tron have total angular momenta of 3/2 and 1/2, respec- +tively, the angular momentum projection of a |⇑↓⟩ (|⇓↑⟩) +pair along this axis is +1 (-1) [25]. Consequently, op- +tical recombination of the |⇑↓⟩ and |⇓↑⟩ pairs results in +a right-handed (R) and left-handed (L) circularly po- +larized photon emission, respectively. +The anisotropic +electron-hole exchange interaction in this QD lifts the +degeneracy of the above basis by δ1 ≈ 30µeV [26] thus +forming new eigenstates, +√ +2 |±x⟩ = |⇑↓⟩ ± |⇓↑⟩. Recom- +bination of those excitonic eigenstates results in either +horizontal, +√ +2H = R + L, or vertical +√ +2V = R − L +rectilinear photon emission, enabling a one-to-one corre- +spondence between the X0 +BE’s two-level system and the +two-dimensional space of light polarization. +The dark exciton (X0 +DE) is another electron-hole state, +but with parallel spins +√ +2 |ψDE⟩1,2 = |⇑↑⟩ ± |⇓↓⟩ . In +general, this state is optically inactive because the an- +gular momentum is not conserved upon recombination +[8]. However, small optical activity of the X0 +DE was mea- +sured [27–29]. Zielinski et al. explained it by small mix- +ing of the X0 +DE and X0 +BE eigenstates [28], which can be +enhanced by applying an in-plane magnetic field perpen- +dicular to the QD optical axis (Voigt configuration). For +a magnetic field parallel to the symmetry axis (Faraday +configuration) no additional mixing occurs, and the X0 +DE +barely emits [30]. +III. +RESULTS +A. +Measuring single-carrier g-factors +The Zeeman interaction between an externally applied +magnetic field and QD confined carriers’ spin removes +the Kramers’ degeneracy between the confined carriers +spin state which is parallel and anti-parallel to the field +direction. The Zeeman interaction linearly depends on +the magnetic field magnitude. This dependence is most +generally expressed in terms of a 3 × 3 g-factor tensor +[31]. For simplicity we assume here that this tensor is +diagonal and have only two different components: along +the symmetry axis (gz +e and gz +h) and perpendicular to it +(g⊥ +e and g⊥ +h ) [32]. +In the first part of the experiment, we measured the +confined electron and hole g-factors tensor components +along the z-axis. This was done by measuring the Zeeman +splitting of the X0 +BE and X0 +DE under B-field in the ˆz +direction. Assuming that the absolute magnitude of the +g-factors of those transitions are given by the sum and +difference of the absolute magnitudes of the single-carrier +g-factors +|gz +BE(DE)| = |gz +e| ± |gz +h| +(1) +[33, 34], we were able to extract gz +e and gz +h from the mea- +sured gz +BE and gz +DE [30, 32, 35]. Eq. 1 is derived from +the parallel and anti-parallel spin nature of the dark and +bright excitons, using the sign convention given by the +Zeeman Hamiltonian +H = −µBgz +eBzSz + 1 +3µBgz +hBzJz . +(2) +Here, µB is the Bohr magneton, Sz and Jz are the angu- +lar momentum z-projections ± 1 +2 and ± 3 +2, and Bz is the +magnetic field. We readily measured the Zeeman split- +ting of the X0 +BE since its spectral doublet appears bright +in the PL, as shown in figure 1. In contrary, the low op- +tical activity of the X0 +DE made its Zeeman splitting mea- +surement more challenging. To overcome this problem, +we added a 1.5T Voigt component to our measurement +(in-plane B-field), enhancing the X0 +DE emission to a mea- +surable amount. Although the total B-field direction was +no longer in the ˆz direction, we found that the in-plane +field effect on the measured ˆz direction Zeeman splitting +could be safely neglected. We veified it by measuring the +influence of the in-plane field on the bright exciton split- +ting (X0 +BE), and found that it stayed unaffected within +our experimental precision. +One can also notice in Figure 1 that the X0 +DE cross- +polarized doublet is not equally intense: at 0 Tesla, its +horizontally (H) polarized component is much stronger +than the vertically (V) polarized one, a phenomenon ob- +served and explained in previous publications [36–38]. +Adding magnetic field in Faraday configuration enhances +the weaker component and gradually adds cross-circular +polarization terms to the X0 +DE doublet. However, up to +the maximal field strength of 1.5 Tesla, the two X0 +DE’s +components remain unequal. Nonetheless, we extracted +the g-factors of the X0 +BE and X0 +DE by fitting their mea- +sured Zeeman splittings to the following expression: +∆EBE(DE) = +� +δ2 +1,2 + (µBgBE(DE)B)2 +(3) +, where δ1,2 are the fine-structure splittings of the X0 +BE +and X0 +DE at 0 Tesla, respectively. We summarize the val- + +3 +Normalized Intensity +Split [meV] +𝐵� = 0𝑇 +𝐵� = 0.4𝑇 +𝐵� = 0.8𝑇 +𝐵� = 1.2𝑇 +𝑋�� +� +𝑋�� +� +Figure 1. Polarization-sensitive magneto-PL of the bright ex- +citon (X0 +BE) and dark exciton (X0 +DE), for various magnetic +field strengths in the ˆz direction. A constant, ˆx-directional +magnetic field of 1.5T was applied during all the measure- +ments to allow the X0 +DE optical transition. Inset: Zeeman +splitting of the X0 +BE and X0 +DE versus Bz-field . The g-factors +of the two transitions are extracted by fitting the measured +splitting with ∆E = +� +δ2 + (µBgB)2. +Spectral line +δ[µeV ] +gz-factor +α[ µeV +T 2 ] +X0 +BE +31.0 ± 1.8 −0.81 ± 0.01 8.44 ± 0.14 +X0 +DE +1.4∗ ± 0.1 −0.29 ± 0.02 +7.0 ± 1.4 +gz +e +−0.55 ± 0.02 +gz +h +−0.26 ± 0.02 +∆0 [µeV ] +270 ± 10 +Table I. Summary of the measured excitonic fine structure and +Zeeman parameters. δ is the natural splitting at B = 0, and +α is the diamagnetic shift coefficient capturing the quadratic +dependence of the energy in B (αB2). gz +(e/h) is the measured +g-factor of the electron and hole in Faraday configuration, re- +spectively. +∗The dark exciton (X0 +DE) splitting is too small +to be directly observed in PL measurement, but can be mea- +sured using time-resolved spectroscopy [8]. For reference, we +also added the X0 +DE − X0 +BE splitting denoted by ∆0. +ues of the measured excitonic and single-carrier g-factors +in table I. +B. +Estimating the QD effective composition from +the measured electronic g-factor +The isotropic electronic g-factor in bulk semiconduc- +tors can be analytically calculated by the Roth’s formula +[39]: +ge = 2 − 2 +3 +Ep∆ +Eg(Eg + ∆) +(4) +where Eg is the band gap energy between the valence +and conduction bands, ∆ is the split-off gap (between +the valence band and the spin-orbit band) at k = 0, and +Ep is the Kane energy defined as Ep ≡ 2ℏ2 +m |⟨s| ∂x |x⟩|2 , +where |s⟩ and |x⟩ are the crystal Bloch functions of the +electron in the conduction band and in one of the three +p−like degenerate valence band, respectively. +QDs are not bulk semiconductors, and applying Roth’s +formula to them requires adjustment [40]. In principle, +the confinement effect of the QD breaks the periodicity of +the electronic wavefunctions, and the derivation of Roth’s +formula collapses. Nevertheless, as long as the confine- +ment energy is much smaller than the parameters ∆, Ep +and Eg, we expect Roth’s formula to be a good approx- +imation. Indeed, a typical separation between confine- +ment energy levels in our QD is of order 10 − 30 meV s +[41], much smaller than ∆, Ep and Eg (see table II). +Since our QD comprises two semiconductors, GaAs +and InAs, we averaged the values of ∆ and Ep over the +two bulk materials. We introduced a weight parameter +x to quantify the composition of the QD, InxGa1−xAs, +and define: +∆x = x∆In+(1−x)∆Ga, +Ex +p = xEp(In)+(1−x)Ep(Ga) +For the band gap, Eg, we preferred to use the directly +measured value of the X0 +BE emission, as it takes into +account the confinement and lattice mismatch strain ef- +fects, omitted in the simple average [42]. We defined a +corrected band gap � +Eg by adding ∼ 50meV to the X0 +BE +emission energy, thus accounting for the binding energy +of the exciton [25]. Combining all three parameters, we +obtained an x-dependent Roth formula +ge(x) = 2 − 2 +3 +Ex +p∆x +� +Eg(� +Eg + ∆x) +(5) +that we can fit to the electronic g-factor as measured in +the experiment. Fitting ge(x) to the measured value of +−0.55 yields x ≈ 0.75. +We summarize the parameter +values of that calculation in table II. +C. +Measuring a negative diamagnetic shift for X+2 +In Figure 2, we present a full Magneto-PL measure- +ment in Faraday configuration of the various spectral +lines of our QD. We present it for two average charge +states of the QD: negative and positive. The charge state +is apparent in each case by considering the emission ratio +between the positive and negative trions, X+ and X−. +Many identified lines are marked in the PL following pre- +vious studies [24]. + +BZEnergy +mey4 +Figure 2. Polarization-sensitive magneto-PL spectra in Faraday configuration for various magnetic field strengths, for negatively +(a) and positively (b) charged QD. The upper panel shows polarization-sensitive magneto-PL spectra (except at B = 0T, where +the rectilinear polarization is shown). The panels below show the degree of circular and rectilinear polarizations (given by the +color bars to the right) as a function of the photon energy and the externally applied magnetic field strength. The identified +spectral lines are marked: X0 - the exciton, XX0 - the biexciton, X+ - (X−-) positively (negatively) charged trion, XX0 +T0(T3) +metastable biexcitons with the two holes in T0(T3) spin Triplet configurations. X+ +T0(T3), and XX+ +T0(T3) are similar positively +charged excitons and biexcitons. The X+2 lines result from the recombination of the doubly positively charged exciton, leaving +behind two holes which can form either a singlet S0 or one of the triplets, T±3 or T0. Note the negative diamagnetic shift of +the X+2 +S0 (marked with an oval dash line). The energy scale is relative to the X0 +BE spectral line at zero magnetic field. +GaAs +InAs In0.75Ga0.25As +Ep[eV ] +28.8 +22.2 +23.85 +Eg(4K)[eV ] +1.519 0.418 +1.334* +∆[eV ] +0.341 0.371 +0.363 +ge Calculated -0.317 -14.65 +−0.55 +ge Measured -0.484 -14.9 +−0.55 +Table II. Comparison between measured electronic g-factors +to calculated values using Roth’s formula. The measured val- +ues for bulk GaAs and InAs semiconductors are taken from +Ref. [43]. The parameters Ep and ∆ for In0.75Ga0.25As are +weighted averages of their values in GaAs and InAs. +∗The +corrected band gap � +Eg (see text). +On top of the Zeeman splitting of the spectral lines, +they undergo a quadratic-in-B diamagnetic shift, which +we characterize by the coefficient α in the term αB2 +added to the Hamiltonian (Eq. 2). For each spectral line +in Figure 2, the shift is attributed to its spectral “center +of mass” (the spectral center of the doublet), which in +most cases shifts towards higher energy (hence the ter- +minology of “diamagnetic” versus “paramagnetic” shift). +We explain this tendency by considering the areas of the +initial and final states of each optical transition. It is a +well-known result for quantum wells that the diamagnetic +shift of a neutral exciton is proportional to its wavefunc- +tion area [43] in a plane which is normal to the direction +of the magnetic field: +α = +e2 +8µ∥c2 ⟨f|ˆρ2|f⟩ = π +4 +e2 +µ∥c2 +� +f 2 (ρ) ρ3dρ +(6) +Here, ρ is the relative in-plane coordinate between the +electron and hole, f (ρ) is the excitonic envelope wave- +function, µ∥ = memh/ (me + mh) is the in-plane reduced +mass of the electron and hole, and e and c are the elec- +tron charge and the speed of light, respectively. The final +state of the QD after the excitonic recombination is just +the vacuum, possessing no magnetic dependence, so the +overall diamagnetic shift is positive. Extending the area +interpretation to other optical transitions, it seems that +in most cases the radiative recombination results in a fi- +nal configuration with a reduced area. As a result, most +lines follow positive diamagnetic shift. +Quantitatively, +plugging in Eq. +6 mh = 0.25m0, me = 0.065m0 (the +effective masses of the hole and electron in the quantum +dot [44] with m0 the free electron mass), and the mea- +sured diamagnetic coefficient α = 8.44 ± 0.14 µeV/T 2, +one calculates the exciton Bohr radius to be ∼ 3.7 nm – +a compatible result with the ∼ 30 nm estimated diameter +of our QD. +Figure 3 summarizes the diamagnetic shifts of several +selected lines. One can see that many lines, including +the X0 +BE, XX0 and the trions, X− and X+, exhibit very +similar diamagnetic shifts of ∼ 8µeV/T 2. We explain this + +(a) +(b) +nsity +5T +4T +3T +2T +1T +Xx +XX% +XX3 +X+2* +Bz =:OT +0.6 +H +0.2 +5 +-0.2 +V +-0.6 +R +0.5 +3 +0 +2 +-0.5 L +.6 +-5 +-4 +-3 +-2 +0 +-6 +-3 +2 +0 +Energy [meV] +Energy [meV]5 +0 +5 +10 +15 +20 +25 +2 +2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +X0 +XX0 +X+ +X- +XX0 +T3 +XX0 +T0 +XX+ +T0 +X+ +T0 +XX+ +T3 +X+ +T3 +XX- +T1 +X- +T1 +XX- +T0 +X- +T0 +X+2 +T0 +X+2 +T3 +X+2 +S0 +Figure 3. Measured energy shifts of various optical transitions +as a function of B2. One spectral line is a prominent exception +- the X+2 +S0 +similarity by arguing that in all those transitions both +the initial and final states contain only charge carriers +occupying the QD ground-level. We observe that when +additional charge carriers occupy higher confined levels, +the diamagnetic shift coefficients change (see for example +X− +T ±1, X+ +T ±3). +Interestingly enough, one prominent line that we at- +tribute to the doubly charged exciton transition X+2 +S0 ex- +hibits a distinctive negative shift. In what follows, we +will try to explain this observation in terms of the ex- +change interaction between the two heavy holes of the +X+2 transitions’ final states. Let us start by describing +those transitions in detail. +D. +The doubly charged exciton X+2 +Figure 4 schematically describes the energy levels +and the optical transitions associated with the doubly +charged exciton, X+2. This exciton comprises one elec- +tron in the ground-level 1e1, and three holes: two of them +forming a singlet in the s-orbital ground-level, 1h2, and +the third one occupies the first excited p-level 2h1. Here, +npm means: n - the energy level order, p - the particle +type (e or h), and m - the number of particles occupying +this level (either 1 or 2). The exchange interaction be- +tween the unpaired electron (in the 1st level) and hole (in +the 2nd level) removes the degeneracy between the four +possible two-carriers’ spin configurations, forming four +distinct eigenstates similar to the case of the neutral ex- +citon (X0). As such, we borrow the exciton “bright” and +“dark” terminology to describe the eigenstates of the X+2 +as well. States with anti-parallel e-h spins would be called +“bright-like”, while states with parallel spins - “dark-like” +(see Figure 4). We emphasize that the dark and bright +𝛿� +���� +𝛿� +���� +𝛿� +���� +𝑋"������"� +�� +𝑋"����"± +�� +𝑆� +𝑇� +𝑇±� +V +H +H +V +R +L +𝑋"������"� +�� +𝑬 +𝑋�� +�� +𝑋�±� +�� +𝑋�� +�� +𝚫𝑬 +𝛿� +���� +Figure 4. Schematic description of the energy levels and opti- +cal transitions associated with the doubly positively charged +exciton X+2. +The configuration of each state is presented +on the left, where thin blue arrows represent electrons with +spin +1 +2, and thick arrows represent heavy-holes with spin +3 +2. +The polarization selection rules are marked by colored down- +ward arrows. H (V ) marks the horizontal (vertical) rectilinear +polarization, while R (L) marks right (left) circular polariza- +tion. A schematic description of the emitted PL is drawn at +the bottom. The X+2 +T±3 spectral line is drawn in green with a +pink edge, symbolizing that the H and V polarizations over- +lap such that the emission is unpolarized. +states are both optically active since the optical recombi- +nation occurs between the unpaired s−electron and one +of the s−level singlet holes, rather than the unpaired +p−hole. +The final states of the X+2 recombination contain two +holes - one in the ground level and one in the first ex- +cited level. As identical particles, they form either one +singlet spin state denoted by S1h2h +0 +or three triplet states +denoted by +� +T 1h2h +0 +, T 1h2h +±3 +� +, respectively. The two initial +bright-like exciton states can only recombine to the sin- +glet S1h2h +0 +or triplet T 1h2h +0 +final states (but not to the +T 1h2h +±3 +), resulting in two pairs of cross-rectilinearly polar- +ized doublets [45]; the dark-like states can only recom- +bine to the T 1h2h +±3 +states. Since in the absence of external +magnetic field (B = 0) both the dark-like and the T 1h2h +±3 +states are almost degenerate, the recombination results +in a single, unpolarized, strong spectral line. We label the +X+2 optical transitions by their final states, specified by +the subscripts: X+2 +T0 , X+2 +T±3 and X+2 +S0 . The latter transi- +tion, X+2 +S0 , is the one exhibiting a negative diamagnetic +shift. We note that in the absence of external field, the +unpolarized X+2 +T±3 spectral line is positioned exactly in be- +tween the two cross linearly polarized components of the +X+2 +T0 line. This indicates that δ1e2h +0 +, denoting the split- + +X土 +4土 +. +1← +土6 +Spectral line δ1[µeV ] +g-factor +α[ µeV +T 2 ] +Model +X0 +BE +31.0 ± 1.8 −0.81 ± 0.01 8.44 ± 0.14 gz +1e + gz +1h +X+2 +T3 +0 +−0.97 ± 0.05 6.90 ± 0.25 +X+2 +T0 +56.3 ± 2.3 +0.55 ± 0.13 +7.6 ± 0.5 +gz +1e + gz +2h +X+2 +S0 +69.0 ± 6.8 +0.65 ± 0.08 +−5.8 ± 0.7 +X0 +DE +0 +−0.29 ± 0.02 +7.0 ± 1.4 +gz +1e − gz +1h +Table III. Summary of the measured g-factors for the X+2 +transitions, compared to those of the bright and dark excitons. +The lines are classified by a simple model which assumes that +the g-factor of a given transition can be decomposed to the +sum of the comprising charge carrier g-factors of the initial +and final states of that transition. gz +n(e/h) denotes the g-factor +of the electron (hole) in the n energy level of the QD, where +n = 1 is the ground state. +ting between the dark-like and bright-like X+2 states, +is equal to δ1h2h +T +, the splitting between the holes’ triplet +states T 1h2h +0 +and T 1h2h +±3 +. The reason why these two terms, +one due to isotropic e-h exchange and the other due to +h-h anisotropic exchange, are almost equal, remains an +open question. +The measured diamagnetic shifts and g-factors of the +X+2 transitions are summarized in table III. To qualita- +tively explain the measured g-factors, we assume that a +g-factor of a state can be deduced by summing up its indi- +vidual single carrier component’s g-factors, and that the +total g-factor of a transition results from the difference +between its initial and final states. By further assuming +that charge carriers in the well-defined symmetry config- +urations S0 and T0 do not exhibit Zeeman splitting, we +conclude that X+2 +T3 ’s g-factor behaves as the bright ex- +citation’s (X0 +BE) factor, while the g-factors of X+2 +T0 and +X+2 +S0 depend on the excited hole’s g-factor, gz +2h. More +measurements justifying the above classification as a gen- +eral result can be found in the authors’ theses [46, 47]. +It is interesting to note that plugging into the gz +1e + gz +2h +sum the measured gz +1e-factor (∼ −0.55), and using for +the sum an averaged value of 0.6±0.1 (see table III), one +finds that gz +2h = 1.15±0.10. This value is opposite in sign +compared to the ground state g-factors of the hole and +the electron (−0.26 and −0.55, respectively, according to +Table I). +A detailed polarization-sensitive magneto-PL spectra +of the X+2 spectral lines are presented in figure 5. One +can see that while the triplet lines shift towards higher +energy with increasing B-field, the singlet lines shift to- +wards lower energy. Since the initial states of the X+2 +S0 +and X+2 +T0 transitions are the same (the bright-like exci- +ton states), we conclude that the difference in the sign of +the diamagnetic shift between the two transitions stems +Figure 5. Rectilinear polarization-sensitive PL spectra of the +X+2 spectral lines relative to the neutral exciton state a) at +zero magnetic field, b) as function of the externally applied +field in Faraday configuration, and c) in magnetic field of 5T. +The transitions are marked by their final spin configurations +(S0,T0, T±3). The energy difference between the X+2 +T0 and the +X+2 +S0 doublets (marked) equals twice the hole-hole exchange +interaction. +from the different influence that the external magnetic +field has on the final states. The h-h singlet final state +rises in energy faster than the initial state such that the +overall spectral shift is negative. On the other hand, the +h-h triplet state rises in energy slower than the initial +state, and thus the total spectral shift is positive. +E. +Magnetic field dependence of the exchange +integral +We now use a simple harmonic oscillator model to +quantitatively describe how the hole-hole exchange in- +teraction is affected by the magnetic field. The exchange +integrals between various states confined by a 2D har- +monic potential with circular symmetry were calculated +in Ref [48]. For two identical particles, one in an s-shell +and one in a p-shell the exchange energy is: +Ksp,0 = 1 +4 +�π +2 +e2 +4πϵ0ϵr +1 +l0 +(7) +where e is the electron charge, ϵ0 is the vacuum per- +mittivity and ϵr is the relative permittivity of the QD +material. The effective length l0 characterizes the extent +of the harmonic potential and is equal to: +l0 = +� +ℏ +mω0 +(8) +where m is the in-plane effective mass of the charge carri- +ers, in our case holes, and ω0 is the harmonic frequency of + +2000 ++2 +400 ++2 +±3 +1500 +300 ++2 +Bz = 5T +(c) +1000 +8 200 +100 +500 +0 +0 +5 +0.8 +2(Ksp,0 + βB2) +H +0.4 +(b) +2.5 +0 +Bz +V +-0.4 +2Ksp.0 +0 +-0.8 +500 +4000 +X+2 +400 +H +. +±3 +H +3000 +V +V +300 +(a +2000 +B, += OT +200 +1000 +100 +0 +0 +-6.4 +-6.2 +-6 +-5.8 +-0.4 +-0.2 +0 +Eenrgy [meV] +Eenrgy [meV]7 +the confining potential. To include the effect of the mag- +netic field, we replace ω0 with ω ≡ +� +ω2 +0 + e2B2z +4m2 , obtained +by adding a magnetic field hamiltonian to the harmonic +oscillator one and solving for the eigenenergies (harmonic +spectrum + Landau levels spectrum). The expression for +the effective length then becomes: +l = +l0 +� +1 + +� eB +2mω0 +�2�1/4 +(9) +Inserting Eq. 9 into Eq. 7 yields an expression for the +field dependence of the exchange energy: +Ksp(B) = Ksp,0 +� +1 + +� eB +2mω0 +�2�1/4 +(10) +Furthermore, mω0 can be expressed in terms of Ksp,0 by +using Eq. 7 and 8: +mω0 = 512 +e4 πℏϵ2 +0ϵ2 +r(Ksp,0)2 +(11) +Inserting this expression into Eq. 10, we obtain an ex- +pression for the field dependence of the hole-hole ex- +change energy +Ksp(B) = Ksp,0 +� +1 + +� +e5B +1024πℏϵ2 +0ϵ2r(Ksp,0)2 +�2�1/4 +(12) +When the magnetic energy is much smaller than the zero- +field exchange energy, we can expand this expression to +the first non-vanishing order in B: +Ksp(B) ≈ Ksp,0 + βB2 +(13) +where: +β(theory) = +e10 +222π2ℏ2ϵ4 +0ϵ4r(Ksp,0)3 +(14) +To compare this result to the measured diamagnetic +shift coefficients, we further extract Ksp,0 and ϵr from our +measurements: Ksp,0 equals half of the energy separation +between the X+2 +S0 and X+2 +T0 optical transitions at B = 0 +(see Figure 5). From the magneto-PL in Figure 5, we +obtain Ksp,0 = 2.79(1)meV . To estimate ϵr, we average +its value over the InAs and GaAs constituents of the QD, +ϵx +r = xϵx +r(In) + (1 − x)ϵx +r(Ga), like we did for the ∆ and +Ep parameters in Section III B. Using x ≈ 0.75, as we +found by fitting Roth formula to the measured g-factor, +we obtain ϵ0.75 +r +≈ 14.25. +Combining those values, we +conclude βtheory ≈ 6.6 µeV +T 2 . +In the experiment, β is directly measured as half the +difference between the diamagnetic shifts of the X+2 +T0 and +X+2 +S0 spectral lines. To see this, note that the initial states +of X+2 +T0 and X+2 +S0 transitions are the same and thus cancel +Figure 6. +The calculated electronic g-factor ge(x) and the +calculated relative diamagnetic shift β(x) between the X+2 +singlet and triplet transitions as function of the composition +ratio x. Note that the calculation of the g-factor uses constant +band gap energy as measured in the experiment, and thus at +x = 0, 1 it does not reproduce the values for pure InAs and +GaAs bulk materials. +out upon subtraction. The only contribution, then, is the +final hole-hole state which is either a singlet or a triplet. +We observe: +βmeasured = +αX+2 +T0 − αX+2 +S0 +2 += 7.6 µeV +T 2 − (−5.8 µeV +T 2 ) +2 += 6.7(4)µeV +T 2 +(15) +The calculated and the measured values of β agree up +to 0.1 µeV +T 2 , well within our experimental error. We see +this compliance as a strong validation of the hole-hole +exchange interaction model. For further conviction, we +test the sensitivity of our result to the value of x. This +is shown in Figure 6. One can see that the dependencies +g (x) and β (x) are close to linear, and that a deviation +of x by more than 0.02 would cause those parameters to +miss the measured values. +F. +Diamagnetic shifts of the singlet and triplet +lines +After explaining the relative diamagnetic shift between +the X+2 +S0 and X+2 +T0 transitions, we proceed by calculating +the absolute values of those shifts. We find that we can +figure those values using the measured emission energy +of the X+2 spectral lines relative to the neutral exciton +X0, and its diamagnetic shift coefficient. In those calcu- +lations, we use the Hartree-Fock approximation to evalu- +ate the energies of the optical transitions’ initial and final +many-body states. The energy of a many-body state is +calculated, within this approximation, by summing over +its individual-particle confining energies and the interac- +tions between all the particle pairs involved. +Under this approximation, the energy of the ground +state neutral exciton is: +EX0 = Ee +s + Eh +s − Jeh +ss +(16) + +(X) +calculatecg mesaurec0.25X) ( + calculatecmesatrec178 +where Ee(h) +s += +1 +2ℏωe(h) is the energy of the electron +(hole) in the s-level and Jeh +ss is the direct Coulomb in- +teraction between the electron and the hole. +This ex- +pression describes the exciton center-of-splitting (includ- +ing at B ̸= 0) as it does not include the e-h exchange +Coulomb interaction. In the following derivation, we use +Jp1p2 +n1n2 (Kp1p2 +n1n2) to describe the direct (exchange) Coulomb +interaction between orbitals n1 and n2 of the charge car- +riers p1 and p2, the latter representing either holes (h) or +electrons (e). +The emission energies of the X+2 +S0 +and X+2 +T0 +optical +transitions are given by the difference between the en- +ergies of the final and initial states (see figure 4). +Einital +X+2 +T0(S0) = Ee +s + 2Eh +s + Eh +p + Jhh +ss + 2Jhh +sp − 2Jeh +ss − Jeh +sp +Efinal +X+2 +T0(S0) = Eh +s + Eh +p + Jhh +sp ∓ Khh +sp +(17) +Note that the hole singlet Efinal +X+2 +S0 +is higher in energy than +the triplet Efinal +X+2 +T0 +(Khh +sp > 0) due to the different symme- +tries of the associated spatial wavefunctions upon particle +exchange. Thus, for the optical transition we have: +EX+2 +T0(S0) = Efinal +X+2 +T0(S0) − Einital +X+2 +T0(S0) +(18) +=Ee +s + Eh +s + Jhh +ss + Jhh +sp − 2Jeh +ss − Jeh +sp ± Khh +sp +We can cast this in terms of the Hartree-Fock neutral +exciton transition energy, EX0 = Ee +s + Eh +s − Jeh +ss , as: +EX+2 +T0(S0) = EX0 + Jhh +ss + Jhh +sp − Jeh +ss − Jeh +sp ± Khh +sp +(19) +All the elements in this equation can be expressed by the +effective length of the electron (le = +� +ℏ +meωe ) and hole +(lh = +� +ℏ +mhωh ), using Ref [48]. Summing them up gives +the following expression: +EX+2 +T0(S0) = EX0 + (a ± 1) Khh +sp +(20) +where we defined the constant a as +a ≡ 7 − 4 +� +2 +1 + γ2 +�1/2 +− (1 + 2γ2) +� +2 +1 + γ2 +�3/2 +(21) +and γ ≡ +le +lh is the ratio between the effective lengths. +As Khh +sp is the exchange energy between two holes, we +have already found in the previous section that it can be +expressed as Khh +sp (B) ≈ Ksp,0 + βB2. Using this, Eq. 20 +becomes: +EX+2 +T0(S0)(B) = EX0(B)+(a±1)Ksp,0 +β(a±1)B2 (22) +The ratio γ can not be determined directly, as we lack +the knowledge about the ratio between the effective in- +plane masses of the electron and hole, or the ratio be- +tween their related harmonic confinement frequencies ωe +and ωh. However, we can extract γ from a, since it is a +function of directly measured quantities. At zero mag- +netic field: +EX+2 +T0 = EX0 + (a + 1)Ksp,0 ⇒ a = +EX+2 +T0 − EX0 +Ksp,0 +− 1 +EX+2 +S0 = EX0 + (a − 1)Ksp,0 ⇒ a = +EX+2 +S0 − EX0 +Ksp,0 ++ 1 +(23) +.EX+2 +T0 − EX0 and EX+2 +S0 − EX0 are the energies of the +X+2 transitions relative to the neutral exciton, which at +zero magnetic field, according to figure 5, equal respec- +tively −0.38(1)meV and −5.95(1)meV . Using these val- +ues and the previously obtained Ksp,0, both expressions +in 23 yield a ≈ −1.13, which in turn implies γ ≈ 0.17. +Finally, we are ready to calculate the absolute diamag- +netic shifts of the hole-hole triplet and singlet lines. Ac- +cording to Eq. +22, the diamagnetic coefficients of the +X+2 transitions are: +αX+2 +T0(S0) = αX0 + β(a ± 1) +(24) +where αX0 = 8.4(2) µeV +T 2 +is the diamagnetic shift coef- +ficient of the exciton, and β = 6.7(4) µeV +T 2 +is the relative +X+2 diamagnetic shift calculated in the previous section. +We find: +αX+2 +T0 = αX0 − 0.13β = 7.5(2)µeV +T 2 +αX+2 +S0 = αX0 − 2.13β = −5.9(8)µeV +T 2 +(25) +which agree with our measured values 7.6(4) µeV +T 2 +and +−5.8(7) µeV +T 2 , respectively. +IV. +SUMMARY +We performed magneto - PL spectroscopy on a well- +characterized InxGa1−xAs/GaAs QD in Faraday config- +uration. +From the measurements we extracted the g- +factors and the diamagnetic shifts of many excitonic tran- +sitions. In particular, we observed an anomalous negative +diamagnetic shift of spectral lines resulting from the ra- +diative recombination of a doubly charged exciton (X+2 +S0 ). +Our results are explained using simple models for the Zee- +man interaction and for the measured diamagnetic shifts. +For both interactions we use one free parameter: x, the +effective relative Indium content of the ternary QD. We +use this parameter to linearly interpolate the QD elec- +tronic g-factor and permittivity, from those of its binary +components GaAs and InAs. +By analysis of the measured g-factors of various opti- +cal transitions we show that while the g-factors of the +electron in the first and second level have the same sign, +the g-factors of the hole in these levels are opposite in +sign. + +9 +We explain the difference between the diamagnetic +shifts of the optical transitions of the doubly positively +charged exciton which result in the remaining holes in a +singlet (X+2 +S0 ) and that in which they form a triplet (X+2 +T0 ) +using a simple circular harmonic potential model. The +model describes in analytical form the hole-hole exchange +interaction including the influence of the externally ap- +plied magnetic field. Finally, using the Hartree-Fock ap- +proximation we calculate the absolute diamagnetic shifts +of these spectral lines using the measured diamagnetic +shift of the neutral exciton (X0). +ACKNOWLEDGMENTS +We thank Dr. J. Tilchin, Professor E. Lifshitz, and +Professor E. L. Ivchenko for their help and valuable dis- +cussions. The support of the Israeli Science Foundation +(ISF), and that of the German Israeli Research Coop- +eration—DIP (DFG-FI947-6-1) are gratefully acknowl- +edged. +[1] E. Dekel, D. Gershoni, E. Ehrenfreund, J. M. Garcia, and +P. M. Petroff, Physical Review B 61, 11009 (2000). +[2] P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. +Petroff, L. Zhang, E. HU, and A. Imamoglu, Science 290, +2282 (2000). +[3] N. Tomm, A. Javadi, N. O. Antoniadis, D. Najer, M. C. +Loebl, A. R. Korsch, R. Schott, S. R. Valentin, A. D. +Wieck, A. Ludwig, and R. J. Warburton, Nature Nan- +otechnology 16, 399 (2021). +[4] C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Ya- +mamoto, Physical Review Letters 86, 1502 (2001). +[5] J. Berezovsky, M. H. Mikkelsen, N. G. Stoltz, L. A. Col- +dren, and D. D. Awschalom, Science 320, 349 (2008). +[6] D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto, Na- +ture 456, 218 (2008). +[7] Y. Kodriano, I. Schwartz, E. Poem, Y. Benny, R. Pres- +man, T. A. Truong, P. M. Petroff, and D. Gershoni, Phys- +ical Review B 85, 241304 (2012). +[8] I. +Schwartz, +E. +Schmidgall, +L. +Gantz, +D. +Cogan, +E. Bordo, Y. Don, M. Zielinski, and D. Gershoni, Physi- +cal Review X 5 (2015). +[9] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, +J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, +physica status solidi (b) 243, 3900 (2006). +[10] E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, +L. Childress, M. V. G. Dutt, A. S. SÞrensen, P. R. +Hemmer, A. S. Zibrov, and M. D. Lukin, Nature 466, +730 (2010). +[11] M. Ghali, K. Ohtani, Y. Ohno, and H. Ohno, Nature +Communications 3, 661 (2012). +[12] W. B. Gao, P. Fallahi, E. Togan, J. Miguel-Sanchez, and +A. Imamoglu, Nature 491, 426 (2012). +[13] K. De Greve, L. Yu, P. L. McMahon, J. S. Pelc, C. M. +Natarajan, N. Y. Kim, E. Abe, S. Maier, C. Schneider, +M. Kamp, S. Hoefling, R. H. Hadfield, A. Forchel, M. M. +Fejer, and Y. Yamamoto, Nature 491, 421 (2012). +[14] J. R. Schaibley, A. P. Burgers, G. A. McCracken, L.- +M. Duan, P. R. Berman, D. G. Steel, A. S. Bracker, +D. Gammon, and L. J. Sham, Phys. Rev. Lett. 110, +167401 (2013). +[15] I. Schwartz, D. Cogan, E. R. Schmidgall, Y. Don, +L. Gantz, O. Kenneth, N. H. Lindner, and D. Gershoni, +Science 354, 434 (2016). +[16] G. Peniakov, Z.-E. Su, A. Beck, D. Cogan, O. Amar, and +D. Gershoni, Physical Review B 101, 245406 (2020). +[17] D. Cogan, G. Peniakov, O. Kenneth, Y. Don, and D. Ger- +shoni, Physical Review Applied 18, 024055 (2022). +[18] Y. J. Fu, S. D. Lin, M. F. Tsai, H. Lin, C. H. Lin, H. Y. +Chou, S. J. Cheng, and W. H. Chang, Physical Review +B 81, 113307 (2010). +[19] M. M. Glazov, E. L. Ivchenko, O. Krebs, K. Kowalik, and +P. Voisin, Physical Review B 76, 193313 (2007). +[20] Y. H. Shin, B. K. Choi, Y. Kim, J. D. Song, D. Naka- +mura, Y. H. Matsuda, and S. Takeyama, Opt. Express +23, 28349 (2015). +[21] S. N. Walck and T. L. Reinecke, Physical Review B 57, +9088 (1998). +[22] C. Schulhauser, D. Haft, R. J. Warburton, K. Kar- +rai, A. O. Govorov, A. V. Kalameitsev, A. Chaplik, +W. Schoenfeld, J. M. Garcia, and P. M. Petroff, Phys- +ical Review B 66, 193303 (2002). +[23] G. Ramon, U. Mizrahi, N. Akopian, S. Braitbart, D. Ger- +shoni, T. Reinecke, B. Gerardot, and P. Petroff, Physical +Review B 73 (2006). +[24] Y. Benny, Y. Kodriano, E. Poem, D. Gershoni, T. A. +Truong, and P. M. Petroff, Physical Review B 86 (2012). +[25] E. Poem, J. Shemesh, I. Marderfeld, D. Galushko, +N. Akopian, D. Gershoni, B. D. Gerardot, A. Badolato, +and P. M. Petroff, Physical Review B 76 (2007). +[26] Y. Benny, S. Khatsevich, Y. Kodriano, E. Poem, R. Pres- +man, D. Galushko, P. M. Petroff, and D. Gershoni, Phys- +ical Review Letters 106 (2011). +[27] I. Schwartz, D. Cogan, E. R. Schmidgall, L. Gantz, +Y. Don, M. Zielinski, and D. Gershoni, Physical Review +B 92 (2015). +[28] Y. Don, M. Zielinski, and D. Gershoni, The optical activ- +ity of the dark exciton (2016), arXiv:1601.05530 [cond- +mat.mes-hall]. +[29] Y. Kodriano, E. R. Schmidgall, Y. Benny, and D. Ger- +shoni, Semiconductor Science and Technology 29, 053001 +(2014). +[30] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. A. Gor- +bunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, +T. L. Reinecke, S. N. Walck, J. P. Reithmaier, F. Klopf, +and F. Schafer, Physical Review B 65 (2002). +[31] P. Y. Yu and M. Cardona, Fundamentals of Semiconduc- +tors (Springer Berlin Heidelberg, 2010). +[32] B. J. Witek, R. W. Heeres, U. Perinetti, E. P. A. M. +Bakkers, L. P. Kouwenhoven, and V. Zwiller, Physical +Review B 84 (2011). +[33] M. Bayer, A. Kuther, A. Forchel, A. Gorbunov, V. B. +Timofeev, F. Schäfer, J. P. Reithmaier, T. L. Reinecke, +and S. N. Walck, Physical Review Letters 82, 1748 +(1999). + +10 +[34] J. G. Tischler, A. S. Bracker, D. Gammon, and D. Park, +Physical Review B 66, 081310 (2002). +[35] L. Gantz, +E. R. Schmidgall, +I. Schwartz, +Y. Don, +E. Waks, G. Bahir, and D. Gershoni, Physical Review +B 94 (2016). +[36] I. Schwartz, E. R. Schmidgall, L. Gantz, D. Cogan, +E. Bordo, Y. Don, M. Zielinski, and D. Gershoni, Physi- +cal Review X 5, 011009 (2015). +[37] Y. D. M. Zielinski and D. Gershoni, Physical Review B +91 (2015). +[38] E. R. Schmidgall, I. Schwartz, D. Cogan, L. Gantz, +Y. Don, and D. Gershoni, Coherent control of dark +excitons in semiconductor quantum dots, in Quantum +Dots for Quantum Information Technologies, edited by +P. Michler (Springer International Publishing, Cham, +2017) pp. 123–164. +[39] L. M. Roth, B. Lax, and S. Zwerdling, Physical Review +114, 90 (1959). +[40] C. E. Pryor and M. E. Flatté, Physical Review Letters +96, 026804 (2006). +[41] Y. Benny, Y. Kodriano, E. Poem, S. Khatsevitch, D. Ger- +shoni, and P. M. Petroff, Physical Review B 84 (2011). +[42] J. van Bree, A. Y. Silov, P. M. Koenraad, M. E. Flatte, +and C. E. Pryor, Physical Review B 85 (2012). +[43] E. Ivchenko, Optical Spectroscopy of Semiconductor +Nanostructures (Alpha Science, 2005). +[44] E. Poem, J. Shemesh, I. Marderfeld, D. Galushko, +N. Akopian, D. Gershoni, B. D. Gerardot, A. Badolato, +and P. M. Petroff, Physical Review B 76, 235304 (2007). +[45] Y. Kodriano, E. Poem, N. H. Lindner, C. Tradonsky, +B. D. Gerardot, P. M. Petroff, J. E. Avron, and D. Ger- +shoni, Physical Review B 82 (2010). +[46] G. Peniakov, Magneto-optical Study of Semiconductor +Quantum Dots, Master’s thesis (2017). +[47] A. Beck, Magneto optic studies of quantum dot confined +charge carriers and excitons, Master’s thesis, Technion - +the Israeli Institute of Technology (2020). +[48] R. J. Warburton, B. T. Miller, C. S. Durr, C. Bode- +feld, K. Karrai, J. P. Kotthaus, G. Medeiros-Ribeiro, +P. M. Petroff, and S. Huant, Physical Review B 58, 16221 +(1998). + diff --git a/ptE2T4oBgHgl3EQf0QjM/content/tmp_files/2301.04140v1.pdf.txt b/ptE2T4oBgHgl3EQf0QjM/content/tmp_files/2301.04140v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3df40970a5c1a8efed1991137a482f2b2ee3ceab --- /dev/null +++ b/ptE2T4oBgHgl3EQf0QjM/content/tmp_files/2301.04140v1.pdf.txt @@ -0,0 +1,160 @@ +High Resolution On-Chip Thin-Film Lithium +Niobate Single-Photon Buffer +Cagin Ekici,1 Yonghe Yu,1 Jeremy C. Adcock,1 Alif Laila Muthali,1 Heyun Tan,2 Hao +Li,2 Leif Katsuo Oxenløwe,1 Xinlun Cai,2 and Yunhong Ding1,* +1 Center for Silicon Photonics for Optical Communication (SPOC), Department of Electrical and Photonics +Engineering, Technical University of Denmark, Lyngby, Denmark +2State Key Laboratory of Optoelectronic Materials and Technologies, School of Electronics and Information +Technology, Sun Yat-sen University, Guangzhou 510275, China +*yudin@dtu.dk +Abstract: +We experimentally demonstrate a room-temperature, voltage controlled, short- +term quantum photonics memory on a lithium niobate chip. Our chip is capable of resolving +100 ps time steps with 0.74 dB loss per round-trip. © 2023 The Author(s) +1. +Introduction +Short-term quantum photonics memories or single-photon buffers are essential for quantum technologies, since +they provide a synchronization scheme for matching independent systems functioning at different speeds. In order +to optimize two-photon interference from distant sources in a quantum network, photon buffers having high reso- +lution configurability is needed to store one photon until the other is transmitted [1]. In addition, entangling quan- +tum operations in photonics are generally probabilistic, and such short-term memories play a crucial role to buffer +gates of probabilistic nature. Furthermore, approaching ideal single-photon sources based on parametric sponta- +neous pair generation through temporal multiplexing requires low-loss and controllable photon storage [2,3]. +To date, optical buffers based on delay lines [4], slow light [5], and Bragg scattering four-wave mixing [6] have +been introduced. All these techniques either have an excessive loss which is not suitable for quantum applications +or are overly sophisticated. Although atomic cloud optical memories are main contenders, they are difficult to +integrate, and only operate at specific wavelengths. Therefore, to fulfill the requirements of a single-photon buffer, +thin-film lithium niobate (TFLN) based integrated photonics platforms are ideal candidates, since they offer volt- +age controlled, low-loss and high-speed interferometric switching. In this paper, we experimentally demonstrate +an on-chip TFLN single-photon buffer based on recirculating 1 cm-long loop with a round-trip time of 100 ps, i.e. +the overall delay can be controlled with 100 ps time resolution, and storage times of up to 1.4 ns (14 round trips). +2. +Experimental Setup and Results +The TFLN single-photon buffer was fabricated on a commercial lithium niobate on insulator (LNOI) platform +with top LN thickness of 600 nm. The switch consists of 4.5 mm long LN phase modulator on push-pull mode, +exhibiting bandwidth more than 40 GHz, as shown in Fig. 1 (b), and the whole chip insertion loss is less than 6.2 +dB (including the coupling loss). +Fig. 1. +(a) Schematics of the experimental setup with a real image of the TFLN chip consist- +ing several buffers. (b) Electro-optic bandwidth (S21) measurement. Abbreviations: FPGA: Field- +Programmable Gate Array, VOA: Variable Optical Attenuator, UC: Ultrafast Comparator, EA: Elec- +tronic Amplifier, TFLN S-PB: TFLN Single-Photon Buffer. +arXiv:2301.04140v1 [quant-ph] 10 Jan 2023 + +Time-tagger +FPGA +UC +EA +(H)m +VOA +t=100ps +TELN S-PB +(a)10 +(dB) +5 +0 +Normalized E +.5 +-10 +-15 +-20 +10 +20 +30 +40 +50 +Frequency (GHz) +(b)The experimental setup is shown in Fig. 1 (a). We conduct the experiments utilizing heavily-attenuated light +from a laser (1550 nm, 40 fs pulse duration), i.e. weak coherent state, with 100 MHz repetition rate instead of +true single-photon quantum states. The switch control signals are generated via an FPGA and are fed into an +ultrafast comparator to obtain a fast fall-rise time. Afterwards, the fast signals are amplified to the Vπ of the TFLN +switch, and are applied to the chip through high speed radio frequency (RF) probes using micropositioners. After +storage and read-out for a delay, photons are detected by a superconducting nanowire single-photon detector(s) +and recorded by a time-tagger which produces a real-time histogram of the detection event. +Fig. 2. The experimental results of single-photon storage: (a) Normalized histogram counts as a +function of different storage time. (b) The peak amplitudes of the normalized histogram counts +revealing the loss for different storage times. (c) Second-order correlation function g(2)(0) of the +read-out single photons for different storage times with the error bars. +The experimental results of single photon storage with our TFLN chip is shown in Fig. 2. Normalized histogram +counts for the first 5 round-trip are depicted in Fig. 2 (a). The round-trip loss performance of the chip as a function +of time is exhibited in Fig. 2 (b). Each peak value after a round-trip has been fitted the line with slope 0.74 dB. +Accordingly, we measure the second-order correlation function g(2)(0) after each round-trip by adding a 50/50 +fiber optic beam splitter before the detection, see Fig. 2 (c). As expected g(2)(0) ≈ 1, since our TFLN photonics +chip is illuminated by a weak coherent state. As a result of constant g(2)(0) ≈ 1 for every round-trip, it can be +inferred that the statistics do not change significantly as a function of a storage time and there is no substantial +optical background noise owing to the absence of an optical pump beam [7]. +3. +Conclusion +We present an experimental study of a recirculating on-chip TFLN single-photon buffer enabling single photons +to be captured, stored, and read-out at will with 100 ps time step resolution in a reliable way. Our promising chip +is a robust and scalable architecture working at room-temperature with low-loss around 0.74 dB per round-trip. +References +1. K. Azuma, K. Tamaki, and H.-K. Lo , “All-photonic quantum repeaters,” Nat. Commun. 7, 6787 (2015). +2. J. C. Adcock, D. Bacco, and Y. Ding, “Temporal Multiplexing Enhancement with a Silicon Waveguide Single Photon +Source,” CLEO, JTu3B. 1 (2022). +3. J. C. Adcock, D. Bacco, and Y. Ding, “Enhancement of a silicon waveguide single photon source by temporal multi- +plexing,” Quantum Science and Technology 7, 025025 (2022). +4. E. F. Burmeister, D. J. Blumenthal, and J. E. Bowers, “A comparison of optical buffering technologies,” Opt. Switching +Networking 6, 10–18 (2008). +5. R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: Capabilities and fundamental limitations,” +J. Lightwave Technol. 23, 4046–4066 (2005). +6. S. Clemmen, A. Farsi, S. Ramelow, and A. L. Gaeta, “All-optically tunable buffer for single photons,” Opt. Lett. 43, +2138 (2018). +7. C. Kupchak, T. Mittiga, B. Jordaan, M. Namazi, C. N¨olleke, and E. Figueroa “Room-Temperature Single-photon level +Memory for Polarization States,” Sci. Rep. 5, 7658 (2015). + +1.0 +0.5 +0.0 +1.0 +0.5 +0.0 + count +1.0 +0.5 +Normalized +0. 0 +1.0 +0.5 +0.0 +1.0 +0.5 +0.0 +1.0 +0. 5 +0.0 +0 +200 +400600 +Time +(ps) +a0 +Experimental data +Fit (-0.74dB per 100ps) +-- +4 +Loss (dB) +-6 +-8 +-10 +-12 +0 +200 +400 +600 +800 +1000 1200　1400 +Storage time (ps) +(b)1.0 +0. 8 +0.6 +ao +0. 4 +0. 2 +0. 0 +0 +200 +400600 +800 1000 1200 1400 +Storage time (ps) +(c) \ No newline at end of file diff --git a/ptE2T4oBgHgl3EQf0QjM/content/tmp_files/load_file.txt b/ptE2T4oBgHgl3EQf0QjM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99c3b4f5cca523db4811dd71fa6c061813f8bfcd --- /dev/null +++ b/ptE2T4oBgHgl3EQf0QjM/content/tmp_files/load_file.txt @@ -0,0 +1,146 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf,len=145 +page_content='High Resolution On-Chip Thin-Film Lithium Niobate Single-Photon Buffer Cagin Ekici,1 Yonghe Yu,1 Jeremy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Adcock,1 Alif Laila Muthali,1 Heyun Tan,2 Hao Li,2 Leif Katsuo Oxenløwe,1 Xinlun Cai,2 and Yunhong Ding1,* 1 Center for Silicon Photonics for Optical Communication (SPOC), Department of Electrical and Photonics Engineering, Technical University of Denmark, Lyngby, Denmark 2State Key Laboratory of Optoelectronic Materials and Technologies, School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 510275, China yudin@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='dk Abstract: We experimentally demonstrate a room-temperature, voltage controlled, short- term quantum photonics memory on a lithium niobate chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Our chip is capable of resolving 100 ps time steps with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='74 dB loss per round-trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' © 2023 The Author(s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Introduction Short-term quantum photonics memories or single-photon buffers are essential for quantum technologies, since they provide a synchronization scheme for matching independent systems functioning at different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' In order to optimize two-photon interference from distant sources in a quantum network, photon buffers having high reso- lution configurability is needed to store one photon until the other is transmitted [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' In addition, entangling quan- tum operations in photonics are generally probabilistic, and such short-term memories play a crucial role to buffer gates of probabilistic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Furthermore, approaching ideal single-photon sources based on parametric sponta- neous pair generation through temporal multiplexing requires low-loss and controllable photon storage [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' To date, optical buffers based on delay lines [4], slow light [5], and Bragg scattering four-wave mixing [6] have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' All these techniques either have an excessive loss which is not suitable for quantum applications or are overly sophisticated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Although atomic cloud optical memories are main contenders, they are difficult to integrate, and only operate at specific wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Therefore, to fulfill the requirements of a single-photon buffer, thin-film lithium niobate (TFLN) based integrated photonics platforms are ideal candidates, since they offer volt- age controlled, low-loss and high-speed interferometric switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' In this paper, we experimentally demonstrate an on-chip TFLN single-photon buffer based on recirculating 1 cm-long loop with a round-trip time of 100 ps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' the overall delay can be controlled with 100 ps time resolution, and storage times of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='4 ns (14 round trips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Experimental Setup and Results The TFLN single-photon buffer was fabricated on a commercial lithium niobate on insulator (LNOI) platform with top LN thickness of 600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' The switch consists of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='5 mm long LN phase modulator on push-pull mode, exhibiting bandwidth more than 40 GHz, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 1 (b), and the whole chip insertion loss is less than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='2 dB (including the coupling loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' (a) Schematics of the experimental setup with a real image of the TFLN chip consist- ing several buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' (b) Electro-optic bandwidth (S21) measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Abbreviations: FPGA: Field- Programmable Gate Array, VOA: Variable Optical Attenuator, UC: Ultrafast Comparator, EA: Elec- tronic Amplifier, TFLN S-PB: TFLN Single-Photon Buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='04140v1 [quant-ph] 10 Jan 2023 Time-tagger FPGA UC EA (H)m VOA t=100ps TELN S-PB (a)10 (dB) 5 0 Normalized E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='5 10 15 20 10 20 30 40 50 Frequency (GHz) (b)The experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' We conduct the experiments utilizing heavily-attenuated light from a laser (1550 nm, 40 fs pulse duration), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' weak coherent state, with 100 MHz repetition rate instead of true single-photon quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' The switch control signals are generated via an FPGA and are fed into an ultrafast comparator to obtain a fast fall-rise time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Afterwards, the fast signals are amplified to the Vπ of the TFLN switch, and are applied to the chip through high speed radio frequency (RF) probes using micropositioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' After storage and read-out for a delay, photons are detected by a superconducting nanowire single-photon detector(s) and recorded by a time-tagger which produces a real-time histogram of the detection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' The experimental results of single-photon storage: (a) Normalized histogram counts as a function of different storage time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' (b) The peak amplitudes of the normalized histogram counts revealing the loss for different storage times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' (c) Second-order correlation function g(2)(0) of the read-out single photons for different storage times with the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' The experimental results of single photon storage with our TFLN chip is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Normalized histogram counts for the first 5 round-trip are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' The round-trip loss performance of the chip as a function of time is exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Each peak value after a round-trip has been fitted the line with slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='74 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Accordingly, we measure the second-order correlation function g(2)(0) after each round-trip by adding a 50/50 fiber optic beam splitter before the detection, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' As expected g(2)(0) ≈ 1, since our TFLN photonics chip is illuminated by a weak coherent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' As a result of constant g(2)(0) ≈ 1 for every round-trip, it can be inferred that the statistics do not change significantly as a function of a storage time and there is no substantial optical background noise owing to the absence of an optical pump beam [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Conclusion We present an experimental study of a recirculating on-chip TFLN single-photon buffer enabling single photons to be captured, stored, and read-out at will with 100 ps time step resolution in a reliable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Our promising chip is a robust and scalable architecture working at room-temperature with low-loss around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='74 dB per round-trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Azuma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Tamaki, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Lo , “All-photonic quantum repeaters,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 7, 6787 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Adcock, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Bacco, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Ding, “Temporal Multiplexing Enhancement with a Silicon Waveguide Single Photon Source,” CLEO, JTu3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Adcock, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Bacco, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Ding, “Enhancement of a silicon waveguide single photon source by temporal multi- plexing,” Quantum Science and Technology 7, 025025 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Burmeister, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Blumenthal, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Bowers, “A comparison of optical buffering technologies,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Switching Networking 6, 10–18 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Tucker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Ku, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Chang-Hasnain, “Slow-light optical buffers: Capabilities and fundamental limitations,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 23, 4046–4066 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Clemmen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Farsi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Ramelow, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Gaeta, “All-optically tunable buffer for single photons,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 43, 2138 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Kupchak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Mittiga, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Jordaan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Namazi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' N¨olleke, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Figueroa “Room-Temperature Single-photon level Memory for Polarization States,” Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 5, 7658 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQf0QjM/content/2301.04140v1.pdf'} +page_content='0 0.' metadata={'source': 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sequential decision making +with multiple objectives that are increasingly im- +portant in many applications. However, the model +is often unknown and must be learned online +while still ensuring the constraint is met, or at +least the violation is bounded with time. Some +recent papers have made progress on this very +challenging problem but either need unsatisfac- +tory assumptions such as knowledge of a safe +policy, or have high cumulative regret. We pro- +pose the Safe PSRL (posterior sampling-based +RL) algorithm that does not need such assump- +tions and yet performs very well, both in terms of +theoretical regret bounds as well as empirically. +The algorithm achieves an efficient tradeoff be- +tween exploration and exploitation by use of the +posterior sampling principle, and provably suffers +only bounded constraint violation by leveraging +the idea of pessimism. Our approach is based +on a primal-dual approach. We establish a sub- +linear ˜O +� +H2.5� +|S|2|A|K +� +upper bound on the +Bayesian reward objective regret along with a +bounded, i.e., ˜O (1) constraint violation regret +over K episodes for an |S|-state, |A|-action, and +horizon H CMDP. +1. Introduction +Markov decision processes (MDPs) (Puterman, 1994) are +used to model many scenarios involving sequential decision- +making. They are used in a wide variety of settings like +robotics, cyber-physical systems, and safety-critical au- +tonomous vehicles. However, a traditional reinforcement +learning (RL) formulation which seeks to maximize a sin- +gle cumulative reward cannot capture many problems that +have multiple objectives. For example, this is the case for a +robot that needs to perform a certain reward-yielding task +while ensuring that the average energy expended is bounded +below a threshold. Such scenarios are well-modeled by con- +strained MDPs (CMDPs) (Altman, 1999), which extend +the MDP formalism by considering additional constraints +on the expected cumulative performance of a policy. In the +CMDP setting, one seeks to find an optimal policy which +maximizes the cumulative objective reward while satisfying +constraints on cost objectives. +In this paper, we consider the problem of online learning +for finite-horizon CMDPs, where an agent interacts with +the environment repeatedly in episodes of fixed length. The +transition probability is not known to the agent, thereby +requiring the agent to learn about the system dynamics by +observing the past states and actions. The performance of +this agent is measured by the notion of cumulative regret, +i.e., the difference between the cumulative reward of the +learning agent and that of the optimal policy. +This online learning problem thus leads to the well-known +trade-off between exploration and exploitation: should the +agent explore the environment to improve future perfor- +mance, or exploit the current knowledge for better short- +term performance? +A common approach to balance this exploration-exploitation +trade-off is the ‘Optimism in the Face of Uncertainty’ (OFU) +principle (Lai & Robbins, 1985). The idea is that each +state and action are assigned optimism bonuses based on +the current knowledge. The agent then chooses a policy +which maximizes the expected return under this optimistic +model. The bonuses are designed to promote exploration +of poorly-understood state-action pairs. This approach has +been widely used for online learning in MDPs (Jaksch et al., +2010; Azar et al., 2017; Jin et al., 2018; Wei et al., 2020; +Kalagarla et al., 2021). +Another alternative for efficient exploration is posterior sam- +pling (also called Thompson sampling) (Thompson, 1933). +In this approach, a posterior distribution is maintained over +the unknown transition probability model based on the prior +distribution and dataset about visited trajectories. At the +beginning of each episode, a model is sampled from this pos- +terior distribution. The agent then chooses a policy which +arXiv:2301.11547v1 [cs.LG] 27 Jan 2023 + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +is optimal with respect to the sampled model and follows +it for the duration of the episode. The advantages of poste- +rior sampling over OFU stem from the fact that (i) known +information about the model can be incorporated into the +algorithm through the prior distribution, and (ii) posterior +sampling algorithms have demonstrated superior empirical +performance for online learning over OFU-type algorithms +including in the RL setting (Osband et al., 2013; Ouyang +et al., 2017). +In this paper, we use the posterior sampling approach and in- +troduce the Safe PSRL algorithm for efficient exploration +in the finite-horizon CMDP setting. Our algorithm uses the +primal-dual approach for CMDPs wherein the primal part +performs unconstrained MDP planning with a sampled tran- +sition probability, and the dual part updates the Lagrangian +variable to track the constraint violation. +We achieve bounded constraint violation regret by leverag- +ing the idea of pessimism, introduced earlier in the context +of constrained bandits (Liu et al., 2021b). “Pessimism” is +achieved by tightening the constraint of the CMDP problem +in every episode at decreasing levels. The key is, however, +how this tightening is achieved. By appropriately balanc- +ing exploration via posterior sampling and safe learning +via pessimism, we show that the Safe PSRL algorithm +achieves sub-linear ˜O +� +H2.5 +τ−c0 +� +|S|2|A|K +� +reward regret +while achieving bounded, i.e., ˜O(1)-constraint violation re- +gret for an |S|-state, |A|-action, and an H episode length +CMDP over K number of episodes. τ denotes the desired +threshold for constraint violation, and c0 is a known feasible +expected cumulative constraint cost of the CMDP. +The contributions of this paper are the following: (i) We +present the first PSRL algorithm for CMDPs that not +only achieves ˜O +�√ +K +� +objective reward regret but also +˜O(1) constraint violation regret. Unlike other proposed +algorithms, our algorithm does not need knowledge of a +constraint-satisfying safe policy. (ii) Our Safe PSRL al- +gorithm has better empirical performance than state-of-the- +art OFU-type algorithms for the same setting introduced in +(Liu et al., 2021a; Bura et al., 2022) which need knowledge +of a safe policy. (iii) The algorithm design is simpler than +other state-of-the-art algorithms (Liu et al., 2021a; Bura +et al., 2022) for the problem: The key design choice are +two pessimism parameters. The regret analysis involves a +novel decomposition which allows us to leverage posterior +sampling regret analysis and Lyapunov-drift analysis for the +dual variables. +2. Related Work +Posterior (or Thompson) sampling goes back to the work +of (Thompson, 1933), but attracted less attention for sev- +eral decades until empirical evidence (Chapelle & Li, 2011) +showed its superior performance for online learning. Re- +cently, it has been widely applied to various settings like +multi-armed bandits (Kaufmann et al., 2012; Agrawal & +Goyal, 2012; 2013), MDPs (Osband et al., 2013; Gopalan +& Mannor, 2015; Osband & Van Roy, 2017; Ouyang et al., +2017) and POMDPs (Jafarnia-Jahromi et al., 2021c). +Multi-armed bandits (MAB) are a special case of MDPs +with a single state and unknown reward function. Safe +online learning has been studied for MABs in multiple set- +tings (Amani et al., 2019; Khezeli & Bitar, 2020; Pacchiano +et al., 2021; Liu et al., 2021b). However, in comparison, the +multi-step and multi-state MDP settings are more challeng- +ing due to the unknown transition probability. +In the CMDP setting, several existing works (Efroni et al., +2020; Qiu et al., 2020; Agarwal et al., 2022) leverage OFU +or posterior sampling to provide ˜O( +√ +K) regret for the re- +ward as well as the constraint objective, where K is the +number of episodes. Such an approach, however, can lead +to a large number of constraint violations during learning, +which is unacceptable during various safety-critical tasks +such as driving or power distribution. Thus, the problem of +an online RL algorithm that has sublinear (and hopefully, +˜O( +√ +K)) reward regret while achieving bounded constraint +violation regret, particularly one that also has very good +empirical performance, remains open. +OFU-based algorithms have been widely used for efficient +learning in CMDPs, e.g., in the setting of PAC performance +guarantees for finite-horizon CMDPs (HasanzadeZonuzy +et al., 2021; Kalagarla et al., 2021), or to provide regret +bounds for CMDPs in the finite-horizon setting (Efroni et al., +2020; Brantley et al., 2020) and infinite-horizon average cost +setting (Singh et al., 2020). Policy gradient algorithms for +CMDPs (Ding et al., 2020; 2021) have also been studied. +However, these algorithms do not provide bounded or zero +constraint violation guarantees. +Recently, some OFU-based approaches for safe learning +with bounded or zero constraint violation guarantees have +been proposed (Zheng & Ratliff, 2020; Chen et al., 2022; +Bai et al., 2022; Wei et al., 2022; Liu et al., 2021a). But +these either assume the transition model is known (but re- +ward function is not), or only satisfy the constraint with high +probability, or assume that a safe policy is known to the algo- +rithm (and can be used by it), e.g., in (Liu et al., 2021a; Bura +et al., 2022). The OptPess-PrimalDual algorithm in (Liu +et al., 2021a) is the closest comparable algorithm but our +Safe PSRL algorithm is better in terms of its dependence +on various problem parameters, e.g., it has ˜O(H3|S|1.5) +dependence as opposed to ˜O(H2.5|S|) for ours. +While the use of the posterior sampling principle for con- +strained RL problems is under-explored (despite the promise + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +of better empirical performance), (Agarwal et al., 2022) in- +deed introduces a PSRL algorithm for CMDPs but for the +average setting. Moreover, it only achieves a ˜O( +√ +K) con- +straint violation regret which is worse than our ˜O(1) bound. +3. Preliminaries +3.1. Notation +We denote the probability simplex over set S by ∆S. We +use the notation ˜O which has similar meaning as the usual +O notation but ignores logarithmic factors. +3.2. Finite-Horizon MDPs +An episodic finite-horizon MDP (Puterman, 1994) can be +formally defined by a tuple M = (S, A, H, s1, p, r), where +S and A denote the state and action spaces, respectively. +In this setting, the agent interacts with the environment +in episodes of fixed length H, with each episode starting +with a random initial state denoted s1. The non-stationary +transition probability ph(s′|s, a) is the the probability of +transitioning to state s′ on taking action a at state s at time +step h ∈ [1 : H] of the episode. The non-stationary reward +obtained on taking action a in state s at time step h of an +episode is denoted by a random variable Rh(s, a) ∈ [0, 1], +with mean rh(s, a). We use r as a shorthand to denote the +mean reward vector r1, . . . , rH. +A non-stationary randomized policy π = (π1, . . . , πH) ∈ Π +where πi : S → ∆A, maps each state to a probability +simplex over the action space A. The action ah at time step +h at state sh is taken according to the policy π, ah ∼ πh(sh). +The value function of a non-stationary randomized policy +π, V π +h (s; r, p) (when clear, s, r, and p are omitted) at a state +s ∈ S and time step h ∈ [1 : H] is defined as: +V π +h (s; r, p) := Eπ +� H +� +i=h +ri(si, ai)|sh = s, p +� +, +where the expectation is over the distribution induced by the +environment and policy randomness. Similarly, the Q-value +function of a policy π, Qπ +h(s, a; r, p), for a state s ∈ S, an +action a ∈ A and time step h ∈ [1 : H], is defined as +Qπ +h(s, a; r, p) := rh(s, a)+ +(1) +Eπ +� +H +� +i=h+1 +ri(si, ai)|sh = s, ah = a, p +� +. +We can always find an optimal non-stationary deterministic +policy ˜π (Puterman, 1994) such that V ˜π +h (s) = ˜Vh(s) = +supπV π +h (s) and Q˜π +h(s, a) = +˜Qh(s, a) = supπQπ +h(s, a). +The optimal policy can be computed by using backward +induction on the Bellman optimality equations (Puterman, +1994): +˜ +Vh(s) = maxa∈A +� +rh(s, a) + ph(·|s, a) ˜Vh+1 +� +, +˜Qh(s, a) = rh(s, a) + ph(·|s, a) ˜Vh+1, +(2) +where ˜VH+1(s) = 0 and ˜Vh(s) = maxa∈A ˜Qh(s, a). The +optimal policy ˜π is then greedy with respect to ˜Qh. +3.3. Finite-Horizon Constrained MDPs +A finite-horizon constrained MDP (CMDP) (Altman, 1999) +is a finite-horizon MDP with a required upper bound on +expectation of on a cost function, {c, τ ∈ (0, H]}. The +non-stationary cost obtained on taking action a in state s +at time step h ∈ [1 : H] with respect to the constraint cost +function is denoted by a random variable Ch(s, a) ∈ [0, 1], +with mean ch(s, a). Similar to r, we use c to denote the +mean cost vector c1, . . . , cH. +The total expected reward (cost) of an episode under pol- +icy π with respect to the reward (cost) function r (c) is +the respective value function from the initial state s1, i.e., +V π +1 (s1; r, p)(V π +1 (s1; c, p)) (by definition). Our objective in +this CMDP setting is to find a policy which maximizes the +total expected objective reward under the constraint that the +total expected constraint cost is below a desired threshold. +Formally, +π∗ ∈ argmax +π∈Π +V π +1 (s1; r, p) +s.t. +V π +1 (c, p) ≤ τ. +(3) +The optimal value is denoted by +V ∗(s1; r, p) += +V π∗ +1 (s1; r, p). A deterministic optimal policy may not exist, +hence we need to consider Π to be the class of all random- +ized policies (Altman, 1999). Since the Bellman optimality +equations may not hold due to the constrained nature of the +problem, we cannot leverage dynamic programming-based +backward induction algorithms to find an optimal policy. +However, a linear programming approach can be given that +will find an optimal policy (Altman, 1999). +4. The Learning Problem +We consider the setting where an agent repeatedly interacts +with a finite-horizon CMDP M = (S, A, H, s1, p, r, {c, τ}) +over multiple episodes of fixed length H, starting each +episode from the same initial state s1 and with stationary +transition probability (i.e., ph = p, ∀h). We employ the +Bayesian framework and regard the transition probability p +as random with a prior distribution µ1. The realized tran- +sition probability is unknown to the learning agent. We +consider finite-horizon CMDP whose transition probability +lies in the set Θc0 with the following property: + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +Assumption 1. For all ˆp ∈ Θc0, there exists a policy π ˆp +0 +such that V +π ˆ +p +0 +1 (c, ˆp) ≤ c0 < τ. +Moreover, we assume that the support of the prior distribu- +tion µ1 is a subset of Θc0 and c0 is known. Without loss of +generality,1 we assume that the reward and cost functions r +and c are respectively are known to the learning agent. Note +that the above assumption is not only reasonable but also +necessary to ensure the problem is feasible. +The agent interacts with the environment for K episodes, +each of length H. In each episode, the agent starts from +a state s1 and chooses a Markov policy πk determined by +the information gathered until that episode. This policy is +then executed until the end of the episode, while collecting +the rewards and costs. The main objectives of the learning +agent are to: +(1) Maximize the expected cumulative reward after K +episodes or equivalently, minimize the Bayesian regret with +respect to the reward function defined as: +BR(K; r) := E +� K +� +k=1 +� +V π∗ +1 (s1; r, p) − V πk +1 +(s1; r, p) +�� +. +(4) +(2) Minimize the constraint violation or equivalently, min- +imize the Bayesian regret with respect to the constraint +defined as: +BR(K; c) := E +� K +� +k=1 +� +V πk +1 +(s1; c, p) − τ +�� +. +(5) +With respect to these objectives, we propose an algorithm +which is able to achieve sub-linear regret with respect to +the reward objective while ensuring that regret with respect +to the cost constraint is bounded above by a constant, i.e., +independent of the number of episodes K. +5. The Safe PSRL Algorithm +We propose the Safe Posterior Sampling-based Reinforce- +ment Learning (Safe PSRL) algorithm for the finite- +horizon CMDP model. This algorithm leverages the idea of +posterior sampling to balance exploration and exploitation. +It also takes a primal-dual approach to handle the constraint +cost objective along with reward maximization objective. +We further introduce the idea of pessimism (Liu et al., +2021b) to ensure that the cost regret is bounded. This “pes- +simism” is achieved by considering a “more constrained” +1The complexity of learning the cost and reward functions is +dominated by the complexity of learning the transition probability +(Auer & Ortner, 2005). The algorithm can be readily extended to +the setting of unknown cost and reward functions by using their +empirical estimate in place of the known cost and reward functions. +CMDP problem as compared to the original problem. This +is done by decreasing the threshold by ϵk in each episode k. +Formally, we consider the objective: +max +V π +1 (r, p) +s.t. +V π +1 (c, p) ≤ τ − ϵk. +(6) +This pessimistic term ϵk ensures bounded cost regret and it +decreases as the episode count increases. +The algorithm starts with the prior distribution µ1 on the +transition probability. Then, at every time step t, the learning +agent maintains a posterior distribution µt on the unknown +transition probability p given by µt(Θ) = P(p ∈ Θ|Ft) +for any set Θ ⊆ Θc0. Here Ft is the information available +at time t, i.e., the sigma algebra generated by encountered +states and actions upto time t, (s1, a1, · · · , st−1, at−1, st). +On observing the next state st+1 by taking action at at state +st, the posterior is updated according to Bayes’s rule: +µt+1(dp) = +pt(st+1|st, at)µt(dp) +� +p +′ +t(st+1|st, at)µt(dp′). +(7) +In parallel, the algorithm proceeds as follows: At the begin- +ning of each episode k, transition probability ˆpk is sampled +from the posterior distribution µtk (where tk is the time step +corresponding to beginning of episode k). We then consider +the Lagrangian defined as: +Lk(π, λ) := V π +1 (r, ˆpk) + λk +ηk +(τ − ϵk − V π +1 (c, ˆpk)) , +The learning agent then chooses a Markov policy πk (pri- +mal update) which maximizes the above Lagrangian. We +can find such a policy by applying standard dynamic pro- +gramming with respect to the reward function r − λk +ηk c. The +(dual) parameter λk is updated according to the sub-gradient +algorithm as follows: +λk+1 = (λk + V πk +1 (c, ˆpk) + ϵk − τ)+ +The agent then applies the policy πk for the H steps of +episode k. +We note that while some of the details of the algorithm +are natural (as they are common to PSRL algorithms for +various settings) (Ouyang et al., 2017; Jafarnia-Jahromi +et al., 2021c;a;b), the key novelty in the design are the ϵk +and ηk parameters to be used in conjunction with a primal- +dual approach. Their choice is guided by the regret analysis +presented in Section 6. +The Safe PSRL algorithm is summarized next. + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +Algorithm 1 Safe-PSRL +Input: K, µ1, c0, τ +Initialization: λ1 ← 0 +for episodes k = 1, . . . , K do +ϵk ← +5|H|1.5√ +|S|2|A|(log k|S||A|H+1) +√ +k log k|S||A|H +ηk ← (τ − c0)H +√ +k +tk = (k − 1)H + 1 +Generate ˆpk ∼ µtk(.) +Compute πk ∈ arg maxπ V π +1 (r − λk +ηk c, ˆpk) according +to (2) (Policy Update) +λk+1 ← max(0, λk + V πk +1 (c, ˆpk) + ϵk − τ) (Dual +Update) +for t = (k − 1)H + 1, . . . , kH do +Choose action at ∼ πk(st) +Observe st+1 ∼ p(.|st, at) +Update the posterior distribution µt+1 according to +(7) +end for +end for +The following theorem then establishes that the Safe +PSRL algorithm can achieve sub-linear ˜O( +√ +K) reward re- +gret while achieving bounded constraint violation regret. +Theorem 1. Suppose Assumption 1 holds, then the reward +and cost regret of the Safe PSRL algorithm is upper +bounded as: +BR(K; r) = ˜O +� H2.5 +τ − c0 +� +|S|2|A|K +� +, and +BR(K; c) = ˜O +� +C′′(H − τ) + H1.5� +|S|2|A|C′′ +� += O(1), +where C′′ = O( H3|S|2|A| +(τ−c0)2 ) is independent of K. +Remark 1. (i) We note that the upper bound on BR(K; r) of +the OptPess-PrimalDual algorithm (Liu et al., 2021a) +is ˜O +� +H3� +|S|3|A|K +� +. Thus, our upper bound is the same +in terms of |A|,K and better in terms of |S| and H. Both, +OptPess-PrimalDual and Safe PSRL algorithms achieve +˜O (1) upper bounds on BR(K; c). +(ii) We note that the upper bound on BR(K; r) of the DOPE +algorithm (Bura et al., 2022) is ˜O +� +H3� +|S|2|A|K +� +. Thus, +our upper bound is the same in terms of |A|, K, |S| and +better in terms of H. While our bounds are comparable to +those of DOPE, we shall see that the numerical performance +is much better. Further, the DOPE algorithm guarantees zero +constraint violations with high probability. But, this requires +a strong assumption, i.e, knowledge of a safe policy that can +satisfy the constraint. +(iii) The CMDP-PSRL algorithm (Agarwal et al., 2022) +uses posterior sampling in the average CMDP setting and +achieves ˜O +� +TM|S| +� +|A|K +� +reward objective and the same +constraint violation regret, where TM is the mixing time. +In comparison, we are able to achieve bounded constraint +violation regret. +6. Regret Analysis +We now provide theoretical analysis of the Safe PSRL +algorithm by providing details of the proof of Theorem 1. +We first state some relevant results from the literature on +posterior sampling in the context of reinforcement learning. +A key property of posterior sampling (Osband et al., 2013) +is the posterior sampling lemma, i.e., the transition probabil- +ity ˆpt sampled from the posterior distribution at time t and +transition probability p have the same distribution. +Lemma 1. For any function f, we have E [f(ˆpt)] = +E [f(p)] where p is the transition probability (with the prior +distribution µ1) and ˆpt is the sampled transition probability +from the posterior distribution µt at time t. +The following is a restatement (Osband et al., 2013) of +the sub-linear regret bound achieved when using posterior +sampling for unconstrained finite horizon MDPs. +Lemma 2. (Osband et al., 2013) The Bayesian regret of the +PSRL algorithm for unconstrained MDPs is given by +K +� +k=1 +E +� +V πk +1 +(c, p) − V πk +1 +(c, ˆpk) +� +≤ H1.5� +30|S|2|A|K log(|S||A|KH) + 2H. +(8) +6.1. Cost Constraint Violation Analysis +We first present analysis of the cost constraint violation. We +can decompose the constraint violation regret as follows: +BR(K; c) := E +� K +� +k=1 +� +V πk +1 +(c, p) − τ +�� += +K +� +k=1 +E +� +V πk +1 +(c, p) − V πk +1 +(c, ˆpk) + V πk +1 +(c, ˆpk) − τ +� += +K +� +k=1 +E +� +V πk +1 +(c, p) − V πk +1 +(c, ˆpk) +� ++ +K +� +k=1 +E +� +V πk +1 +(c, ˆpk) − τ +� +≤ +K +� +k=1 +E +� +V πk +1 +(c, p) − V πk +1 +(c, ˆpk) +� ++ +K +� +k=1 +E [λk+1 − λk − ϵk] +(by dual update rule of algorithm) += +K +� +k=1 +E +� +V πk +1 +(c, p) − V πk +1 +(c, ˆpk) +� ++ E [λK+1] − +K +� +k=1 +ϵk +(9) + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +≤ H1.5� +30|S|2|A|K log(|S||A|KH) + 2H ++ E [λK+1] − +K +� +k=1 +ϵk +(10) +where the last upper bound follows by use of Lemma 2 to +upper bound the first term in (9). +We next show that the dual parameter E [λK+1] can be upper +bounded by use of Lyapunov-drift analysis. To that end, +we restate the following lemma (Liu et al., 2021b) which +states the Lyapunov-drift conditions for the boundedness of +a random process. +Lemma 3. (Liu et al., 2021b) Consider a random process +S(t) with a Lyapunov function Φ(k) such that Φ(0) = 0 and +∆(k) = Φ(k + 1) − Φ(k) is the Lyapunov drift. Given an +increasing sequence {ϕk} and constants ρ and νmax with +0 < ρ ≤ νmax, if the expected drift E [∆(k)|S(k) = s] sat- +isfies the following conditions: +(i) There exists constants ρ +> +0 and ϕk +> +0 s.t. +E [∆(k)|S(k) = s] ≤ −ρ when Φ(k) ≥ ϕk, and +(ii) |Φ(k+1)−Φ(k)| ≤ νmax holds with probability 1, then +E +� +eζΦ(t)� +≤ E +� +eζΦ0� ++ 2eζ(νmax+ϕt) +ζρ +, +where ζ = ρ/(ν2 +max + νmaxρ/3). +We divide the episodes into two parts, i.e. k < C′′ and +k ≥ C′′ where C′′ = +80H3|S|2|A| +(τ−c0)2 +. We can clearly see +that for k ≥ C′′, we have ϵk ≤ τ−c0 +2 +. Thus, for k ≥ C′′, +Problem (6) is feasible for all ˆpk ∈ Θc0 by Assumption 1. +For k ≥ C′′, we show that the Lyapunov function Φ(λ) = λ +satisfies the conditions of Lemma 3 and thus provide a +bound on the exponential moment of the dual variable λ. +Lemma 4. For k ≥ C′′, when λ ≥ ϕk, we have, +E [λk+1 − λk|λk = λ] ≤ ρ and +|λk+1 − λk| ≤ H +with probability 1, +where ϕk := 4(H2 + ϵ2 +k + ηkH)/(τ − c0) and +ρ := −(τ − c0)/4. Thus, we have, +E +� +eζλK+1� +≤ E +� +eζλC′′ � ++ 2eζ(H+ϕK+1) +ζρ +, +(11) +where ζ = ρ/(H2 + Hρ/3). The above inequality (11) can +be simplified to +E [λK+1] ≤ 1 +ζ log 11H2 +3ρ2 ++ H + +C′′ +� +1 +ϵk + C′′(H − τ) ++4(H2 + ϵ2 +K+1 + ηK+1H) +(τ − c0) +. +(12) +Next, we bound the sumkϵk term: +K +� +k=1 +ϵk ≥ +� K+1 +1 +ϵudu +≥ 10H1.5� +|S|2|A|Klog|S||A|HK +− 10H1.5� +|S|2|A|log|S||A|H. +(13) +Thus, putting together (10), (12) and (13), the leading terms +of ˜O( +√ +K) cancel out and we get +BR(K; c) = ˜O +� +C′′(H − τ) + H1.5� +|S|2|A|C′′ +� += ˜O(1), +i.e., constraint violation regret is a constant, and does not +grow with K. +6.2. Reward Objective Regret Analysis +We next provide regret analysis of the reward objective. Let +πϵk,∗ be the optimal policy for the pessimistic optimization +problem (where p is the true transition probability of the +MDP): +max +V π +1 (r, p) +(14) +s.t. +V π +1 (c, p) ≤ τ − ϵk. +Let πϵk,ˆpk be the optimal policy for the pessimistic optimiza- +tion problem (where ˆpk is the sampled transition probability +of the MDP): +max +V π +1 (r, ˆpk) +(15) +s.t. +V π +1 (c, ˆpk) ≤ τ − ϵk. +We can decompose the reward regret term as follows: +BR(K; r) = E +� K +� +k=1 +� +V π∗ +1 (r, p) − V πk +1 +(r, p) +�� += +C +′′−1 +� +k=1 +E +� +V π∗ +1 (r, p) − V πk +1 +(r, p) +� ++ +K +� +k=C′′ +E +� +V π∗ +1 (r, p) − V πk +1 +(r, p) +� +(splitting the sum across the sets of episodes) += +C +′′−1 +� +k=1 +E +� +V π∗ +1 (r, p) − V πk +1 (r, p) +� ++ +K +� +k=C′′ +E +� +V π∗ +1 (r, p) − V πϵk,∗ +1 +(r, p) +� ++ +K +� +k=C′′ +E +� +V πϵk,∗ +1 +(r, p) − V πϵk,ˆ +pk +1 +(r, ˆpk) +� + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation ++ +K +� +k=C′′ +E +� +V πϵk,ˆ +pk +1 +(r, ˆpk) − V πk +1 (r, ˆpk) +� ++ +K +� +k=C′′ +E [V πk +1 (r, ˆpk) − V πk +1 (r, p)] +(splitting the second sum into four parts) +≤ C +′′H + +K +� +k=C′′ +E +� +V π∗ +1 (r, p) − V πϵk,∗ +1 +(r, p) +� ++ +K +� +k=C′′ +E +� +V πϵk,∗ +1 +(r, p) − V πϵk,ˆ +pk +1 +(r, ˆpk) +� ++ +K +� +k=C′′ +E +� +V πϵk,ˆ +pk +1 +(r, ˆpk) − V πk +1 (r, ˆpk) +� ++ +K +� +k=C′′ +E [V πk +1 (r, ˆpk) − V πk +1 (r, p)] +≤ C +′′H + +K +� +k=C′′ +E +� +V π∗ +1 (r, p) − V πϵk,∗ +1 +(r, p) +� ++ 0 ++ +K +� +k=C′′ +E +� +V πϵk,ˆ +pk +1 +(r, ˆpk) − V πk +1 (r, ˆpk) +� ++ +K +� +k=C′′ +E [V πk +1 (r, ˆpk) − V πk +1 (r, p)] +(by the posterior sampling property in Lemma 1) +≤ C +′′H + +K +� +k=C′′ +E +� +V π∗ +1 (r, p) − V πϵk,∗ +1 +(r, p) +� ++ +K +� +k=C′′ +E +� +V πϵk,ˆ +pk +1 +(r, ˆpk) − V πk +1 (r, ˆpk) +� ++ H1.5� +30|S|2|A|K log(|S||A|KH) + 2H +(by the regret bound in Lemma 2) +The other terms are bounded as follows. Similar to Lemma +5.7 in (Liu et al., 2021a), we can define a probabilistic mixed +policy of π∗ and πp +0 to prove the following lemma: +Lemma 5. The first summation term above can be bounded +as +K +� +k=C′′ +E +� +V π∗ +1 (r, p) − V πϵk,∗ +1 +(r, p) +� +≤ +K +� +k=C′′ +ϵkH +τ − c0 = ˜O +� H2.5 +τ − c0 +� +|S|2|A|K +� +. +(16) +By optimality of πk and the nature of the update of the dual +parameter λk, we can prove the following lemma: +Lemma 6. +K +� +k=C′′ +E +� +V πϵk,ˆ +pk +1 +(r, ˆpk) − V πk +1 (r, ˆpk) +� += ˜O +� +H +τ − c0 +√ +K +� +The proof of this lemma can be found in the Appendix. +Now, putting together (16), Lemma 5 and lemma 6, we get +that +BR(K; r) = ˜O +� H2.5 +τ − c0 +� +|S|2|A|K +� +. +Remark 2. +We note that we can improve the up- +per bound on BR(K; r) from +˜O +� +H2.5� +|S|2|A|K +� +to ˜O +� +H2.5� +|S||A|K +� +by using the leveraging an im- +proved regret bound (Osband & Van Roy, 2017) i.e., +˜O +� +H1.5� +|S||A|K +� +for the PSRL algorithm and appro- +priate scaling of the ϵk terms. But, this would require an +assumption that the transition probability has an indepen- +dent Dirichlet prior. +7. Experimental Results +In this section, we evaluate the empirical performance of +the Safe PSRL algorithm and compare it with the state- +of-the-art DOPE algorithm (Bura et al., 2022), which has +been shown to perform better than other comparable algo- +rithms (e.g., the OptPess-LP in (Liu et al., 2021a)). The +empirical performance is evaluated with respect to (i) the +objective regret and (ii) the constraint regret. +We consider the setting of a media streaming service (Bura +et al., 2022) from a wireless base station. The base sta- +tion provides the streaming service at two different speeds. +These speeds follow independent Bernoulli distributions +denoted by parameters µ1 = 0.9 and µ2 = 0.1, with µ1 cor- +responding to the faster service. The data packets arriving +at the device are stored in a buffer and sent out according to +a Bernoulli random process with mean γ. The buffer size +sh evolves as sh+1 = min (max (0, sh + Ah − Bh) , N) +where Ah is the number of packet arrivals, Bh is the num- +ber of packet departures, and N = 10 is the maximum size +of the buffer. The device desires to minimize the cost of run- +ning out of packets, i.e., an empty buffer, while restricting +the use of the faster service. We model this scenario as a +finite horizon CMDP with the state representing the buffer +size and actions {1, 2} denoting the choice of speed. We set +the objective cost as r(s, a) = 1{s = 0} and the constraint +cost as c(s, a) = 1{a = 1}. The episode length H is 10 +and the constraint threshold τ is 5. +We evaluate the cumulative regret for the Safe PSRL and +the DOPE algorithm. The transition probability is fixed and +not sampled from a prior distribution. For the Safe PSRL +algorithm, we consider a Dirichlet prior for the transition + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +probability with parameters [0.1, . . . , 0.1]. The Dirichlet +prior is a good choice since it is a conjugate prior for multi- +nomial and categorical distributions. We further scale the ϵk +parameters of the Safe PSRL algorithm by 0.05 to avoid +excessive pessimism. +The performance of our algorithm is compared against the +DOPE algorithm, which requires a known safe policy. We +choose the optimal policy of the given CMDP with a tighter +constraint threshold c0 = 1 as the safe policy. The same c0 +is also used in the Safe PSRL algorithm as the satisfiable +constraint threshold. +The algorithms are evaluated over K = 400, 000 episodes. +All the experiments are performed on a 2019 MacBook Pro +with 1.4 GHz Quad-Core Intel Core i5 processor and 16GB +RAM. +Figure 1. Plots showing (a) cumulative objective regret and (b) +cumulative constraint regret for the Safe PSRL and DOPE algo- +rithms. +Figure 2. Plots showing (a) average objective regret and (b) average +constraint regret for the Safe PSRL and DOPE algorithms. +Fig. 1(a) shows that the Safe PSRL algorithm greatly out- +performs the DOPE algorithm in terms of objective regret. +The objective regret for the DOPE algorithm grows almost +linearly for a very large number of episodes. In compar- +ison, the Safe PSRL attains +√ +K behavior much earlier. +Fig. 2(a) for the average objective regret shows this behavior +more clearly. +In Fig. 1(b), we see that the constraint regrets for both +the Safe PSRL and the DOPE algorithm are negative for +almost all of the episodes. This implies that the constraint +was satisfied in almost all of the episodes and matches with +the theoretical guarantees for both the algorithms. +We observe the initial jumps in Fig. 2(a) and Fig. 2(b) with +respect to the Safe PSRL regret plots because a few initial +policies returned by the Safe PSRL algorithm fail to sat- +isfy the constraint while achieving better reward objective +performance. +This behavior occurs because the dual parameter λ, which +starts from 0, has not yet caught up with the appropriate +value which would ensure optimal objective performance +while satisfying the constraint. We can infer from the regret +plot that this appropriate λ value is reached fairly quickly +by the Safe PSRL algorithm. The DOPE algorithm, on +the other hand, relies on the safe policy for too long before +it starts to explore. +We thus show that the Safe PSRL algorithm is able to +achieve superior objective regret performance while satisfy- +ing the constraint for almost all the episodes. This result is +further achieved without the knowledge of a safe policy. +8. Conclusions +We addressed the problem of safe online learning for +episodic MDPs with constraints and unknown transition +probabilities. The Safe PSRL is the first posterior sam- +pling algorithm that achieves bounded constraint violation +regret while achieving near-optimal cumulative reward re- +gret. The algorithm has better empirical performance than +other state-of-the-art algorithms (e.g., the DOPE algorithm) +for the same setting and does not need to assume knowl- +edge of a safe policy. The algorithm can be extended to +the infinite-horizon setting. Incorporating chance or risk +constraints would be another interesting direction for future +work. + +1e6 +Safe PSRL +1.0 +DOPE +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Episode k +1e51e6 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Safe PSRL +1.75 +DOPE +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Episode k +1e5Average Objective Regret +Safe PSRL +-2 +DOPE +-3 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Episode k +1e5Safe PSRL +DOPE +2 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Episode k +1e5Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +References +Agarwal, M., Bai, Q., and Aggarwal, V. Regret guarantees +for model-based reinforcement learning with long-term +average constraints. In The 38th Conference on Uncer- +tainty in Artificial Intelligence, 2022. +Agrawal, S. and Goyal, N. Analysis of thompson sampling +for the multi-armed bandit problem. In Conference on +learning theory, pp. 39–1. JMLR Workshop and Confer- +ence Proceedings, 2012. +Agrawal, S. and Goyal, N. Thompson sampling for contex- +tual bandits with linear payoffs. In International Confer- +ence on Machine Learning, pp. 127–135. PMLR, 2013. +Altman, E. Constrained Markov Decision Processes, vol- +ume 7. CRC Press, 1999. +Amani, S., Alizadeh, M., and Thrampoulidis, C. Linear +stochastic bandits under safety constraints. Advances in +Neural Information Processing Systems, 32, 2019. +Auer, P. and Ortner, R. Online regret bounds for a new +reinforcement learning algorithm. In Proceedings 1st +Austrian Cognitive Vision Workshop, 2005. +Azar, M. G., Osband, I., and Munos, R. Minimax regret +bounds for reinforcement learning. In Proceedings of +the 34th International Conference on Machine Learning- +Volume 70, pp. 263–272. JMLR. org, 2017. +Bai, Q., Bedi, A. S., Agarwal, M., Koppel, A., and Ag- +garwal, V. Achieving zero constraint violation for con- +strained reinforcement learning via primal-dual approach. +In Proceedings of the AAAI Conference on Artificial In- +telligence, volume 36, pp. 3682–3689, 2022. +Brantley, K., Dudik, M., Lykouris, T., Miryoosefi, S., Sim- +chowitz, M., Slivkins, A., and Sun, W. +Constrained +episodic reinforcement learning in concave-convex and +knapsack settings. Advances in Neural Information Pro- +cessing Systems, 33:16315–16326, 2020. +Bura, A., Hasanzadezonuzy, A., Kalathil, D., Shakkottai, +S., and Chamberland, J.-F. Dope: Doubly optimistic and +pessimistic exploration for safe reinforcement learning. +In Advances in Neural Information Processing Systems, +2022. +Chapelle, O. and Li, L. An empirical evaluation of thompson +sampling. Advances in neural information processing +systems, 24:2249–2257, 2011. +Chen, L., Jain, R., and Luo, H. Learning infinite-horizon +average-reward markov decision processes with con- +straints. arXiv preprint arXiv:2202.00150, 2022. +Ding, D., Zhang, K., Basar, T., and Jovanovic, M. Natu- +ral policy gradient primal-dual method for constrained +markov decision processes. Advances in Neural Informa- +tion Processing Systems, 33:8378–8390, 2020. +Ding, D., Wei, X., Yang, Z., Wang, Z., and Jovanovic, M. +Provably efficient safe exploration via primal-dual policy +optimization. In International Conference on Artificial +Intelligence and Statistics, pp. 3304–3312. PMLR, 2021. +Efroni, Y., Mannor, S., and Pirotta, M. +Exploration- +exploitation in constrained mdps. +arXiv preprint +arXiv:2003.02189, 2020. +Gopalan, A. and Mannor, S. Thompson sampling for learn- +ing parameterized markov decision processes. In Confer- +ence on Learning Theory, pp. 861–898. PMLR, 2015. +HasanzadeZonuzy, A., Bura, A., Kalathil, D., and Shakkot- +tai, S. Learning with safety constraints: Sample complex- +ity of reinforcement learning for constrained mdps. In +Proceedings of the AAAI Conference on Artificial Intelli- +gence, volume 35, pp. 7667–7674, 2021. +Jafarnia-Jahromi, M., Chen, L., Jain, R., and Luo, H. Online +learning for stochastic shortest path model via posterior +sampling. arXiv preprint arXiv:2106.05335, 2021a. +Jafarnia-Jahromi, M., Jain, R., and Nayyar, A. Learning +zero-sum stochastic games with posterior sampling. arXiv +preprint arXiv:2109.03396, 2021b. +Jafarnia-Jahromi, M., Jain, R., and Nayyar, A. +Online +learning for unknown partially observable mdps. arXiv +preprint arXiv:2102.12661, 2021c. +Jaksch, T., Ortner, R., and Auer, P. Near-optimal regret +bounds for reinforcement learning. Journal of Machine +Learning Research, 11(Apr):1563–1600, 2010. +Jin, C., Allen-Zhu, Z., Bubeck, S., and Jordan, M. I. Is +Q-learning provably efficient? In Advances in Neural +Information Processing Systems, pp. 4863–4873, 2018. +Kalagarla, K. C., Jain, R., and Nuzzo, P. +A sample- +efficient algorithm for episodic finite-horizon mdp with +constraints. In Proceedings of the AAAI Conference on +Artificial Intelligence, volume 35, pp. 8030–8037, 2021. +Kaufmann, E., Korda, N., and Munos, R. Thompson sam- +pling: An asymptotically optimal finite-time analysis. In +International conference on algorithmic learning theory, +pp. 199–213. Springer, 2012. +Khezeli, K. and Bitar, E. Safe linear stochastic bandits. In +Proceedings of the AAAI Conference on Artificial Intelli- +gence, volume 34, pp. 10202–10209, 2020. + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +Lai, T. L. and Robbins, H. Asymptotically efficient adaptive +allocation rules. Advances in applied mathematics, 6(1): +4–22, 1985. +Liu, T., Zhou, R., Kalathil, D., Kumar, P., and Tian, C. +Learning policies with zero or bounded constraint viola- +tion for constrained mdps. Advances in Neural Informa- +tion Processing Systems, 34:17183–17193, 2021a. +Liu, X., Li, B., Shi, P., and Ying, L. An efficient pessimistic- +optimistic algorithm for stochastic linear bandits with +general constraints. +Advances in Neural Information +Processing Systems, 34:24075–24086, 2021b. +Osband, I. and Van Roy, B. Why is posterior sampling +better than optimism for reinforcement learning? +In +International Conference on Machine Learning, pp. 2701– +2710. PMLR, 2017. +Osband, I., Russo, D., and Van Roy, B. (more) efficient +reinforcement learning via posterior sampling. Advances +in Neural Information Processing Systems, 26, 2013. +Ouyang, Y., Gagrani, M., Nayyar, A., and Jain, R. Learn- +ing unknown markov decision processes: A thompson +sampling approach. In Advances in Neural Information +Processing Systems, pp. 1333–1342, 2017. +Pacchiano, A., Ghavamzadeh, M., Bartlett, P., and Jiang, +H. Stochastic bandits with linear constraints. In Interna- +tional Conference on Artificial Intelligence and Statistics, +pp. 2827–2835. PMLR, 2021. +Puterman, M. L. Markov Decision Processes: Discrete +Stochastic Dynamic Programming. John Wiley & Sons, +Inc., New York, NY, USA, 1st edition, 1994. +ISBN +0471619779. +Qiu, S., Wei, X., Yang, Z., Ye, J., and Wang, Z. Upper +confidence primal-dual reinforcement learning for cmdp +with adversarial loss. Advances in Neural Information +Processing Systems, 33:15277–15287, 2020. +Singh, R., Gupta, A., and Shroff, N. B. Learning in markov +decision processes under constraints. +arXiv preprint +arXiv:2002.12435, 2020. +Thompson, W. R. On the likelihood that one unknown +probability exceeds another in view of the evidence of +two samples. Biometrika, 25(3/4):285–294, 1933. +Wei, C.-Y., Jafarnia-Jahromi, M., Luo, H., Sharma, H., and +Jain, R. Model-free reinforcement learning in infinite- +horizon average-reward markov decision processes. In In- +ternational Conference on Machine Learning, pp. 10170– +10180. PMLR, 2020. +Wei, H., Liu, X., and Ying, L. Triple-q: A model-free +algorithm for constrained reinforcement learning with +sublinear regret and zero constraint violation. In Interna- +tional Conference on Artificial Intelligence and Statistics, +pp. 3274–3307. PMLR, 2022. +Zheng, L. and Ratliff, L. Constrained upper confidence +reinforcement learning. In Learning for Dynamics and +Control, pp. 620–629. PMLR, 2020. + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation +A. Proofs +A.1. Proof of Lemma 4 +Proof. Now for k ≥ C′′, consider: +λk+1 +2 +2 +− λk +2 +2 += λk(λk+1 − λk) + 1 +2(λk+1 − λk)2 += λk(V πk +1 (c, ˆpk) + ϵk − τ) + 1 +2(V πk +1 (c, ˆpk) + ϵk − τ)2 += λk(V πk +1 (c, ˆpk) + ϵk − τ) − ηkV πk +1 (r, ˆpk) ++ ηkV πk +1 (r, ˆpk) + 1 +2(V πk +1 (c, ˆpk) + ϵk − τ)2 +≤ λk(V πk +1 (c, ˆpk) + ϵk − τ) − ηkV πk +1 (r, ˆpk) ++ ηkH + 1 +2(V πk +1 (c, ˆpk) + ϵk − τ)2 +≤ λk(V πk +1 (c, ˆpk) + ϵk − τ) − ηkV πk +1 (r, ˆpk) ++ ηkH + (V πk +1 (c, ˆpk) − τ)2 + ϵ2 +k +( Using (a + b)2 +2 +≤ a2 + b2) +≤ λk(V πk +1 (c, ˆpk) + ϵk − τ) − ηkV πk +1 (r, ˆpk) ++ ηkH + H2 + ϵ2 +k +≤ λk(V +π +ˆ +pk +0 +1 +(c, ˆpk) + ϵk − τ) − ηkV +π +ˆ +pk +0 +1 +(r, ˆpk) ++ ηkH + H2 + ϵ2 +k +( By optimality of πk in primal update ) +≤ λk(c0 + ϵk − τ) + ηkH + H2 + ϵ2 +k +≤ −λk(τ − c0) +2 ++ ηkH + H2 + ϵ2 +k +( as for k ≥ C′′, ϵk ≤ (τ − c0) +2 +) +Now for λ ≥ ϕk where ϕk := 4(H2 +ϵ2 +k +ηkH)/(τ −c0), +we have: +E [λk+1 − λk|λk = λ] ≤ E +�λ2 +k+1 − λ2 +k +2λk +|λk = λ +� +(Using x − y ≤ x2 − y2 +2y +, for y > 0) += 1 +λE +�λ2 +k+1 − λ2 +k +2 +|λk = λ +� +≤ 1 +λE +� +−λk(τ − c0) +2 ++ ηkH + H2 + ϵ2 +k|λk = λ +� += −(τ − c0) +2 ++ ηkH + H2 + ϵ2 +k +λ +≤ −(τ − c0) +2 ++ (τ − c0) +4 += −(τ − c0) +4 +:= ρ +Further, |λk+1 − λk| = |V πk +1 (c, ˆpk) + ϵk − τ| ≤ H with +probability 1. Thus, by lemma 3, we have : +E +� +eζλK+1� +≤ E +� +eζλC′′ � ++ 2eζ(H+ϕK+1) +ζρ +, +where ζ = ρ/(H2 + Hρ/3). +=⇒ eζE[λK+1] ≤ E +� +eζλC′′ � ++ 2eζ(H+ϕK+1) +ζρ +(By Jensen’s inequality) +=⇒ E [λK+1] ≤ 1 +ζ log +� +E +� +eζλC′′ � ++ 2eζ(H+ϕK+1) +ζρ +� +Further, +λC′′ ≤ λ1 + +C′′−1 +� +1 +(V πk +1 +(c, ˆpk) + ϵk − τ)+ +≤ +C′′ +� +1 +ϵk + C′′(H − τ) +:= λmax +C′′ +Continuing, +E [λK+1] +≤ 1 +ζ log +� +eζλmax +C′′ + 2eζ(H+ϕK+1) +ζρ +� +≤ 1 +ζ log +� +eζλmax +C′′ + 8H2eζ(H+ϕK+1) +3ρ2 +� +( Using ζ ≥ 3(τ − c0) +13H2 +) +≤ 1 +ζ log +�11H2 +3ρ2 eζ(H+ϕK+1+λmax +C′′ ) +� += 1 +ζ log 11H2 +3ρ2 ++ H + ϕK+1 + λmax +C′′ += 1 +ζ log 11H2 +3ρ2 ++ H + +C′′ +� +1 +ϵk + C′′(H − τ) ++ 4(H2 + ϵ2 +K+1 + ηK+1H) +(τ − c0) +A.2. Proof of Lemma 6 +Proof. +K +� +k=C′′ +E +� +V πϵk,ˆ +pk +1 +(r, ˆpk) − V πk +1 (r, ˆpk) +� + +Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation += +K +� +k=C′′ +E +�λk +ηk +� +V πϵk,ˆ +pk +1 +(c, ˆpk) − V πk +1 (c, ˆpk) +�� ++ +K +� +k=C′′ +E +�� +V πϵk,ˆ +pk +1 +(r, ˆpk) − λk +ηk +V πϵk,ˆ +pk +1 +(c, ˆpk) +�� +− +K +� +k=C′′ +E +�� +V πk +1 +(r, ˆpk) − λk +ηk +V πk +1 +(c, ˆpk) +�� +≤ +K +� +k=C′′ +E +�λk +ηk +� +V πϵk,ˆ +pk +1 +(c, ˆpk) − V πk +1 (c, ˆpk) +�� ++ 0 +( By optimality of πk in primal update ) +≤ +K +� +k=C′′ +E +�λk +ηk +(τ − ϵk − V πk +1 (c, ˆpk)) +� +≤ +K +� +k=C′′ +E +� 1 +ηk +((λk(λk+1 − λk) + τ 2) +� +(By update rule for λk) +≤ E +� +K +� +k=C′′ +1 +ηk +(λ2 +k +2 − λ2 +k+1 +2 +) + +K +� +k=C′′ +1 +2ηk +(λk+1 − λk)2 ++ +K +� +k=C′′ +τ 2 +ηk +� +≤ E +�(λC′′)2 +2ηC′′ +� ++ +K +� +k=C′′ +H2 +2ηk ++ +K +� +k=C′′ +H2 +ηk +(As ηk increases with k) +≤ (�C′′ +k=1 ϵk + C +′′(H − τ))2 +2ηC′′ ++ 3H +2 +K +� +C′′ +1 +(τ − c0) +√ +k += ˜O +� +H +τ − c0 +√ +K +� + diff --git a/tdFJT4oBgHgl3EQfdSw-/content/tmp_files/load_file.txt b/tdFJT4oBgHgl3EQfdSw-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8ffbaafb310d35396a19c2d6c25f4f5837db6ce --- /dev/null +++ b/tdFJT4oBgHgl3EQfdSw-/content/tmp_files/load_file.txt @@ -0,0 +1,836 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf,len=835 +page_content='Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation Krishna C Kalagarla, Rahul Jain, Pierluigi Nuzzo Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles Email: kalagarl,rahul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='jain,nuzzo@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='edu Abstract Constrained Markov decision processes (CMDPs) model scenarios of sequential decision making with multiple objectives that are increasingly im- portant in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' However, the model is often unknown and must be learned online while still ensuring the constraint is met, or at least the violation is bounded with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Some recent papers have made progress on this very challenging problem but either need unsatisfac- tory assumptions such as knowledge of a safe policy, or have high cumulative regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We pro- pose the Safe PSRL (posterior sampling-based RL) algorithm that does not need such assump- tions and yet performs very well, both in terms of theoretical regret bounds as well as empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The algorithm achieves an efficient tradeoff be- tween exploration and exploitation by use of the posterior sampling principle, and provably suffers only bounded constraint violation by leveraging the idea of pessimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Our approach is based on a primal-dual approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We establish a sub- linear ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S|2|A|K � upper bound on the Bayesian reward objective regret along with a bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', ˜O (1) constraint violation regret over K episodes for an |S|-state, |A|-action, and horizon H CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Introduction Markov decision processes (MDPs) (Puterman, 1994) are used to model many scenarios involving sequential decision- making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' They are used in a wide variety of settings like robotics, cyber-physical systems, and safety-critical au- tonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' However, a traditional reinforcement learning (RL) formulation which seeks to maximize a sin- gle cumulative reward cannot capture many problems that have multiple objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' For example, this is the case for a robot that needs to perform a certain reward-yielding task while ensuring that the average energy expended is bounded below a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Such scenarios are well-modeled by con- strained MDPs (CMDPs) (Altman, 1999), which extend the MDP formalism by considering additional constraints on the expected cumulative performance of a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In the CMDP setting, one seeks to find an optimal policy which maximizes the cumulative objective reward while satisfying constraints on cost objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In this paper, we consider the problem of online learning for finite-horizon CMDPs, where an agent interacts with the environment repeatedly in episodes of fixed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The transition probability is not known to the agent, thereby requiring the agent to learn about the system dynamics by observing the past states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The performance of this agent is measured by the notion of cumulative regret, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', the difference between the cumulative reward of the learning agent and that of the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This online learning problem thus leads to the well-known trade-off between exploration and exploitation: should the agent explore the environment to improve future perfor- mance, or exploit the current knowledge for better short- term performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' A common approach to balance this exploration-exploitation trade-off is the ‘Optimism in the Face of Uncertainty’ (OFU) principle (Lai & Robbins, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The idea is that each state and action are assigned optimism bonuses based on the current knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The agent then chooses a policy which maximizes the expected return under this optimistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The bonuses are designed to promote exploration of poorly-understood state-action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This approach has been widely used for online learning in MDPs (Jaksch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Azar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Kalagarla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Another alternative for efficient exploration is posterior sam- pling (also called Thompson sampling) (Thompson, 1933).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In this approach, a posterior distribution is maintained over the unknown transition probability model based on the prior distribution and dataset about visited trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' At the beginning of each episode, a model is sampled from this pos- terior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The agent then chooses a policy which arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='11547v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='LG] 27 Jan 2023 Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation is optimal with respect to the sampled model and follows it for the duration of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The advantages of poste- rior sampling over OFU stem from the fact that (i) known information about the model can be incorporated into the algorithm through the prior distribution, and (ii) posterior sampling algorithms have demonstrated superior empirical performance for online learning over OFU-type algorithms including in the RL setting (Osband et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In this paper, we use the posterior sampling approach and in- troduce the Safe PSRL algorithm for efficient exploration in the finite-horizon CMDP setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Our algorithm uses the primal-dual approach for CMDPs wherein the primal part performs unconstrained MDP planning with a sampled tran- sition probability, and the dual part updates the Lagrangian variable to track the constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We achieve bounded constraint violation regret by leverag- ing the idea of pessimism, introduced earlier in the context of constrained bandits (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' “Pessimism” is achieved by tightening the constraint of the CMDP problem in every episode at decreasing levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The key is, however, how this tightening is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' By appropriately balanc- ing exploration via posterior sampling and safe learning via pessimism, we show that the Safe PSRL algorithm achieves sub-linear ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5 τ−c0 � |S|2|A|K � reward regret while achieving bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', ˜O(1)-constraint violation re- gret for an |S|-state, |A|-action, and an H episode length CMDP over K number of episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' τ denotes the desired threshold for constraint violation, and c0 is a known feasible expected cumulative constraint cost of the CMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The contributions of this paper are the following: (i) We present the first PSRL algorithm for CMDPs that not only achieves ˜O �√ K � objective reward regret but also ˜O(1) constraint violation regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Unlike other proposed algorithms, our algorithm does not need knowledge of a constraint-satisfying safe policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (ii) Our Safe PSRL al- gorithm has better empirical performance than state-of-the- art OFU-type algorithms for the same setting introduced in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Bura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) which need knowledge of a safe policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (iii) The algorithm design is simpler than other state-of-the-art algorithms (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Bura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) for the problem: The key design choice are two pessimism parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The regret analysis involves a novel decomposition which allows us to leverage posterior sampling regret analysis and Lyapunov-drift analysis for the dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Related Work Posterior (or Thompson) sampling goes back to the work of (Thompson, 1933), but attracted less attention for sev- eral decades until empirical evidence (Chapelle & Li, 2011) showed its superior performance for online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Re- cently, it has been widely applied to various settings like multi-armed bandits (Kaufmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Agrawal & Goyal, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2013), MDPs (Osband et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Gopalan & Mannor, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Osband & Van Roy, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2017) and POMDPs (Jafarnia-Jahromi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Multi-armed bandits (MAB) are a special case of MDPs with a single state and unknown reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Safe online learning has been studied for MABs in multiple set- tings (Amani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Khezeli & Bitar, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Pacchiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' However, in comparison, the multi-step and multi-state MDP settings are more challeng- ing due to the unknown transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In the CMDP setting, several existing works (Efroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) leverage OFU or posterior sampling to provide ˜O( √ K) regret for the re- ward as well as the constraint objective, where K is the number of episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Such an approach, however, can lead to a large number of constraint violations during learning, which is unacceptable during various safety-critical tasks such as driving or power distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thus, the problem of an online RL algorithm that has sublinear (and hopefully, ˜O( √ K)) reward regret while achieving bounded constraint violation regret, particularly one that also has very good empirical performance, remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' OFU-based algorithms have been widely used for efficient learning in CMDPs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', in the setting of PAC performance guarantees for finite-horizon CMDPs (HasanzadeZonuzy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Kalagarla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021), or to provide regret bounds for CMDPs in the finite-horizon setting (Efroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Brantley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020) and infinite-horizon average cost setting (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Policy gradient algorithms for CMDPs (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2021) have also been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' However, these algorithms do not provide bounded or zero constraint violation guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Recently, some OFU-based approaches for safe learning with bounded or zero constraint violation guarantees have been proposed (Zheng & Ratliff, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' But these either assume the transition model is known (but re- ward function is not), or only satisfy the constraint with high probability, or assume that a safe policy is known to the algo- rithm (and can be used by it), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Bura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The OptPess-PrimalDual algorithm in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a) is the closest comparable algorithm but our Safe PSRL algorithm is better in terms of its dependence on various problem parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', it has ˜O(H3|S|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5) dependence as opposed to ˜O(H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5|S|) for ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' While the use of the posterior sampling principle for con- strained RL problems is under-explored (despite the promise Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation of better empirical performance), (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) in- deed introduces a PSRL algorithm for CMDPs but for the average setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Moreover, it only achieves a ˜O( √ K) con- straint violation regret which is worse than our ˜O(1) bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Notation We denote the probability simplex over set S by ∆S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We use the notation ˜O which has similar meaning as the usual O notation but ignores logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Finite-Horizon MDPs An episodic finite-horizon MDP (Puterman, 1994) can be formally defined by a tuple M = (S, A, H, s1, p, r), where S and A denote the state and action spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In this setting, the agent interacts with the environment in episodes of fixed length H, with each episode starting with a random initial state denoted s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The non-stationary transition probability ph(s′|s, a) is the the probability of transitioning to state s′ on taking action a at state s at time step h ∈ [1 : H] of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The non-stationary reward obtained on taking action a in state s at time step h of an episode is denoted by a random variable Rh(s, a) ∈ [0, 1], with mean rh(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We use r as a shorthand to denote the mean reward vector r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' , rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' A non-stationary randomized policy π = (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' , πH) ∈ Π where πi : S → ∆A, maps each state to a probability simplex over the action space A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The action ah at time step h at state sh is taken according to the policy π, ah ∼ πh(sh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The value function of a non-stationary randomized policy π, V π h (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) (when clear, s, r, and p are omitted) at a state s ∈ S and time step h ∈ [1 : H] is defined as: V π h (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) := Eπ � H � i=h ri(si, ai)|sh = s, p � , where the expectation is over the distribution induced by the environment and policy randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Similarly, the Q-value function of a policy π, Qπ h(s, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p), for a state s ∈ S, an action a ∈ A and time step h ∈ [1 : H], is defined as Qπ h(s, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) := rh(s, a)+ (1) Eπ � H � i=h+1 ri(si, ai)|sh = s, ah = a, p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We can always find an optimal non-stationary deterministic policy ˜π (Puterman, 1994) such that V ˜π h (s) = ˜Vh(s) = supπV π h (s) and Q˜π h(s, a) = ˜Qh(s, a) = supπQπ h(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The optimal policy can be computed by using backward induction on the Bellman optimality equations (Puterman, 1994): ˜ Vh(s) = maxa∈A � rh(s, a) + ph(·|s, a) ˜Vh+1 � , ˜Qh(s, a) = rh(s, a) + ph(·|s, a) ˜Vh+1, (2) where ˜VH+1(s) = 0 and ˜Vh(s) = maxa∈A ˜Qh(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The optimal policy ˜π is then greedy with respect to ˜Qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Finite-Horizon Constrained MDPs A finite-horizon constrained MDP (CMDP) (Altman, 1999) is a finite-horizon MDP with a required upper bound on expectation of on a cost function, {c, τ ∈ (0, H]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The non-stationary cost obtained on taking action a in state s at time step h ∈ [1 : H] with respect to the constraint cost function is denoted by a random variable Ch(s, a) ∈ [0, 1], with mean ch(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Similar to r, we use c to denote the mean cost vector c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' , cH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The total expected reward (cost) of an episode under pol- icy π with respect to the reward (cost) function r (c) is the respective value function from the initial state s1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', V π 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p)(V π 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c, p)) (by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Our objective in this CMDP setting is to find a policy which maximizes the total expected objective reward under the constraint that the total expected constraint cost is below a desired threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Formally, π∗ ∈ argmax π∈Π V π 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' V π 1 (c, p) ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (3) The optimal value is denoted by V ∗(s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) = V π∗ 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' A deterministic optimal policy may not exist, hence we need to consider Π to be the class of all random- ized policies (Altman, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Since the Bellman optimality equations may not hold due to the constrained nature of the problem, we cannot leverage dynamic programming-based backward induction algorithms to find an optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' However, a linear programming approach can be given that will find an optimal policy (Altman, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The Learning Problem We consider the setting where an agent repeatedly interacts with a finite-horizon CMDP M = (S, A, H, s1, p, r, {c, τ}) over multiple episodes of fixed length H, starting each episode from the same initial state s1 and with stationary transition probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', ph = p, ∀h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We employ the Bayesian framework and regard the transition probability p as random with a prior distribution µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The realized tran- sition probability is unknown to the learning agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We consider finite-horizon CMDP whose transition probability lies in the set Θc0 with the following property: Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' For all ˆp ∈ Θc0, there exists a policy π ˆp 0 such that V π ˆ p 0 1 (c, ˆp) ≤ c0 < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Moreover, we assume that the support of the prior distribu- tion µ1 is a subset of Θc0 and c0 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Without loss of generality,1 we assume that the reward and cost functions r and c are respectively are known to the learning agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Note that the above assumption is not only reasonable but also necessary to ensure the problem is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The agent interacts with the environment for K episodes, each of length H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In each episode, the agent starts from a state s1 and chooses a Markov policy πk determined by the information gathered until that episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This policy is then executed until the end of the episode, while collecting the rewards and costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The main objectives of the learning agent are to: (1) Maximize the expected cumulative reward after K episodes or equivalently, minimize the Bayesian regret with respect to the reward function defined as: BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) := E � K � k=1 � V π∗ 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) − V πk 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r, p) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (4) (2) Minimize the constraint violation or equivalently, min- imize the Bayesian regret with respect to the constraint defined as: BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c) := E � K � k=1 � V πk 1 (s1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c, p) − τ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (5) With respect to these objectives, we propose an algorithm which is able to achieve sub-linear regret with respect to the reward objective while ensuring that regret with respect to the cost constraint is bounded above by a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', independent of the number of episodes K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The Safe PSRL Algorithm We propose the Safe Posterior Sampling-based Reinforce- ment Learning (Safe PSRL) algorithm for the finite- horizon CMDP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This algorithm leverages the idea of posterior sampling to balance exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' It also takes a primal-dual approach to handle the constraint cost objective along with reward maximization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We further introduce the idea of pessimism (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021b) to ensure that the cost regret is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This “pes- simism” is achieved by considering a “more constrained” 1The complexity of learning the cost and reward functions is dominated by the complexity of learning the transition probability (Auer & Ortner, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The algorithm can be readily extended to the setting of unknown cost and reward functions by using their empirical estimate in place of the known cost and reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' CMDP problem as compared to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This is done by decreasing the threshold by ϵk in each episode k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Formally, we consider the objective: max V π 1 (r, p) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' V π 1 (c, p) ≤ τ − ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (6) This pessimistic term ϵk ensures bounded cost regret and it decreases as the episode count increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The algorithm starts with the prior distribution µ1 on the transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Then, at every time step t, the learning agent maintains a posterior distribution µt on the unknown transition probability p given by µt(Θ) = P(p ∈ Θ|Ft) for any set Θ ⊆ Θc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Here Ft is the information available at time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', the sigma algebra generated by encountered states and actions upto time t, (s1, a1, · · · , st−1, at−1, st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' On observing the next state st+1 by taking action at at state st, the posterior is updated according to Bayes’s rule: µt+1(dp) = pt(st+1|st, at)µt(dp) � p ′ t(st+1|st, at)µt(dp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (7) In parallel, the algorithm proceeds as follows: At the begin- ning of each episode k, transition probability ˆpk is sampled from the posterior distribution µtk (where tk is the time step corresponding to beginning of episode k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We then consider the Lagrangian defined as: Lk(π, λ) := V π 1 (r, ˆpk) + λk ηk (τ − ϵk − V π 1 (c, ˆpk)) , The learning agent then chooses a Markov policy πk (pri- mal update) which maximizes the above Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We can find such a policy by applying standard dynamic pro- gramming with respect to the reward function r − λk ηk c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The (dual) parameter λk is updated according to the sub-gradient algorithm as follows: λk+1 = (λk + V πk 1 (c, ˆpk) + ϵk − τ)+ The agent then applies the policy πk for the H steps of episode k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We note that while some of the details of the algorithm are natural (as they are common to PSRL algorithms for various settings) (Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jafarnia-Jahromi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='b), the key novelty in the design are the ϵk and ηk parameters to be used in conjunction with a primal- dual approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Their choice is guided by the regret analysis presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The Safe PSRL algorithm is summarized next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation Algorithm 1 Safe-PSRL Input: K, µ1, c0, τ Initialization: λ1 ← 0 for episodes k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' , K do ϵk ← 5|H|1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5√ |S|2|A|(log k|S||A|H+1) √ k log k|S||A|H ηk ← (τ − c0)H √ k tk = (k − 1)H + 1 Generate ˆpk ∼ µtk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=') Compute πk ∈ arg maxπ V π 1 (r − λk ηk c, ˆpk) according to (2) (Policy Update) λk+1 ← max(0, λk + V πk 1 (c, ˆpk) + ϵk − τ) (Dual Update) for t = (k − 1)H + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' , kH do Choose action at ∼ πk(st) Observe st+1 ∼ p(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='|st, at) Update the posterior distribution µt+1 according to (7) end for end for The following theorem then establishes that the Safe PSRL algorithm can achieve sub-linear ˜O( √ K) reward re- gret while achieving bounded constraint violation regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Suppose Assumption 1 holds, then the reward and cost regret of the Safe PSRL algorithm is upper bounded as: BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) = ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5 τ − c0 � |S|2|A|K � , and BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c) = ˜O � C′′(H − τ) + H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S|2|A|C′′ � = O(1), where C′′ = O( H3|S|2|A| (τ−c0)2 ) is independent of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (i) We note that the upper bound on BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) of the OptPess-PrimalDual algorithm (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a) is ˜O � H3� |S|3|A|K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thus, our upper bound is the same in terms of |A|,K and better in terms of |S| and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Both, OptPess-PrimalDual and Safe PSRL algorithms achieve ˜O (1) upper bounds on BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (ii) We note that the upper bound on BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) of the DOPE algorithm (Bura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) is ˜O � H3� |S|2|A|K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thus, our upper bound is the same in terms of |A|, K, |S| and better in terms of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' While our bounds are comparable to those of DOPE, we shall see that the numerical performance is much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Further, the DOPE algorithm guarantees zero constraint violations with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' But, this requires a strong assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e, knowledge of a safe policy that can satisfy the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (iii) The CMDP-PSRL algorithm (Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) uses posterior sampling in the average CMDP setting and achieves ˜O � TM|S| � |A|K � reward objective and the same constraint violation regret, where TM is the mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In comparison, we are able to achieve bounded constraint violation regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Regret Analysis We now provide theoretical analysis of the Safe PSRL algorithm by providing details of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We first state some relevant results from the literature on posterior sampling in the context of reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' A key property of posterior sampling (Osband et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2013) is the posterior sampling lemma, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', the transition probabil- ity ˆpt sampled from the posterior distribution at time t and transition probability p have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' For any function f, we have E [f(ˆpt)] = E [f(p)] where p is the transition probability (with the prior distribution µ1) and ˆpt is the sampled transition probability from the posterior distribution µt at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The following is a restatement (Osband et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2013) of the sub-linear regret bound achieved when using posterior sampling for unconstrained finite horizon MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (Osband et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2013) The Bayesian regret of the PSRL algorithm for unconstrained MDPs is given by K � k=1 E � V πk 1 (c, p) − V πk 1 (c, ˆpk) � ≤ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� 30|S|2|A|K log(|S||A|KH) + 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (8) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Cost Constraint Violation Analysis We first present analysis of the cost constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We can decompose the constraint violation regret as follows: BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c) := E � K � k=1 � V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − τ �� = K � k=1 E � V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − τ � = K � k=1 E � V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + K � k=1 E � V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − τ � ≤ K � k=1 E � V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + K � k=1 E [λk+1 − λk − ϵk] (by dual update rule of algorithm) = K � k=1 E � V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + E [λK+1] − K � k=1 ϵk (9) Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation ≤ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� 30|S|2|A|K log(|S||A|KH) + 2H + E [λK+1] − K � k=1 ϵk (10) where the last upper bound follows by use of Lemma 2 to upper bound the first term in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We next show that the dual parameter E [λK+1] can be upper bounded by use of Lyapunov-drift analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' To that end, we restate the following lemma (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021b) which states the Lyapunov-drift conditions for the boundedness of a random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021b) Consider a random process S(t) with a Lyapunov function Φ(k) such that Φ(0) = 0 and ∆(k) = Φ(k + 1) − Φ(k) is the Lyapunov drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Given an increasing sequence {ϕk} and constants ρ and νmax with 0 < ρ ≤ νmax, if the expected drift E [∆(k)|S(k) = s] sat- isfies the following conditions: (i) There exists constants ρ > 0 and ϕk > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' E [∆(k)|S(k) = s] ≤ −ρ when Φ(k) ≥ ϕk, and (ii) |Φ(k+1)−Φ(k)| ≤ νmax holds with probability 1, then E � eζΦ(t)� ≤ E � eζΦ0� + 2eζ(νmax+ϕt) ζρ , where ζ = ρ/(ν2 max + νmaxρ/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We divide the episodes into two parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' k < C′′ and k ≥ C′′ where C′′ = 80H3|S|2|A| (τ−c0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We can clearly see that for k ≥ C′′, we have ϵk ≤ τ−c0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thus, for k ≥ C′′, Problem (6) is feasible for all ˆpk ∈ Θc0 by Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' For k ≥ C′′, we show that the Lyapunov function Φ(λ) = λ satisfies the conditions of Lemma 3 and thus provide a bound on the exponential moment of the dual variable λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' For k ≥ C′′, when λ ≥ ϕk, we have, E [λk+1 − λk|λk = λ] ≤ ρ and |λk+1 − λk| ≤ H with probability 1, where ϕk := 4(H2 + ϵ2 k + ηkH)/(τ − c0) and ρ := −(τ − c0)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thus, we have, E � eζλK+1� ≤ E � eζλC′′ � + 2eζ(H+ϕK+1) ζρ , (11) where ζ = ρ/(H2 + Hρ/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The above inequality (11) can be simplified to E [λK+1] ≤ 1 ζ log 11H2 3ρ2 + H + C′′ � 1 ϵk + C′′(H − τ) +4(H2 + ϵ2 K+1 + ηK+1H) (τ − c0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (12) Next, we bound the sumkϵk term: K � k=1 ϵk ≥ � K+1 1 ϵudu ≥ 10H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S|2|A|Klog|S||A|HK − 10H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S|2|A|log|S||A|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (13) Thus, putting together (10), (12) and (13), the leading terms of ˜O( √ K) cancel out and we get BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' c) = ˜O � C′′(H − τ) + H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S|2|A|C′′ � = ˜O(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', constraint violation regret is a constant, and does not grow with K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Reward Objective Regret Analysis We next provide regret analysis of the reward objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Let πϵk,∗ be the optimal policy for the pessimistic optimization problem (where p is the true transition probability of the MDP): max V π 1 (r, p) (14) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' V π 1 (c, p) ≤ τ − ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Let πϵk,ˆpk be the optimal policy for the pessimistic optimiza- tion problem (where ˆpk is the sampled transition probability of the MDP): max V π 1 (r, ˆpk) (15) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' V π 1 (c, ˆpk) ≤ τ − ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We can decompose the reward regret term as follows: BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) = E � K � k=1 � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) �� = C ′′−1 � k=1 E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � + K � k=C′′ E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � (splitting the sum across the sets of episodes) = C ′′−1 � k=1 E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � + K � k=C′′ E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � + K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation + K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + K � k=C′′ E [V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p)] (splitting the second sum into four parts) ≤ C ′′H + K � k=C′′ E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � + K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + K � k=C′′ E [V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p)] ≤ C ′′H + K � k=C′′ E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � + 0 + K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + K � k=C′′ E [V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p)] (by the posterior sampling property in Lemma 1) ≤ C ′′H + K � k=C′′ E � V π∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) − V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='∗ 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' p) � + K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � + H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� 30|S|2|A|K log(|S||A|KH) + 2H (by the regret bound in Lemma 2) The other terms are bounded as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Similar to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='7 in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a), we can define a probabilistic mixed policy of π∗ and πp 0 to prove the following lemma: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The first summation term above can be bounded as K � k=C′′ E � V π∗ 1 (r, p) − V πϵk,∗ 1 (r, p) � ≤ K � k=C′′ ϵkH τ − c0 = ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5 τ − c0 � |S|2|A|K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (16) By optimality of πk and the nature of the update of the dual parameter λk, we can prove the following lemma: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' K � k=C′′ E � V πϵk,ˆ pk 1 (r, ˆpk) − V πk 1 (r, ˆpk) � = ˜O � H τ − c0 √ K � The proof of this lemma can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Now, putting together (16), Lemma 5 and lemma 6, we get that BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) = ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5 τ − c0 � |S|2|A|K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We note that we can improve the up- per bound on BR(K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' r) from ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S|2|A|K � to ˜O � H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S||A|K � by using the leveraging an im- proved regret bound (Osband & Van Roy, 2017) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', ˜O � H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5� |S||A|K � for the PSRL algorithm and appro- priate scaling of the ϵk terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' But, this would require an assumption that the transition probability has an indepen- dent Dirichlet prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Experimental Results In this section, we evaluate the empirical performance of the Safe PSRL algorithm and compare it with the state- of-the-art DOPE algorithm (Bura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022), which has been shown to perform better than other comparable algo- rithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', the OptPess-LP in (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2021a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The empirical performance is evaluated with respect to (i) the objective regret and (ii) the constraint regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We consider the setting of a media streaming service (Bura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', 2022) from a wireless base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The base sta- tion provides the streaming service at two different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' These speeds follow independent Bernoulli distributions denoted by parameters µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='9 and µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='1, with µ1 cor- responding to the faster service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The data packets arriving at the device are stored in a buffer and sent out according to a Bernoulli random process with mean γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The buffer size sh evolves as sh+1 = min (max (0, sh + Ah − Bh) , N) where Ah is the number of packet arrivals, Bh is the num- ber of packet departures, and N = 10 is the maximum size of the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The device desires to minimize the cost of run- ning out of packets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', an empty buffer, while restricting the use of the faster service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We model this scenario as a finite horizon CMDP with the state representing the buffer size and actions {1, 2} denoting the choice of speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We set the objective cost as r(s, a) = 1{s = 0} and the constraint cost as c(s, a) = 1{a = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The episode length H is 10 and the constraint threshold τ is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We evaluate the cumulative regret for the Safe PSRL and the DOPE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The transition probability is fixed and not sampled from a prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' For the Safe PSRL algorithm, we consider a Dirichlet prior for the transition Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation probability with parameters [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The Dirichlet prior is a good choice since it is a conjugate prior for multi- nomial and categorical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We further scale the ϵk parameters of the Safe PSRL algorithm by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='05 to avoid excessive pessimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The performance of our algorithm is compared against the DOPE algorithm, which requires a known safe policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We choose the optimal policy of the given CMDP with a tighter constraint threshold c0 = 1 as the safe policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The same c0 is also used in the Safe PSRL algorithm as the satisfiable constraint threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The algorithms are evaluated over K = 400, 000 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' All the experiments are performed on a 2019 MacBook Pro with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='4 GHz Quad-Core Intel Core i5 processor and 16GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Plots showing (a) cumulative objective regret and (b) cumulative constraint regret for the Safe PSRL and DOPE algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Plots showing (a) average objective regret and (b) average constraint regret for the Safe PSRL and DOPE algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 1(a) shows that the Safe PSRL algorithm greatly out- performs the DOPE algorithm in terms of objective regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The objective regret for the DOPE algorithm grows almost linearly for a very large number of episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In compar- ison, the Safe PSRL attains √ K behavior much earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2(a) for the average objective regret shows this behavior more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 1(b), we see that the constraint regrets for both the Safe PSRL and the DOPE algorithm are negative for almost all of the episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This implies that the constraint was satisfied in almost all of the episodes and matches with the theoretical guarantees for both the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We observe the initial jumps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2(b) with respect to the Safe PSRL regret plots because a few initial policies returned by the Safe PSRL algorithm fail to sat- isfy the constraint while achieving better reward objective performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This behavior occurs because the dual parameter λ, which starts from 0, has not yet caught up with the appropriate value which would ensure optimal objective performance while satisfying the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We can infer from the regret plot that this appropriate λ value is reached fairly quickly by the Safe PSRL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The DOPE algorithm, on the other hand, relies on the safe policy for too long before it starts to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' We thus show that the Safe PSRL algorithm is able to achieve superior objective regret performance while satisfy- ing the constraint for almost all the episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' This result is further achieved without the knowledge of a safe policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Conclusions We addressed the problem of safe online learning for episodic MDPs with constraints and unknown transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The Safe PSRL is the first posterior sam- pling algorithm that achieves bounded constraint violation regret while achieving near-optimal cumulative reward re- gret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The algorithm has better empirical performance than other state-of-the-art algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', the DOPE algorithm) for the same setting and does not need to assume knowl- edge of a safe policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' The algorithm can be extended to the infinite-horizon setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Incorporating chance or risk constraints would be another interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 1e6 Safe PSRL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='0 DOPE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='0 Episode k 1e51e6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='50 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Regret guarantees for model-based reinforcement learning with long-term average constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In The 38th Conference on Uncer- tainty in Artificial Intelligence, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Agrawal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Goyal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Analysis of thompson sampling for the multi-armed bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Conference on learning theory, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 39–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' JMLR Workshop and Confer- ence Proceedings, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Agrawal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Goyal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thompson sampling for contex- tual bandits with linear payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In International Confer- ence on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 127–135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Altman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Constrained Markov Decision Processes, vol- ume 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' CRC Press, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Amani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Alizadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Thrampoulidis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Linear stochastic bandits under safety constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Auer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Ortner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Online regret bounds for a new reinforcement learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Proceedings 1st Austrian Cognitive Vision Workshop, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Azar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Osband, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Munos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Minimax regret bounds for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Proceedings of the 34th International Conference on Machine Learning- Volume 70, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 263–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' org, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Bai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Bedi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Agarwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Koppel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Ag- garwal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Achieving zero constraint violation for con- strained reinforcement learning via primal-dual approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial In- telligence, volume 36, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 3682–3689, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Brantley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Dudik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Lykouris, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Miryoosefi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Sim- chowitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Slivkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Constrained episodic reinforcement learning in concave-convex and knapsack settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Information Pro- cessing Systems, 33:16315–16326, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Bura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Hasanzadezonuzy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Kalathil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Shakkottai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Chamberland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Dope: Doubly optimistic and pessimistic exploration for safe reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Chapelle, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' An empirical evaluation of thompson sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in neural information processing systems, 24:2249–2257, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Learning infinite-horizon average-reward markov decision processes with con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='00150, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Ding, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Basar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Jovanovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Natu- ral policy gradient primal-dual method for constrained markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Informa- tion Processing Systems, 33:8378–8390, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Ding, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Wei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Jovanovic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Provably efficient safe exploration via primal-dual policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In International Conference on Artificial Intelligence and Statistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 3304–3312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Efroni, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Mannor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Pirotta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Exploration- exploitation in constrained mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='02189, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Gopalan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Mannor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thompson sampling for learn- ing parameterized markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Confer- ence on Learning Theory, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 861–898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' HasanzadeZonuzy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Bura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Kalathil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Shakkot- tai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Learning with safety constraints: Sample complex- ity of reinforcement learning for constrained mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 7667–7674, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jafarnia-Jahromi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Online learning for stochastic shortest path model via posterior sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='05335, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jafarnia-Jahromi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Nayyar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Learning zero-sum stochastic games with posterior sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='03396, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jafarnia-Jahromi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Nayyar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Online learning for unknown partially observable mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='12661, 2021c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jaksch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Ortner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Auer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Near-optimal regret bounds for reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Journal of Machine Learning Research, 11(Apr):1563–1600, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Jin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Allen-Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Bubeck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Is Q-learning provably efficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 4863–4873, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Kalagarla, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Nuzzo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' A sample- efficient algorithm for episodic finite-horizon mdp with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 8030–8037, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Kaufmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Korda, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Munos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thompson sam- pling: An asymptotically optimal finite-time analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In International conference on algorithmic learning theory, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 199–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Khezeli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Bitar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Safe linear stochastic bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 10202–10209, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation Lai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Robbins, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Asymptotically efficient adaptive allocation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in applied mathematics, 6(1): 4–22, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Zhou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Kalathil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Kumar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Tian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Learning policies with zero or bounded constraint viola- tion for constrained mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Informa- tion Processing Systems, 34:17183–17193, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Shi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Ying, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' An efficient pessimistic- optimistic algorithm for stochastic linear bandits with general constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:24075–24086, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Osband, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Van Roy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Why is posterior sampling better than optimism for reinforcement learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2701– 2710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Osband, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Russo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Van Roy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' (more) efficient reinforcement learning via posterior sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 26, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Ouyang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Gagrani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Nayyar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Learn- ing unknown markov decision processes: A thompson sampling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 1333–1342, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Pacchiano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Ghavamzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Bartlett, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Jiang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Stochastic bandits with linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Interna- tional Conference on Artificial Intelligence and Statistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 2827–2835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Puterman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Markov Decision Processes: Discrete Stochastic Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', New York, NY, USA, 1st edition, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ISBN 0471619779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Qiu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Wei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Upper confidence primal-dual reinforcement learning for cmdp with adversarial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:15277–15287, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Shroff, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Learning in markov decision processes under constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' arXiv preprint arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='12435, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thompson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' On the likelihood that one unknown probability exceeds another in view of the evidence of two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Biometrika, 25(3/4):285–294, 1933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Wei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Jafarnia-Jahromi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Sharma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Jain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Model-free reinforcement learning in infinite- horizon average-reward markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In In- ternational Conference on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 10170– 10180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Wei, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=', and Ying, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Triple-q: A model-free algorithm for constrained reinforcement learning with sublinear regret and zero constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Interna- tional Conference on Artificial Intelligence and Statistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 3274–3307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Zheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' and Ratliff, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Constrained upper confidence reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' In Learning for Dynamics and Control, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' 620–629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Proof of Lemma 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Now for k ≥ C′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' consider: λk+1 2 2 − λk 2 2 = λk(λk+1 − λk) + 1 2(λk+1 − λk)2 = λk(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ) + 1 2(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ)2 = λk(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ) − ηkV πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ηkV πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + 1 2(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ)2 ≤ λk(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ) − ηkV πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ηkH + 1 2(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ)2 ≤ λk(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ) − ηkV πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ηkH + (V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − τ)2 + ϵ2 k ( Using (a + b)2 2 ≤ a2 + b2) ≤ λk(V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ) − ηkV πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ηkH + H2 + ϵ2 k ≤ λk(V π ˆ pk 0 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ) − ηkV π ˆ pk 0 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ηkH + H2 + ϵ2 k ( By optimality of πk in primal update ) ≤ λk(c0 + ϵk − τ) + ηkH + H2 + ϵ2 k ≤ −λk(τ − c0) 2 + ηkH + H2 + ϵ2 k ( as for k ≥ C′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ϵk ≤ (τ − c0) 2 ) Now for λ ≥ ϕk where ϕk := 4(H2 +ϵ2 k +ηkH)/(τ −c0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' we have: E [λk+1 − λk|λk = λ] ≤ E �λ2 k+1 − λ2 k 2λk |λk = λ � (Using x − y ≤ x2 − y2 2y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' for y > 0) = 1 λE �λ2 k+1 − λ2 k 2 |λk = λ � ≤ 1 λE � −λk(τ − c0) 2 + ηkH + H2 + ϵ2 k|λk = λ � = −(τ − c0) 2 + ηkH + H2 + ϵ2 k λ ≤ −(τ − c0) 2 + (τ − c0) 4 = −(τ − c0) 4 := ρ Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' |λk+1 − λk| = |V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ| ≤ H with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Thus, by lemma 3, we have : E � eζλK+1� ≤ E � eζλC′′ � + 2eζ(H+ϕK+1) ζρ , where ζ = ρ/(H2 + Hρ/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' =⇒ eζE[λK+1] ≤ E � eζλC′′ � + 2eζ(H+ϕK+1) ζρ (By Jensen’s inequality) =⇒ E [λK+1] ≤ 1 ζ log � E � eζλC′′ � + 2eζ(H+ϕK+1) ζρ � Further,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' λC′′ ≤ λ1 + C′′−1 � 1 (V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) + ϵk − τ)+ ≤ C′′ � 1 ϵk + C′′(H − τ) := λmax C′′ Continuing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' E [λK+1] ≤ 1 ζ log � eζλmax C′′ + 2eζ(H+ϕK+1) ζρ � ≤ 1 ζ log � eζλmax C′′ + 8H2eζ(H+ϕK+1) 3ρ2 � ( Using ζ ≥ 3(τ − c0) 13H2 ) ≤ 1 ζ log �11H2 3ρ2 eζ(H+ϕK+1+λmax C′′ ) � = 1 ζ log 11H2 3ρ2 + H + ϕK+1 + λmax C′′ = 1 ζ log 11H2 3ρ2 + H + C′′ � 1 ϵk + C′′(H − τ) + 4(H2 + ϵ2 K+1 + ηK+1H) (τ − c0) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' Proof of Lemma 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' K � k=C′′ E � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) � Safe Posterior Sampling for Constrained MDPs with Bounded Constraint Violation = K � k=C′′ E �λk ηk � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) �� + K � k=C′′ E �� V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − λk ηk V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) �� − K � k=C′′ E �� V πk 1 (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − λk ηk V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) �� ≤ K � k=C′′ E �λk ηk � V πϵk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content='ˆ pk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk) �� + 0 ( By optimality of πk in primal update ) ≤ K � k=C′′ E �λk ηk (τ − ϵk − V πk 1 (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} +page_content=' ˆpk)) � ≤ K � k=C′′ E � 1 ηk ((λk(λk+1 − λk) + τ 2) � (By update rule for λk) ≤ E � K � k=C′′ 1 ηk (λ2 k 2 − λ2 k+1 2 ) + K � k=C′′ 1 2ηk (λk+1 − λk)2 + K � k=C′′ τ 2 ηk � ≤ E �(λC′′)2 2ηC′′ � + K � k=C′′ H2 2ηk + K � k=C′′ H2 ηk (As ηk increases with k) ≤ (�C′′ k=1 ϵk + C ′′(H − τ))2 2ηC′′ + 3H 2 K � C′′ 1 (τ − c0) √ k = ˜O � H τ − c0 √ K �' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFJT4oBgHgl3EQfdSw-/content/2301.11547v1.pdf'} diff --git a/udFLT4oBgHgl3EQfjC8Y/content/tmp_files/2301.12109v1.pdf.txt b/udFLT4oBgHgl3EQfjC8Y/content/tmp_files/2301.12109v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b157e2975bdc985a7e16d640129dd7106a9fd71 --- /dev/null +++ b/udFLT4oBgHgl3EQfjC8Y/content/tmp_files/2301.12109v1.pdf.txt @@ -0,0 +1,3059 @@ +MNRAS 000, 1–16 (2022) +Preprint 31 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The TIME Table: Rotation and Ages of Cool Exoplanet Host Stars +Eric Gaidos★1,2,3, Zachary Claytor4,5, Ryan Dungee6, Aleezah Ali4, Gregory A. Feiden7 +1Department of Earth Sciences, University of Hawai’i at M¯anoa, Honolulu, HI 96822, USA +2Institute for Astronomy, University of Vienna, 1180 Wien, Austria +3Institute for Particle Physics & Astrophysics, ETH Zürich, 8093 Zürich, Switzerland +4Institute for Astronomy, University of Hawai’i at M¯anoa, Honolulu, HI 96822 USA +5Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611 USA +6Institute for Astronomy, University of Hawai’i at Hilo, Hilo, HI 96720 USA +7Department of Physics and Astronomy, University of North Georgia, Dahlonega, GA 30597 USA +Submitted, accepted +ABSTRACT +Age is a stellar parameter that is both fundamental and difficult to determine. Among middle- +aged M dwarfs, the most prolific hosts of close-in and detectable exoplanets, gyrochronology +is the most promising method to assign ages, but requires calibration by rotation-temperature +sequences (gyrochrones) in clusters of known ages. We curated a catalog of 249 late K- +and M-type (𝑇eff=3200-4200K) exoplanet host stars with established rotation periods, and +applied empirical, temperature-dependent rotation-age relations based on relevant published +gyrochrones, including one derived from observations of the 4 Gyr-old open cluster M67. We +estimated ages for 227 of these stars, and upper limits for 8 others, excluding 14 which are +too rapidly rotating or are otherwise outside the valid parameter range of our gyrochronology. +We estimated uncertainties based on observed scatter in rotation periods in young clusters, +error in the gyrochrones, and uncertainties in temperature and non-solar metallicity. For those +stars with measured metallicities, we provide but do not incorporate a correction for the +effects of deviation from solar-metallicity. The age distribution of our sample declines to +near zero at 10 Gyr, the age of the Galactic disk, with the handful of outliers explainable +by large uncertainties. Continued addition or extension of cluster rotation sequences to more +thoroughly calibrate the gyrochronology in time and temperature space, more precise and +robust measurement of rotation periods, and more accurate stellar parameter measurements +will enable continued improvements in the age estimates of these important exoplanet host +stars. +Key words: exoplanets – stars: evolution – stars: late-type – stars: low-mass – stars: rotation +– planetary systems +1 +INTRODUCTION +Over the past three decades, thousands of planets have been discov- +ered around other stars. Exoplanet surveys have revealed that M- +type dwarfs, the least massive but most numerous stars, host more +planets on close-orbits than their solar-mass counterparts (Mulders +et al. 2015; Hardegree-Ullman et al. 2019; Hsu et al. 2019). These +include Earth-size, rocky planets that orbit within the compact hab- +itable zone of these intrinsically faint stars, and which are more +feasible to study, e.g. with JWST, because of the host stars’ lower +mass, radius, and luminosity. +Precise characterization of the host star (e.g., radius, mass, +luminosity, metallicity) is essential to obtain properties of its plan- +ets, but is challenging for very low-mass stars for which methods +tuned to the Sun do not apply. For this reason, empirical approaches +★ Contact e-mail: gaidos@hawaii.edu +have proven useful for M dwarfs, enabled by the advent of the Gaia +astrometry mission, space- and ground-based photometric surveys, +and advances and expansion in spectroscopic instrumentation. For +example, interferometry can directly measure the angular radii of +very nearby M dwarfs; pairing with trigonometric parallaxes yields +physical radii. Combined with a bolometric luminosity from a flux- +calibrated spectral energy distribution (SED), this allows the effec- +tive temperature 𝑇eff to be derived using the Stefan-Boltzmann law +(Boyajian et al. 2012). Spectra of stars with a range of established +𝑇eff can then serve as templates to estimate the 𝑇eff of more distant +stars using spectra and, combining with SEDs, their radii (Mann +et al. 2015). Metallicities can be calibrated using binaries where the +solar-type companion has an established metallicity (relative to the +Sun) (e.g., Mann et al. 2013, 2014; Montes et al. 2018; Souto et al. +2020). +Age is a fundamental property of planets but is difficult to accu- +rately estimate for most systems (Christensen-Dalsgaard & Aguirre +© 2022 The Authors +arXiv:2301.12109v1 [astro-ph.EP] 28 Jan 2023 + +2 +E. Gaidos et al. +2018). Planets and their atmospheres are expected to evolve under +the influence of their host star and their own internal thermodynam- +ics and compositional change (Kite et al. 2009; Lammer 2013). For +temperate, Earth-like planets, changes in atmospheric composition +will be the backdrop against which biosignatures will be searched +for. Observations of disk lifetimes, planet formation theory, and +isotope-ages of Solar System bodies indicate that the age of a star +should be no more than a few tens of Myr older than that of its +planets (Helled & Morbidelli 2021), but ages of most host stars +remain poorly constrained. Planets are difficult to detect around +young stars (e.g., Miyakawa et al. 2022) and most planet hosts are +older and no longer members of (relatively) well-dated clusters and +young-moving groups. +Ages of isolated field stars have been estimated by (1) compari- +son of stellar parameters to stellar models (e.g., in a color-magnitude +diagram); (2) asteroseismic measurement of increasing density due +to the conversion of H into He and heavy elements in stellar in- +teriors; (3) the abundance of lithium, which is destroyed in stellar +interiors; (4) metallicity and the overall age-metallicity relation of +the Galactic disk; (5) increase in the peculiar motion of stars with +time with respect to the overall orbital motion of the Galactic disk +as a result of perturbations from molecular clouds and other stars; +and (6) rotation and rotation-driven magnetic activity that decline +as angular momentum (AM) is lost through a magnetized wind. +But M dwarfs are resistant to most age-dating techniques: They +evolve imperceptibly on the main sequence (MS; Adams et al. 2005) +and the pulsations that are the grist of asteroseismology are below +current detection thresholds (Rodríguez-López 2019, and regardless +the stars’ densities do not change) . M dwarfs consume their lithium +within ∼50 million years (Binks & Jeffries 2014) and thus this proxy +cannot be used at later ages, and metallicity and peculiar motions +are only meaningful in a statistical sense for stellar populations, +not individual stars; the age-metallicity relation appears flat for +disk stars (Rebassa-Mansergas et al. 2021) and age-abundance ratio +relations do not appear to apply universally (Casali et al. 2020). +This leaves gyrochronology, the application of relations be- +tween age and rotation (and its proxies) brought about by stellar +spin-down, as a viable method to age-date main-sequence M dwarfs +in the field (Barnes 2007). Such stars are potentially well-suited for +this approach because they have a smaller or no inner radiative core +which can rotate quasi-independently of the convective envelope, +and instead could have behavior similar to simple “solid-body" ro- +tation. However, spin-down driven by magnetic activity scales with +the Rossby number 𝑅𝑜 ≡ 𝑃rot/𝜏𝑐, where 𝜏𝑐 is the local turnover +time in the convective envelope, and the longer 𝜏𝑐 of M dwarfs +compared to solar-type stars means their rotational evolution will +be distinct. Using co-eval bainries, Otani et al. (2022) tested several +color-dependent spin-down models for internal consistency, and de- +rived internal errors (i.e., only those arising from errors in 𝑃rot and +color) of 5-10% for relatively young early M-type stars. +Pioneering ground-based observations of open clusters of co- +eval stars, followed by the revolutionary wide-field surveys of the +Kepler and TESS space telescopes, document the formation of tight +sequences in rotation vs. color or 𝑇eff diagrams that extend to cooler +temperatures and longer 𝜏𝑐 with time (Gallet & Bouvier 2015; +Curtis et al. 2020). The formation of the sequence among solar-mass +stars is thought to be the result of a change in the braking law (i.e., +the rotation-rate dependence of the torque) as stars transition from a +“saturated" (less rotation rate-dependent) phase to an “unsaturated" +(more rotation rate-dependent) phase of stellar activity at a critical +value of 𝑅𝑜 (Matt et al. 2012; Curtis et al. 2019a). After a star enters +this regime the strong rotation rate dependence effectively erases the +effect of initial conditions, allowing gyrochronologic relations to be +applied. +Among cooler dwarf stars the situation is more complex. The +outer convective envelope—the part of the star that is both observ- +able and feels the decelerating torque from the magnetized wind—is +more substantial, and the coupling timescale between the the en- +velopes and the radiative core is longer. Among K dwarfs, this can +lead to early differential rotation, with the core spinning faster than +the envelope, and later “stalling" as the core transfers AM to the +envelope and the observed spin-down temporarily slows or halts +(Denissenkov et al. 2010; Curtis et al. 2019b, 2020). Stalling could +contribute to the formation of a rotational sequence, but also delays +the epoch at which gyrochronology is useful. Among yet cooler M +dwarfs the radiative core is smaller or absent, the coupling timescale +is expected to get longer, and the magnitude of the stalling could +diminish/disappear (Lu et al. 2022). +The mechanism responsible for core-envelope coupling has not +been established, nor has a theoretical model that is quantitatively +consistent with the observations and has predictive power over a +range of 𝑇eff been constructed. Moreover, it is not certain if braking +laws developed for solar-type stars apply to M dwarfs. For these +reasons, observations of stars in clusters of known ages are im- +perative for identifying rotational sequences (vs. 𝑇eff), constructing +empirical rotation-age relations, and calibrating successful models. +But M dwarfs are intrinsically faint and their rotation evolves more +slowly, and thus deeper observations of older (and statistically more +distant) clusters are required. M dwarfs in these clusters are beyond +the range of current space telescopes both in terms of signal (limited +by the modest telescope aperture) and spatial resolution (limited by +the large pixel size). K2 monitoring of the oldest nearby cluster +(Ruprecht 147, ≈2.7 Gyr; Curtis et al. 2013) captured rotation peri- +odsfor only a handful of M dwarfs near the K-M boundary. Similar +observations of the nearest old cluster (M67, ≈4 Gyr Richer et al. +1998; VandenBerg & Stetson 2004; Schiavon et al. 2004; Sarajedini +et al. 2009) failed to yield useful results (Esselstein et al. 2018). +Ground-based observatories can go deeper and with high res- +olution. Dungee et al. (2022) carried out a Sloan 𝑖-band monitor- +ing campaign of late K and early M dwarf members of M67 with +the MegaCAM wide-field camera at the prime focus of the 3.6-m +Canada France Hawaii Telescope (CFHT) (Boulade 1998), obtain- +ing 294 rotation periods and identifying a rotational sequence that +ranged from ≈25 days at 4200K to 125 days at 3200K. Dungee et al. +(2022) found that the “warm" end of the M67 sequence could be ex- +plained by the overlapping “cool" end of the Ruprecht 147 sequence +identified by Curtis et al. (2020), plus Skumanich-like power-law +spin-down 𝑃 ∝ 𝑡𝑛 Skumanich (1972) with an index 𝑛 = 0.62. This +indicates that (a) the rotation sequence of M dwarfs extends close +to the fully convective boundary (near 𝑇eff=3200K) by no later than +4 Gyr; (b) spin-down among middle-aged M dwarfs seems to obey +a relatively simple braking law. Both of these findings bode well for +the gyrochronology of very cool dwarfs. +In this work, we curate a catalog of rotation periods of late +K and early M-type dwarfs known to host validated or confirmed +planets1, and apply empirical rotation-age relations based on the +M67 gyrochrone of Dungee et al. (2022) and previously published +gyrochrones (Curtis et al. 2020) to estimate ages. The rotation pe- +riods of many host stars have been established either using the +1 A “validated" planet is one for which known false positive scenarios +are highly unlikely; a “confirmed" planet has been detected by a second, +independent method. +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +3 +same space-based photometry (i.e., Kepler, K2, TESS) used to iden- +tify their transiting planets, or by data obtained from the ground- +or space as part of the validation/confirmation of candidate planets. +There are also collections of rotation periods of field stars (including +planet hosts) based on data from Kepler (Santos et al. 2019, 2021), +K2 (Reinhold & Hekker 2020), TESS (Canto Martins et al. 2020), +and ground-based surveys (Oelkers et al. 2018; Newton et al. 2018; +Christy et al. 2022). We also identify additional candidate rotational +signatures directly in the photometric datasets. We emphasize that +some rotation periods are tentative and that very cool dwarf gy- +rochronology is a work in progress and makes assumptions which +will be borne out or refuted by future observations. +2 +SOURCES OF ROTATION PERIODS +We identified all host stars of validated or confirmed exoplanets with +𝑇eff of 3200–4200 K in the NASA Exoplanet Archive as of August +2022. This included 112 Kepler host stars or Kepler Objects of +Interest (KOIs) having rotation periods 𝑃rot in Santos et al. (2019). +From the list of the non-Kepler host stars we removed evolved +(giant), T Tauri, and pre-main sequence (PMS) stars, as rotation of +these stars obviously does not follow the gyrochronology of dwarfs, +as well as members of star-forming regions and young moving +groups that have ages estimated by other techniques, leaving 215 +non-Kepler stars. +Rotation periods for these stars were obtained from the litera- +ture; these were determined using ground- or space-based photome- +try, e.g. by WASP (Pollacco et al. 2006), MEarth (Berta et al. 2012), +ASAS-SN (Shappee et al. 2014), K2(Howell et al. 2014), and TESS +(Ricker et al. 2014) and/or time-series spectroscopy of indicators +of active regions and magnetic fields, notably by the HARPS (Pepe +et al. 2000) and CARMENES Doppler RV surveys for exoplanets +(Reiners et al. 2018). +We revisited the K2 data by matching all stars against the EPIC +catalog (Huber et al. 2016) and downloaded all Pre-search Data Con- +ditioning Simple Aperture Photometry (PDCSAP) lightcurves from +the MAST archive (including many K2-detected transiting exoplanet +host stars). The lightcurves were further de-trended with a best-fit +second-order polynomial before a Lomb-Scargle analysis (Scargle +1982) to search for signals with periods of 0.4-40 days, an interval +chosen to avoid the pervasive 6-hr thruster firing signal and for the +typical lightcurve to span at least two periods. 115 lightcurves of +95 EPIC stars contained peaks exceeding a 𝑝 = 0.001 false-positive +level. Of these 20 were not previously published, and in one other +case (K2-345) we replaced the Reinhold & Hekker (2020) value +as being obviously erroneous. We did not revise other Reinhold & +Hekker (2020) values. In 11 cases (K2-5, 14, 83, 124, 125, 129, +151, 288B, 315, 322, 377) we judged that the peak was an upper +harmonic and doubled the period and its error based on inspection +of the lightcurve. The de-trended lightcurves and periodograms are +shown in Figs. A1-A4. +We retrieved lightcurves from the Zwicky Transient Facility +(ZTF, Masci et al. 2019) using the Python wrapper of the InfraRed +Science Archive (IRSA) API query (Rigault 2018) and a matching +criteria of 5". We obtained 319 ZTF lightcurves for 102 stars; each +star has up to three (𝑔𝑟𝑖) lightcurves and sometimes the 5" search +cone contained more than one star. We constructed Lomb-Scargle +periodograms of the 145 lightcurves with at least 100 points (the +maximum was 1092) and identified 25 signals among 22 stars with +peaks with a false alarm probability < 0.01. (We were less stringent +than for K2 because of the availability of data in multiple filters for +comparison). For three stars with two periodic signals (in different +filters), none had matching periods. Seven signals were rejected +based on similarity to the lunar synodic period (29.5 day) or its +seasonal alias (27.3 and 32.1 days). Marginal, long term (> 100-day) +signals seen in the lightcurves of K2-123 and K2-124 were rejected +based on detection of shorter rotation periods by K2. ZTF periods +for two other systems (TOIs-2136 and 3174) were already reported +in the literature. We assigned tentative periods to four other stars +based on these data; none of these periods are close to the 29.5-day +lunar synodic period or its annual alias, or harmonics of 1-day (Fig. +A5). All other periodograms contained no clearly visible peaks or a +forest of peaks of roughly equal (in)significance. We also retrieved +𝑔- and 𝑉-band lightcurves from the All-Sky Automated Search for +Super-Novae (ASAS-SN Shappee et al. 2014; Kochanek et al. 2017) +on which we performed a similar analysis. We identified 16 and 15 +host stars with significant (𝑝 < 0.01) periodic signals in the 𝑔- +and 𝑉-band data, respectively. Among these are four stars that have +matching 𝑔- and 𝑉-band periods that are not at the lunar synodic +period, its aliases, or 1-day harmonics. The 𝑔-band photometry for +these is shown in Fig. A6. In the case of GJ 486, the detected signals +at 13.7 days differ markedly from the published 𝑃rot of 49.9 ± 5.5 +days (Caballero et al. 2022) (which we retain). +We included 𝑃rot values determined from time-series measure- +ments of spectroscopic indicators of stellar activity such as Ca ii HK +and H𝛼, but we excluded estimates based on correlations with indi- +cators of stellar activity, e.g., the overall level of Ca ii HK emission +(e.g., Astudillo-Defru et al. 2017b). +3 +STELLAR PARAMETERS +We retrieved values for 𝑇eff and metallicities ([Fe/H]) of KOIs from +the literature via the NASA Exoplanet Archive tables (Thompson +et al. 2018; Coughlin et al. 2016; Mullally et al. 2015; Rowe et al. +2015; Burke et al. 2014; Batalha et al. 2013). For each star, we used +the most recent KOI table available. However, the present character- +ization of cool exoplanet host stars is markedly heterogeneous, with +many stars lacking published metallicities, and multiple values for +the same star can differ by much more than the published uncertain- +ties. For those stars without effective temperature solutions from +the NASA Exoplanet Archive, we used their absolute 𝐾𝑠 magnitude +to calculate an effective temperature with the Mann et al. (2015) +empirical relations. +We identified stars in multiple systems using the literature, as +well as stars with Gaia astrometric renormalized unit weight error +(RUWE) values >1.4 indicative of unresolved multiplicity (Be- +lokurov et al. 2020). We flagged but did not exclude these systems, +cautioning that binaries that are unresolved in the data used to ob- +tain rotational signals could be assigned incorrect rotation periods, +and a single-star gyrochronology is expected to be more erroneous +or fail in sufficiently close binaries (see Section 7.1). +We searched for additional, unpublished stellar companions re- +solved by Gaia (separations ≳1′′) and identifiable based on similar +parallaxes and proper motions in the DR3 catalog (Gaia Collab- +oration et al. 2016, 2022). This was performed by calculating a +Bayesian probability that a candidate companion’s astrometry is the +same as a given star, relative to the probability that this occurs in a +“background" population. We identified five stars with FAP < 0.01, +but all are previously known binaries. +We searched the Hipparcos-Gaia (EDR3) Catalog of Acceler- +ations (Brandt 2021) and found 5 matches, a paucity that reflects +the magnitude limit of the Hipparcos catalog. Of these, only two +MNRAS 000, 1–16 (2022) + +4 +E. Gaidos et al. +3200 +3400 +3600 +3800 +4000 +4200 +Teff (K) +101 +102 +period (days) +Galactic Disk age +Rossby number = 0.13 +lunar synodic +half Kepler quarter +Other +binaries +Kepler +Figure 1. Rotation periods of late-K and early M-type hosts of known +exoplanets from the literature and this work. Members of multi-star systems +are indicated in red. The horizontal dashed and dotted lines mark one half +the Kepler rotation interval (close to one half the K2 campaign interval) and +the lunar synodic period, near which ground-based detection of 𝑃rot values +are limited. The magenta curve is the 𝑃rot above which a star’s Rossby +number exceeds the critical value (based on convective turnover times from +the Dartmouth magnetic model) and activity leaves the “saturated" phase, +and the blue curve is the 𝑃rot predicted for stars with the age of the Galactic +disk using the Dungee et al. (2022) gyrochrone and power-law evolution. +Errors in 𝑇eff are not shown but are typically 75K. +have 𝜒2 fits approaching but not reaching a formal 0.3% false-alarm +probability threshold of 11.8: HD 238090 (𝜒2 = 9.1) is the primary +of a mid-M-type companion at 14.6′′ (224 au) that is unlikely to be +the source of any acceleration, and HIP 71135 (𝜒2=9.3) which has +no known stellar companion. The RV-detected planets are inferred +to have (sub)-Neptune-like masses (Feng et al. 2019; Stock et al. +2020), and could not produce detectable acceleration. Finally, we +searched the catalog of the Robo-AO M-dwarf Multiplicity Survey +(Lamman et al. 2020), which identified candidate companions with +separations of 0.1-4′′ of nearby M dwarfs from the Lépine & Shara +(2005) catalog. We identified 20 overlapping stars, none of which +had AO-identified companions. +Figure 1 plots 𝑃rot vs. 𝑇eff for the catalog of host stars, with Ke- +pler- and non-Kepler hosts marked by different points and members +of known binaries shown in red. The horizontal dashed line marks +90 days, the cadence at which Kepler performed a role maneuver +that introduced systematics in lightcurves, only slightly longer than +the 80-day interval which the spacecraft could observe a field dur- +ing the K2 mission. Recovery of rotation periods longer than these +intervals is inhibited by these systematics. The horizontal dotted +line is the lunar synodic period of 29.5 days near which the ground- +based detection of rotation periods is inhibited. The magenta curve +is the critical 𝑃rot at which 𝑅𝑜 = 0.13 using the Jeffries et al. (2011) +prescription for 𝜏𝑐. Nearly all stars are above this line and are thus +in the “unsaturated" regime of dynamo-driven activity, although not +necessarily yet in Skumanich-like spin-down. The blue curve is the +expected 𝑃rot for 10 Gyr, the approximate age of the Galactic disk +(Kilic et al. 2017, but see Xiang & Rix (2022)). +4 +AGE ESTIMATION +We estimated the age of each star with an established 𝑃rot (Sec. +2 by comparing it to available empirical cluster gyrochrones that +(partially) include the 𝑇eff range of interest, i.e., that of M67 Dungee +et al. (2022), but also that of the 2.7 Gyr-old Ruprecht 147 (Curtis +et al. 2020), the 1.4 Gyr-old NGC 752 (Agüeros et al. 2018), the +1 Gyr-old NGC 6811 (Curtis et al. 2019a), and the 0.67 Gyr-old +Praesepe (Douglas et al. 2019), as filtered for binaries and fit with +polynomials by Curtis et al. (2020). +Dungee et al. (2022) fit the M67 rotation sequence with a +fourth-order polynomial with 𝑇eff over 3200-4200K. +𝑃4Gyr = 9.66 × 10−10 · (𝑇eff − 4000)4 + 8.25 × 10−7 · (𝑇eff − 4000)3 ++ 2.69 × 10−4 · (𝑇eff − 4000)2 + 0.016 · (𝑇eff − 4000) + 25.9, +(1) +𝑇eff of M67 members were estimated from their Pan-STARRS 𝑟 − 𝑖 +colors using a polynomial relation based on synthetic photometry of +nearby M dwarf standards (Mann et al. 2015). Curtis et al. (2020) fit +the other rotational sequences as polynomials with Gaia 𝐵𝑃 − 𝑅𝑃 +color (see their Appendix B), which we converted to 𝑇eff using the +empirical MS of Pecaut & Mamajek (2013). +We estimated ages by simple power-law interpolation between +gyrochrone calibration points (linear interpolation in a log-log plot +of period vs. age). Calibration points were calculated from Eqn. 1 +and Curtis et al. (2020) for a given stellar 𝑇eff. Figure 2 plots the +derived period vs. age tracks for a range of representative 𝑇eff. The +flattening of the tracks at ages younger than Ruprecht 147 could be +“stalling" of spin-down as a faster rotating radiative core adds AM +to the convective envelope (Curtis et al. 2020) (see Sec. 7.1). The +bunching of the hotter tracks reflects the weak dependence of 𝑃rot +on 𝑇eff among co-eval K dwarfs (Curtis et al. 2020). +A calibration point was used only if (1) 𝑇eff falls within the +range of a gyrochrone; (2) for the particular 𝑇eff and gyrochrone +rotation period, the star would not be in the “saturated" phase of +activity (a condition for a rotational sequence); and (3) the star would +not be on the PMS and still affected by contraction and spin-up at +that gyrochrone age. The condition for saturated activity is a Rossby +number 𝑅𝑜 =𝑃rot/𝜏cwhere 𝜏𝑐 is the local convective turnover time, +to be less than a critical value. We adopted the relation of 𝜏𝑐 with +luminosity of Jeffries et al. (2011) and critical 𝑅𝑜 of 0.13 (Wright +et al. 2018). PMS durations were taken from the Dartmouth standard +models of stellar evolution (Dotter et al. 2008). +If the only available calibration point was that of M67 (4 Gyr) +then we calculate the age of the star using the Skumanich-like +spin-down law that Dungee et al. (2022) found by comparing the +rotational sequence of M67 with that of 2.7 Gyr-old Ruprecht 147 +(Curtis et al. 2020) +𝑡 = 𝑡0 (𝑃/𝑃0)1/𝑛 , +(2) +with 𝑛 = 0.62. However, if the age derived in this manner was <2.7 +Gyr (i.e., that of Ruprecht 147) we took this to be an upper limit. +If additional calibration points were available we derived an age +as above; if that was <4 Gyr we then consequently derived ages +by power-law interpolation between successfully younger pairs of +neighboring calibration points until the derived age lay between +the ages of the gyrochrones. However, if the age determined from +the 4 Gyr and next youngest calibration points was >4 Gyr, we +adopted that age. This allowed for the occasional pathological cases +where the Dungee et al. (2022) gyrochrone produced an age that was +slightly <4 Gyr, but the gyrochrone pair produced an age slightly +>4 Gyr. If no self-consistent age was produced (i.e., the age was +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +5 +1 +2 +3 +5 +10 +age (Gyr) +10 +20 +30 +50 +100 +200 +rotation period (days) +M67 +Rup 147 +NGC 752 +NGC 6811 +Praesepe +Teff +3200 +3300 +3400 +3500 +3600 +3700 +3800 +3900 +4000 +4100 +Figure 2. Rotation period vs. age tracks constructed from five cluster gy- +rochrones (vertically labelled). The dashed parts of the trajectory are based +on the Skumanich-like relation derived by Dungee et al. (2022), with extrap- +olation beyond 4 Gyr. Dotted lines are the periods above which the Rossby +number exceeds the critical value for unsaturated activity (≈0.13) at each +𝑇eff. +younger than the youngest calibration point) we adopted the age of +the youngest valid gyrochrone as an upper limit. +Monte Carlo (MC) realizations of these calculations were per- +formed to estimate the uncertainty in the age, incorporating error in +rotation period, stellar parameters, the gyrochronology, and initial +conditions (see Sec. 5). The mean and standard error of the age +distributions were adopted as the nominal age and its uncertainty +reported in Table 6. If the MC realizations produced only upper +limits, the 95 percentile value of the distribution was reported as an +upper limit. If neither values nor upper limits were produced, or the +resulting age would place the star on the PMS, no age was assigned. +5 +AGE ERROR ANALYSIS +Error in gyrochronologic age assignment arises from (1) the formal +uncertainty in 𝑃rot, as well as systematic errors in 𝑃rot not included +in the formal error (i.e., aliasing and confusion with harmonics); +(2) the formal error in the gyrochrones as well as uncertainty in +the ages of the calibrator clusters; (3) errors in 𝑇eff used to apply +the calibration; (4) variation in initial conditions, i.e the angular +momentum or rotation rate at an early time; and (5) stellar rotational +evolution that deviates from the assumed behavior due to differences +in the internal structure of a star or the wind torque. +Epstein & Pinsonneault (2014) quantified the effect of uncer- +tainty in 𝑃rot (#1 above) and scatter in rotation period at a fixed age +(#4) by projecting “initial" rotation conditions found in the ∼500 +Myr-old cluster M37 forward in time with a rotational model. They +found that the age uncertainty among middle-aged M dwarf stars is +very large, dominated by the scatter in initial conditions, and as a +result the age precision is very poor (factors of two or more between +minimum and maximum ages) because, essentially, these stars have +not yet spun down to form a rotational sequence, but this depends +on the details of the model, including the assumed braking law. +Here we use Monte Carlo methods to calculate probabilistic +distributions of ages — an approach used by Otani et al. (2022) — +and take the standard deviation as the error. Where data are available +we determine variation empirically; otherwise we rely on models of +stellar interior and rotational evolution. +5.1 +Period Error +We report and use the formal uncertainties in rotation period either +from the literature, or from our periodogram analysis. In the latter +case, Gaussian functions are set to the periodogram peak and the +standard deviation is taken to be the Gaussian width 𝜎. This uncer- +tainty can arise from the finite observation baseline relative to the +rotation period, particularly for older stars with longer rotation peri- +ods. Significant evolution in the spots causing rotational variability +can occur over a few rotation periods (Basri et al. 2022), causing +drift in the phase of the overall photometric signal and broadening +of the peak. Differential rotation, combined with changes in the +latitude of spots, will also broaden a periodogram peak or even pro- +duce distinct neighboring peaks (e.g., Reinhold et al. 2013; Balona +& Abedigamba 2016). +The first (upper) harmonic might be confused for the true pe- +riod as a result of the distribution of star-spots (e.g., Suto et al. +2022), causing the star to appear much younger than it is. Ground- +based observations may incur much larger systematic error as a +result of aliasing caused by sampling bias on nocturnal, lunar syn- +odic, or seasonal timescales. This can distribute the power in single +periodic signal into multiple weaker peaks in a periodogram (Van- +derPlas 2018). These peaks can either be “failure modes" if the +peak is far from the true period, or if close to the true period and +unresolved due to limiting sampling, broaden the peak and resulting +uncertainty. Given the diverse data sets and sources, we point out +but do not attempt to quantify such errors in this work. +5.2 +Gyrochrone error +Due to finite sample size and error in 𝑇eff and 𝑃rot, the gyrochrones +themselves have uncertainties. These are not typically published, +but since most of the ages in TIME-Table rely heavily on the 4 +Gyr-old M67 gyrochrone, we determined uncertainties in the best- +fit polynomial coefficients using Monte-Carlo simulations. This is +shown as the dashed black lines in Fig. 3. The ages of the clusters +used to calibrate the gyrochronology themselves have significant +standard errors and systematics; full consideration of these is beyond +the scope of this work but should be included when considering +rotation-based ages in an absolute sense (Sandquist et al. 2021). +5.3 +Error in effective temperature +Effective temperature 𝑇eff is usually used as the independent vari- +able of a gyrochronology since it is set by radiated energy per unit +area and is thus related to the eddy velocity and magnetodynamo +strength in the convective envelope of a star. Precise and accurate +determination of 𝑇eff is a classic problem in stellar astrophysics, +and a particularly acute issue for the coolest stars. Recent calibra- +tions of the temperature scale of M dwarfs based on interferometric +measurement of stellar radii, (Gaia) parallaxes, and the Stefan- +Boltzmann law (Boyajian et al. 2012; Mann et al. 2015) permit +precision as good as 50 K, but most host stars have 𝑇eff with pre- +cision no better than 75 K. The M67 gyrochrone of Dungee et al. +(2022) was fit to the 𝑔-𝑖 colors of stars, and converted to 𝑇eff using +the temperature scale of Mann et al. (2015). This combined with +the slope of the M67 gyrochrone will produce formal uncertainties +MNRAS 000, 1–16 (2022) + +6 +E. Gaidos et al. +3200 +3400 +3600 +3800 +4000 +4200 +Teff (K) +0 +20 +40 +60 +80 +100 +120 +P4Gyr (days) +1 gyrochrone +from 75K error +Figure 3. Intrinsic dispersion in the 4 Gyr-old M67 gyrochrone (dashed +black lines) established by Dungee et al. (2022) and dispersion produced by +errors in 𝑇eff of 75K (red lines). +of 0-17% in age, with the largest value at the cool (3200K) end, +decreasing to zero near 3900K, and rising to 12% at 4200K. +5.4 +Initial Rotation +A major contributor to age error is departure of a star’s behavior +from a rotation-age relation due to variation in initial conditions, +i.e angular momentum/rotation rate at an early age. A star that +is initially spinning slower/faster than the mean of the population +used to construct the gyrochronology will have an estimated age +that is erroneously older/younger. Surveys of star-forming regions +and very young clusters show that low-mass members emerge from +their disk-hosting phase with a wide range of rotation rates at a given +mass (Cody & Hillenbrand 2010; Rebull et al. 2016; Venuti et al. +2017; Rebull et al. 2018; Kounkel et al. 2019; Serna et al. 2021), +perhaps due to variations of disk lifetime brought on by differences +in the tidal and ultraviolet environment of stars (Kraus et al. 2012; +Roquette et al. 2021). Since it is not possible to know the initial +rotation of an individual star, this variation can induce significant +uncertainty in derived ages. Epstein & Pinsonneault (2014) used +rotational evolution models to show that this greatly limited the +utility of rotation-based ages among cooler, older dwarfs. +The scatter in initial rotation rates can be estimated from ob- +servations of the PMS members of very young clusters. If AM loss +through winds is small over the PMS interval, the fractional dis- +persion in rotation rate at a given mass should be conserved even +as the stars contract and spin up. The fractional dispersion is also +conserved during the saturated phase of magnetic activity because +the torque is proportional to rotation rate and the spin-down is ex- +ponential with a rate-independent time constant. In the 130 Myr-old +Pleiades, no well-defined rotational sequence exists for our range +of 𝑇eff/colors but there is a strong trend of decreasing 𝑃rot (6–0.6 +days) with decreasing 𝑇eff (Rebull et al. 2016). An iterative fit to +this over the equivalent de-reddened 𝑉 − 𝐾 color range (3.22–5.29) +yields a fractional dispersion of 45% (see also Fig. 18 in Stauffer +et al. (2016)), but this value does not reflect the numerous outliers, +some of which may be binaries. Likewise, the rotation periods of +members of the ∼150 Myr-old cluster NGC 2516 (Fritzewski et al. +2020) within this same color range have a scatter of ≳50%, about +one order of magnitude larger than what is observed at later epochs. +This large scatter in initial (ZAMS) rotation rates 𝑃0 will be +compressed once the stars decelerate to the unsaturated regime at +𝑃crit ∼ 0.13𝜏𝑐 (Wright et al. 2018) where the torque becomes +highly rotation rate-dependent and period-time trajectories flatten. +In the saturated regime, 𝑃(𝑡) = 𝑃0𝑒𝑡/𝑇 , where 𝑇 is the spin-down +timescale (Matt et al. 2015), thus a star with a ZAMS rotation period +that differs by Δ𝑃0 ≪ 𝑃0, will reach 𝑃crit at a time that differs by +Δ𝑡 = −𝑇 log +� 𝑃0 + Δ𝑃0 +𝑃0 +� +. +(3) +Because subsequent rotational evolution proceeds from the con- +dition 𝑃 = 𝑃crit, independent of 𝑃0, the star will experience the +same rotation history, but delayed by Δ𝑡. Thus this is the error in +gyrochronological age induced by Δ𝑃0. 𝑇 depends on the brak- +ing torque parameter and moment of inertia of the star. Somers +et al. (2017) found that stars in mass bins of 0.25 − 0.40M⊙ and +0.40 − 0.60M⊙ lose 25% and 39%, respectively, of their AM in +the ≈ 115 Myr interval between the age of the Upper Scorpius +star-forming region and the Pleiades cluster, implying a spin-down +timescale of 225-400 Myr. Somers et al. (2017) also found a disper- +sion of 0.21 and 0.44 dex in the specific (mass-normalized) AM of +Pleiades stars for stars in these respective mass bins. Application of +Eqn. 3 produces an age error of about 200 Myr, which for a nominal +4 Gyr-old star is 5%. +Finally, the intrinsic dispersion in the rotation sequences of +cluster stars can be used to empirically estimate age error due to +rotation rate dispersion. For stars undergoing power-law spin-down +(𝑃rot∝ 𝑡𝜒, Skumanich 1972), the dispersion in age consistent with a +given 𝑃rot is related to the dispersion in period for a given age (i.e., +in the co-eval population of a cluster): +Δ𝑡 +𝑡 = 1 +𝜒 +Δ𝑃 +𝑃 +(4) +We measured the outlier-excluded dispersion Δ𝑃 around the best- +fit rotational sequence of M67 to be 1.6 days, but this is consistent +with the 10% measurement errors alone, and the intrinsic dispersion +could be smaller. The dispersion in other, younger (≲1 Gyr) clusters +like those observed by K2 can be better determined, but with the +caveat that Skumanich-like spin-down only applies to times much +later than the era of saturated magnetic activity. A well-defined +rotational sequence is apparent in the Praesepe cluster (upper right- +hand panel of Fig. 4), estimated to be 600 Myr-old (Gossage et al. +2018). We fit a second-order polynomial with iterative 3𝜎 outlier +rejection to 3500-4200K members with rotation periods cataloged +by Douglas et al. (2019). (Cooler stars have yet to converge to +an identifiable sequence.) The scatter is 0.97 days (N=121). The +same analysis applied to the much sparser Hyades sample yields the +same dispersion: 0.95 days (N=13), a fractional dispersion of about +5% (bottom left panel of Fig. 4). The data on M dwarfs in other, +older clusters are much sparser: Curtis et al. (2019a) estimated a +dispersion of ±10% for the coolest stars in NGC 6819 (≈2.5 Gyr) +and Ruprecht 147 (≈2.7 Gyr) but of these, only five stars have +Gaia 𝐵𝑝 − 𝑅𝑝 colors that correspond to our 𝑇eff range. Using the +Godoy-Rivera et al. (2021) catalog of members of M37 (≈470 Myr +Fragkou et al. 2022) and NGC 6811 (≈950 Myr) with rotation +periods we estimate dispersions of 0.8 days (n=20) and 0.6 days +(n=12) respectively (upper left and bottom right panels of Fig. 4). +The existence of a rotation sequence among M37 M dwarfs is not +clear. Some of the observed dispersion in these clusters is probably +the result of errors in 𝑇eff: for a standard error of 75 K, this is ∼0.5-1 +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +7 +day, depending on cluster and 𝑇eff (purple dotted lines in Fig. 4). +Thus the actual period dispersion could be substantially lower than +the observed values. +We adopted a conservative fractional dispersion in 𝑃0 of 5%, +with the caveat that current data on the establishment of a rotational +sequence at these epochs only extends to a spectral type of M2 +(𝑇eff∼ 3550 K). For 𝑛 = 0.62, this corresponds to an age dispersion +of 8% (Eqn. 4), which we incorporate in our MC realizations of age +estimates by adopting a Gaussian distribution. These calculations +assume that a rotational sequence has developed by the epoch of +interest for the relevant 𝑇eff, i.e., that stars have spun down into +the unsaturated regime. This is not the outcome for M dwarfs in +the models of Epstein & Pinsonneault (2014), which explains their +large predicted age errors, but it is the case at least by 4 Gyr for this +𝑇eff range (see Sec. 7.2). +5.4.1 +Metallicity +The rotational evolution of cool dwarf stars is expected to be +metallicity-dependent through the effects of opacity and mean +atomic weight on the interior structure and dynamics of the star, +which in turn govern the moment of inertia, and the scale of con- +vection that drives the dynamo responsible for star’s magnetic field, +activity, and mass loss through a wind. Moreover, at a fixed 𝑇eff, a +more metal-rich MS star will have a higher mass. The [Fe/H] of the +nearby stellar clusters used to construct gyrochronologies is close to +solar: M67, Ruprecht 147, and NGC 6811 are within uncertainties +of solar (Pasquini et al. 2008; Bragaglia et al. 2018; Molenda- +Żakowicz et al. 2014), while both Praesepe and the Hyades have +[Fe/H] = +0.15 (Cummings et al. 2017). However, stars in the solar +neighborhood have metallicities ranging from −1 to +0.5 (Toyouchi +& Chiba 2018), and the M dwarf stars observed by Kepler have a +[Fe/H] distribution that is approximately Gaussian with a mean of +−0.09 dex and standard deviation of ±0.22 dex (Gaidos et al. 2016). +We predict the effect of non-solar metallicity on the duration +of the stellar PMS phase and the subsequent spin-down on the +MS, which we model as exponential “saturated" spin-down from an +initial rotation period 𝑃0 to the critical value 𝑃crit = 0.13𝜏𝑐 at which +the Rossby number exceeds a critical value 0.13 (Wright et al. 2018), +and unsaturated, Skumanich-like power-law spin-down thereafter. +The error in an age based on solar-metallicity gyrochrones induced +by a non-solar metallicity is the sum of the variation in these two +intervals. We used the Dartmouth standard (non-magnetic) models +to compute these for a range of masses and [Fe/H] in 0.1 dex intervals +from −0.7 to +0.5. The MS saturated spin-down interval is taken to +be: +𝑇sat = 𝐼 +𝑛 log 𝑃crit +𝑃0 +, +(5) +where 𝐼 is the moment of inertia (nearly constant for MS M dwarfs) +and Γ is the (constant) torque parameter. Both 𝐼 and 𝑃crit are +metallicity-dependent and were calculated using the Dartmouth +models. We adopted the scaling relationship for the torque parame- +ter from van Saders & Pinsonneault (2013), which is based on Matt +et al. (2012): +Γ ∝ 𝑅3.1 +∗ +𝐿0.56 +∗ +𝑀−0.22 +∗ +𝑝0.44 +phot +(6) +where 𝑝phot is the pressure at the photosphere. 𝑝phot will be propor- +tional to gravity and inversely proportional to the specific opacity 𝜅. +The mass inferred for a given MS 𝑇eff will also vary with [Fe/H] be- +cause the radius and hence luminosity changes. Using the 𝑀∗−𝑀𝐾𝑠 +(mass-luminosity) relations of Mann et al. (2019), and exploiting the +fact that the bolometric correction for the 𝐾𝑠-band is only weakly +dependent on [Fe/H] (Mann et al. 2015) and thus will be approx- +imately fixed at a given 𝑇eff, 𝐿∗ ≈ 𝑀2.7 +∗ +in this range. With that +relation and the Stefan-Boltzmann law, at a given mass and 𝑇eff, +Eqn. 6 becomes: +Γ ∝ 𝑅4.53 +∗ +𝜅−0.44 +(7) +In the interiors of cool stars where bound-free opacity dominates, the +opacity will scale linearly with the metal abundance 𝑍 ∝ 10[𝐹𝑒/𝐻]. +We calculated for Γ for the solar-metallicity case by finding the age +at which Skumanich-like rotational evolution marched backward +from the M67 gyrochrone (Dungee et al. 2022) gives 𝑃rot=𝑃crit +and then setting Γ so that exponential spin-down from 𝑃rot=𝑃0 also +reach 𝑃crit at this age, i.e: +Γ = +𝐼0 +4Gyr +� 𝑃4 +𝑃crit +�1/0.62 +log 𝑃crit +𝑃0 +(8) +The PMS interval of M dwarfs is not readily defined since these +stars gradually approach the MS. Since we are concerned only with +the effect of stellar contraction on spin-up, we define the interval +at which the timescale for contraction and spin-up (taken to be the +logarithmic change in the momentum of inertia with time) greatly +exceeds the timescale for spin-down by saturated magnetic activity. +Here, we adopted “greatly" to be 10× but our estimates are not +sensitive to the exact figure. +The top panel of Fig. 5 plots the variation in PMS duration rel- +ative to the solar-metallicity cases vs. [Fe/H] for the same mass/𝑇eff +cases as the top panel. The middle panel of Fig. 5 plots variation +of MS 𝑇sat relative to the solar-metallicity value vs. [Fe/H] for 40 +different mass tracks with MS 𝑇eff falling within 3200-4200K. As +one metric of the age error due to non-solar [Fe/H] we added the +PMS and saturated spin-down durations and performed linear re- +gression with [Fe/H] over the range of −0.3 to +0.3, where most +M dwarfs fall. This slope of this regression vs. 𝑇eff is the light- +colored curve in the bottom panel of Fig. 5. The sensitivity of the +age estimates to [Fe/H] increases from 0.05 Gyr/dex at 3200K, peak- +ing at 0.2 Gyr/dex at around 3900K, and declining in the K dwarf +regime (Fig. 5). This behavior is almost entirely a consequence of +the metallicity dependence of the braking torque (van Saders & Pin- +sonneault 2013), with a lesser contribution from changes in mass +with metallicity at a fixed 𝑇eff. +Although the power-law index of the Skumanich-like spin +down observed on long time-scales is not considered metallicity- +dependent, metallicity-dependent torque could still impart addi- +tional deviation from predictions based on solar-metallicity gy- +rochrones during the transition to purely power-law behavior. To +quantify this, we performed calculations of rotation evolution using +the models of Claytor et al. (2020b), which use the torque scaling of +Eqn. 6. We used the stellar model interpolation and Markov Chain +Monte Carlo (MCMC) tools in kiauhoku (Claytor et al. 2020a) +to make 𝑃rot-based age estimates of 4 Gyr-old model stars with a +given 𝑇eff and varying [Fe/H], but assuming solar [Fe/H]. We per- +formed a linear regression of the inferred age minus the “true" age +(4 Gyr) versus [Fe/H] for different values of 𝑇eff, and the slope is +plotted as the black curve in the bottom panel of Fig. 5). (These +calculations do not go below 3500 K because of incompleteness in +the model grid.) This curve has the same overall shape as our curve +of PMS+MS age sensitivity (light colored curve) but is generally +larger in magnitude, as expected. +Based on a comparison of the curves in Fig. 5a, we approxi- +mately incorporate the metallicity dependence of spin-down during +the transition to pure Skumanich-like behavior by doubling the off- +MNRAS 000, 1–16 (2022) + +8 +E. Gaidos et al. +3000 +3250 +3500 +3750 +4000 +4250 +0 +5 +10 +15 +20 +25 +30 +M37 (470 Myr) +3000 +3250 +3500 +3750 +4000 +4250 +0 +5 +10 +15 +20 +25 +30 +Praesepe (600 Myr) +3000 +3250 +3500 +3750 +4000 +4250 +effective temperature (K) +0 +5 +10 +15 +20 +25 +30 +rotation period (days) +Hyades (700 Myr) +3000 +3250 +3500 +3750 +4000 +4250 +0 +5 +10 +15 +20 +25 +30 +NGC 6811 (1.0 Gyr) +Figure 4. Fits of the cool dwarf rotational sequences of four open clusters. (The existence of a rotation sequence in M37 M dwarfs is unclear). Black points +indicate stars used in iterative, outlier-rejection fits. The solid red line is the fit, the dashed red lines are the ±2.5𝜎 rejection boundaries, and the dotted magenta +lines show the extent of the scatter induced solely by an error of 𝑇eff of 75K. The blue line is the critical rotation period for 𝑅𝑜 = 0.13 using the relation +between convective turnover time and luminosity of Jeffries et al. (2011), with luminosities from the Dartmouth stellar evolution models (Dotter et al. 2008; +Feiden 2016). Stars below this line will have saturated magnetic fields and experience exponential spin-down. +set during the MS saturated and adding it to the PMS offset. This is +the heavy colored curve in the bottom panel. We multiply the slope +by a typical uncertainty of ±0.1 dex in [Fe/H] and add this (up to +±150 Myr) to our error budget. We also use kiauhoku to calculate +individual [Fe/H]-dependent corrections for the age of each star +with known metallicity and 𝑇eff>3500K that can be added to age +estimated from our solar-metallicity gyrochronology. +6 +RESULTS: PLANET HOST STAR AGES +Table 6 provides the 𝑇eff and [Fe/H] (if available) that were used +for the gyrochrone calculations, the rotation period, the method +and instruments used to obtain it and the reference, and the es- +timated age and uncertainty. If a metallicity is available, we also +provide, but do not incorporate, a kiauhoku-calculated value for +the [Fe/H]-dependent correction. Only the first 50 entries are shown; +the complete machine-readable table is provided on Zenodo (DOI: +10.5281/zenodo.7578269). Figure 6 compares the empirical ages +of host stars to ages using generated with the kiauhoku model +(Claytor et al. 2020a). There is good agreement among the warmer +stars in the sample, but a clear trend of older kiauhoku-based es- +timates for cooler 𝑇eff, where the models have not been calibrated. +The systematic offset of model-derived ages for the coolest stars +further illustrates the need for calibrators across the full range of +temperature and age. +The distribution of ages assigned to Monte Carlo realizations +is plotted in Fig. 7, where we have plotted the KOIs and all other +host stars with separate curves as distinct in terms of sensitivity +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +9 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +[Fe/H] +0.10 +0.05 +0.00 +0.05 +0.10 + age [Gyr] +variation in PMS duration +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +[Fe/H] +0.4 +0.2 +0.0 +0.2 +0.4 + age [Gyr] +variation in MS time to Pcrit +3200 +3400 +3600 +3800 +4000 +4200 +Teff [K] +0.0 +0.5 +1.0 +1.5 +2.0 + age [Gyr] per dex +age sensitivity to [Fe/H] +3200 +3400 +3600 +3800 +4000 +4200 +Teff +Figure 5. Difference in actual age relative to gyrochrone age due to non-solar +metallicity due to variation in the PMS duration (a) and MS interval required +for spin-down sufficiently for Ro to exceed the critical value 0.13 and the star +to leave the saturated phase of activity and follow power-law Skumanich- +like spin-down (middle). Positive values means that a star will be older than +its gyrochronologic age. Each color corresponds to a different mass track +in Dartmouth standard model calculations, converted to 𝑇eff on the main- +sequence using the empirical relation of Pecaut & Mamajek (2013). The light +colored line in the bottom panel is the slope of the summed intervals vs. +[Fe/H] obtained at each value of 𝑇eff. The black curve is the slope calculated +from a full model of metallicity-dependent spin-down over a representative +age of 4 Gyr. The heavy colored line is the PMS interval plus twice the MS +interval used as an approximation for the actual sensitivity that reproduces +the shape and magnitude of the model simulations. +and systematics as well as (potentially) stellar populations, and +compare these to the isochrone-based distribution for all Kepler +stars from Berger et al. (2020). For upper limits, ages were drawn +from a uniform distribution from zero to the upper limit. We did +not exclude known binary stars from these distributions since the +effect of binaries depends on semi-major axis in a manner that is +still being actively investigated (e.g., Messina 2019). +To help discern between actual structure and systematics in +the age distribution, we created a mock stellar population with a +uniform age distribution, 𝑇eff drawn with replacement from the +actual catalog, and 𝑃rot calculated with a simple model of the spin- +0 +2 +4 +6 +8 +10 +12 +14 +model age +0 +2 +4 +6 +8 +10 +12 +14 +empirical age +3200 +3400 +3600 +3800 +4000 +4200 +equilibrium temperature +Figure 6. Empirical ages of host stars based on the rotation-age relations in +Fig. 2 vs. the estimates using the kiauhoku model (Claytor et al. 2020a). +0 +2 +4 +6 +8 +10 +12 +14 +age (Gyr) +0.00 +0.05 +0.10 +0.15 +0.20 +probability density +other +Kepler +Kepler (Berger+2020) +Figure 7. Distribution of estimated ages for KOIs and all other known +M dwarf host stars, accounting for the uncertainties. The grey curve is +the isochrone-based age distribution for all Kepler stars from Berger et al. +(2020). +down of stellar population. The model assumed solid-body rotation +and power-law spin-down with index 𝛾 = 0.62 for 𝑅𝑜 > 0.13, i.e., +at MS ages 𝑡 > 𝑡crit where the condition 𝑃rot > 𝑃crit, as defined +before, is satisfied. For main-sequence ages 𝑡 < 𝑡crit, stars undergo +exponential spin-down with a time constant set such that at 𝑡 = 0, +𝑃rot is equal to an initial value 𝑃0. The initial period is derived from +the specific momentum distribution of among ∼10 Myr-old M dwarf +members of the Upper Scorpius star-forming region (Somers et al. +2017), and the moments of rotational inertia from the Dartmouth +stellar evolution models. Main sequence ages were a random draw +from a uniform 0-10 Gyr distribution, minus the PMS duration +as taken from Dartmouth solar-metallicity models (Dotter et al. +2008). (The 𝑃rot of PMS stars is fixed at 𝑃0, but this choice is not +important since these stars were subsequently excluded.) We added +the Gaussian-distributed error of 7.4% to the periods, the median +of the distribution of actual error, and then derived the ages and +MNRAS 000, 1–16 (2022) + +10 +E. Gaidos et al. +0 +2 +4 +6 +8 +10 +12 +14 +age (Gyr) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +probability density +actual +uniform +Figure 8. Distribution of actual ages vs. those inferred from a “mock" +population of stars with the same 𝑇eff distribution as the exoplanet host stars, +but a uniform 0-10 Gyr age distribution (see text for details). +errors with the same routines used for the actual exoplanet host star +catalog. The actual and mock distributions are compared in Fig. 8. +There is a marked deficit and marked structure at <3 Gyr in the +age distribution of stars in both the Kepler and non-Kepler samples +(Fig. 7), but this feature also appears in the simulation of a uniform +distribution of age (Fig. 8), showing that the inferred age distribu- +tion is heavily affected by systematics, i.e. the incomplete working +gyrochronology over the entire 𝑇eff range at young ages. Monte +Carlo realizations that fall in these gaps are assigned upper limits +and thus not correctly represented in this distribution. Both the in- +ferred Kepler and non-Kepler distributions decline with age, and +more rapidly than that inferred from the mock uniform-age popula- +tion. A declining rotation-based age distribution was also inferred +for solar-type Kepler host stars and is in part a bias caused by the +difficulty of detecting the lower amplitude, longer period rotational +signals that are more prevalent around older stars (Walkowicz & +Basri 2013). This pattern is also mimicked by an age distribution of +Kepler target stars based on isochrone analysis (Berger et al. 2020, +grey curve in Fig. 7); this is also biased towards younger (and more +massive stars) that evolve more quickly. The Kepler distribution +peaks at older ages than the non-Kepler sample, which could be +due to the greater sensitivity and longer monitoring interval of the +prime mission, but perhaps also because Kepler was observing a +field centered at 𝑏 = 13 deg containing a slightly older population +further above the Galactic plane. The Kepler distribution terminates +at 10 Gyr, about the age of the Galactic disk. The distribution of +non-Kepler host stars has a tail that extends well beyond 10 Gyr but +this is largely due to host stars with significant uncertainties in 𝑃rot, +along with a handful of binaries (see below). +Binaries: Twenty-six of the 249 planet host stars are known +to have stellar companions. This 11% fraction is much lower than +26.8±1.4% among field M dwarfs (Winters et al. 2019). Since exo- +planet hosts are comparatively well-studied among cool field stars, +this is very unlikely due to limited characterization of these stars. +Instead, it probably reflects survey/detection bias where binary stars +are avoided in exoplanet surveys because it is usually more difficult +to detect planets around them (Kraus et al. 2016; Ziegler et al. 2018, +2021; Su et al. 2021; Clark et al. 2022), and because contamina- +0 +2 +4 +6 +8 +10 +12 +14 +age (Gyr) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +probability density +singles +binaries +Figure 9. Distribution of estimated ages for single vs. binary/multiple host +stars, accounting for the uncertainties. The irregularity of the binary distri- +bution is sampling noise: only 26 systems constitute the binary sample. +tion of the host star signal by other stars is detrimental to precise +measurement of RVs (Cunha et al. 2013). +The rotational history of stars in multiple systems can differ +greatly from that of single stars. Stars with stellar companions are +more likely to be rapidly rotating relative to single stars of the same +age/mass (Kraus et al. 2012; Simonian et al. 2019). Very close (≪ 1 +au) binaries transfer orbital AM to rotational AM via tidal torques +(Fleming et al. 2019). At moderate separations (∼100 au) a stellar +companion will truncate a disk and shorten its viscous lifetime +(Cieza et al. 2009; Kraus et al. 2012; Rosotti & Clarke 2018). +This removes a sink of stellar angular momentum, allowing the +star to spin-up unimpeeded during pre-main sequence contraction +(Messina 2019). +We computed ages based on rotation without regard to mul- +tiplicity, but warn that in the case of binaries, such ages could be +seriously in error. In this sample, at least, the distribution of rotation +periods does not appear remarkably different from single stars (Fig. +1) nor does the age distribution of known binaries appear remark- +able (Fig. 9), although it is greatly limited by the small sample size. +This could be because the smaller fraction of binaries that do appear +in the catalog tend to be very wide, and the effects on rotation and +hence age are negligible. +Anomalously old stars: Four host stars (GJ 27.1, GJ 667 C, +HD 238090, and HIP 70849) are assigned problematic ages that are +> 2𝜎 older than 10 Gyr, the nominal age of the Galactic disk. These +are unlikely to be Galactic halo or former globular cluster members +because unusual abundances and peculiar motion characteristic of +such stars would have been noted. Three of the stars are in binaries, +in which stellar companions could directly or indirectly affect the +rotation evolution. The fourth, GJ 27.1, has a 140±10 day rotation +period estimated from ASAS-SN photometry, far longer than that +expected for an early-type M dwarf in the Galactic disk, and the +age is obviously unphysical: 35 ± 9 Gyr. The star’s metallicity has +not been reported. Potentially the rotation period is an artifact, e.g., +confusion with another star (the survey’s resolution is 15"). +Young stars: No PMS-ages were assigned to our stars, which +is expected since we removed all known disk-hosting and PMS stars +from our catalog. Fourteen stars can only be assigned upper limits +for ages, and of these 8 are younger than 1 Gyr at the 95-percentile +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +11 +Table 1. TIME-Table: Catalog of Cool Host Stars with Established Rotation Periods.1 +Name +𝑇eff +Fe/H +Period (unc) +Method2 (Instruments3) +Reference +Age (unc) +Corr4 +Note5 +[K] +[dex] +[days] +Gyr +Gyr +EPIC 201170410 +3650 +-0.05 +20.16 (1.6) +P (K2) +this work +1.77 (0.58) ++0.00 +binary +EPIC 211822797 +3856 ++0.14 +15.88 (0.72) +P (K2) +Reinhold & Hekker (2020) +1.18 (0.24) ++0.03 +G 264-012 +3326 ++0.10 +100.0 (6.0) +PS (TE/ME/AN/CA) +Amado et al. (2021) +6.66 (3.26) ++0.12 +G 9-40 +3713 ++0.04 +34.08 (11.46) +P (K2) +Reinhold & Hekker (2020) +5.35 (2.57) ++0.00 +GJ 1132 +3270 +-0.12 +122.3 (5.5) +P (ME) +Cloutier et al. (2017) +6.31 (3.23) +—– +GJ 1148 +3304 ++0.16 +73.5 +P (HA) +Hartman et al. (2011) +4.44 (1.6) ++0.51 +no error +GJ 1214 +3252 ++0.29 +125.0 (5.0) +P (ST) +Mallonn et al. (2018) +5.91 (3.0) +—– +GJ 1252 +3458 ++0.10 +64.0 (4.0) +P (WA) +Shporer et al. (2020) +6.61 (2.34) ++0.58 +GJ 1265 +4052 +-0.04 +29.17 (6.27) +P (K2) +Reinhold & Hekker (2020) +4.44 (1.53) ++0.11 +GJ 15 A +3607 +-0.32 +43.82 (0.56) +PS (Fa/HI) +Howard et al. (2014) +5.87 (1.14) +-0.66 +binary +GJ 176 +3680 ++0.14 +38.92 +P (AS) +Kiraga & Stepien (2007) +5.6 (0.9) ++0.52 +no error +GJ 229 +3790 +—– +27.3 +P (AS) +Suárez Mascareño et al. (2016) +3.77 (0.51) +—– +binary,no error +GJ 251 +3389 +-0.03 +122.1 (2.05) +PS (WA/CA) +Stock et al. (2020) +12.52 (5.86) +—– +GJ 27.1 +3578 +—– +139.0 (3.5) +P (AS) +this work +34.94 (7.93) +—– +too old +GJ 273 +3317 ++0.09 +93.5 (16.0) +S (HA) +Suárez Mascareño et al. (2017) +6.12 (3.17) ++0.09 +GJ 3138 +3899 +-0.30 +42.5 +S (HA) +Astudillo-Defru et al. (2017a) +8.44 (0.95) ++0.14 +no error +GJ 3293 +3600 ++0.11 +41.0 +S (HA) +Astudillo-Defru et al. (2017a) +5.19 (1.06) ++0.40 +no error +GJ 338 B +3770 +-0.03 +10.17 +P (K2) +Magaudda et al. (2020) +<0.91 +—– +binary,no error +GJ 3470 +3611 ++0.20 +20.7 (0.15) +P (Fa) +Biddle et al. (2014) +1.65 (0.54) ++0.16 +GJ 3473 +3347 ++0.11 +168.3 (3.65) +P (ME/TJ) +Kemmer et al. (2020) +16.46 (8.46) +—– +binary +GJ 357 +3480 +-0.12 +77.8 (2.1) +P (AS/AN/NS) +Luque et al. (2019) +9.76 (3.31) +-0.30 +GJ 367 +3687 +-0.01 +48.0 (2.0) +P (WA) +Lam et al. (2021) +7.95 (1.31) ++0.04 +GJ 3779 +3324 ++0.00 +95.0 (5.0) +PS (ME/CA) +Luque et al. (2018) +6.13 (2.87) +—– +GJ 3929 +3369 ++0.00 +122.0 (13.0) +P (HA/AN) +Kemmer et al. (2022) +11.1 (5.76) +-0.36 +GJ 393 +3548 +-0.18 +34.15 (0.22) +P (K2) +Amado et al. (2021) +3.32 (0.87) +-0.24 +GJ 3942 +3850 +-0.04 +16.3 (0.1) +PS (AS/HA) +Suárez Mascareño et al. (2018) +1.18 (0.24) ++0.01 +GJ 3998 +3825 +-0.16 +33.0 +P (AP) +Giacobbe et al. (2020) +5.25 (0.69) +—– +no error +GJ 411 +3719 +-0.36 +56.16 (0.27) +P (Fa) +Díaz et al. (2019) +10.73 (1.52) +-0.52 +GJ 414 A +4120 ++0.24 +40.0 (4.0) +P (KE) +Dedrick et al. (2021) +5.87 (1.77) ++0.51 +binary +GJ 4276 +3440 ++0.12 +6.14 +P (AP) +Giacobbe et al. (2020) +— +—– +no age,no error +GJ 436 +3479 ++0.01 +44.1 (0.2) +P (Fa) +Lothringer et al. (2018) +4.32 (1.09) ++0.00 +GJ 486 +3290 +-0.15 +49.9 (5.5) +P (AS/LC/WA/TJ/OS) +Caballero et al. (2022) +3.51 (0.81) +-0.13 +GJ 514 +3755 +-0.09 +30.0 (0.9) +S (HA) +Suárez Mascareño et al. (2017) +4.11 (0.6) ++0.03 +GJ 581 +3490 +-0.09 +130.0 (2.0) +S (HA) +Robertson et al. (2014) +23.26 (7.7) +—– +GJ 625 +3540 +-0.35 +76.79 (0.13) +P (WA) +Díez Alonso et al. (2019) +11.96 (3.14) +—– +GJ 628 +3305 +-0.02 +89.3 (1.8) +P (Fa) +Kane et al. (2017) +5.27 (2.32) +—– +GJ 649 +3741 ++0.03 +23.8 (0.1) +P (AS) +Díez Alonso et al. (2019) +2.96 (0.51) ++0.11 +GJ 667 C +3755 +—– +103.9 (0.7) +S (HA) +Suárez Mascareño et al. (2015) +30.59 (4.06) +—– +binary,too old +GJ 674 +3453 +-0.28 +35.0 (0.1) +P (AS) +Suárez Mascareño et al. (2016) +2.85 (0.72) +-0.28 +GJ 676 A +3827 ++0.23 +41.2 (3.8) +S (HA) +Suárez Mascareño et al. (2015) +7.58 (1.46) ++0.56 +binary +GJ 685 +3844 ++0.10 +19.3 (0.3) +P (TJ) +Díez Alonso et al. (2019) +2.25 (0.36) ++0.15 +GJ 687 +3439 ++0.05 +61.8 (1.0) +P (Fa) +Burt et al. (2014) +5.82 (1.99) ++0.42 +GJ 720 A +4013 ++0.01 +36.05 (1.41) +S (HA) +González-Álvarez et al. (2021) +6.26 (1.05) ++0.21 +binary +GJ 740 +3832 ++0.08 +35.56 (0.07) +S (CA/HA) +Toledo-Padrón et al. (2021) +5.97 (0.72) ++0.32 +GJ 832 +3522 +-0.30 +45.7 (9.3) +S (HA) +Suárez Mascareño et al. (2015) +5.2 (2.13) +-0.98 +GJ 849 +3551 ++0.25 +39.2 (6.3) +S (HA) +Suárez Mascareño et al. (2015) +4.33 (1.55) ++0.68 +GJ 876 +3472 ++0.17 +31.31 (8.15) +P (K2) +Reinhold & Hekker (2020) +2.7 (1.35) ++0.29 +GJ 96 +3892 ++0.14 +29.5 (0.5) +P (WA) +Díez Alonso et al. (2019) +4.67 (0.52) ++0.47 +GJ 9689 +3880 +-0.11 +39.3 (0.4) +S (HA) +Maldonado et al. (2021) +7.35 (0.82) ++0.18 +Gl 49 +3740 ++0.13 +18.4 (0.7) +S (HA) +Suárez Mascareño et al. (2018) +1.55 (0.41) ++0.07 +binary +1The full table is available as a machine-readable table (DOI:10.5281/zenodo.7578269) +2P = photometric. S = spectroscopic. +3AN=All-Sky Automated Survey for Super-Novae (ASAS-SN, Kochanek et al. 2017), AP=APACHE (Sozzetti et al. 2013); AS=All Sky Automated Survey +(ASAS, Pojmanski 2002); CA=CARMENES (Quirrenbach et al. 2016), ES=ESPRESSO (Pepe et al. 2021); FA=Fairborn (Henry 1999); HA=HARPS (Pepe +et al. 2000); HN=Hungarian Automated Telescope Network (HAT-Net, Bakos et al. 2004); K2=K2 (Howell et al. 2014), KE=Kepler (Borucki et al. 2010); +LC=Las Cumbres Observatory Global Telescope (LCO, Brown et al. 2013); ME=MEarth (Berta et al. 2012), NS=Northern Sky Variability Survey (NSVS, +Woźniak et al. 2004); OS=Observatorio de Sierra Nevada; SP=SPIRou (Donati et al. 2020); ST=STELLA (Strassmeier et al. 2004); TE=TESS (Ricker et al. +2014); TJ=Telescope Joan Oró (TJO, Colomé et al. 2010); WA=Wide Angle Search for Planets (WASP, Pollacco et al. 2006). +4Metallicity-dependent age correction to be added to value for solar-metallicity. +5If no period error is provided, an uncertainty of 1 day is assumed. +MNRAS 000, 1–16 (2022) + +12 +E. Gaidos et al. +level and not known to be binaries. The rotation period of one of +these, TOI-620, is tentative and the star is also a suspected binary +(Reefe et al. 2022). The other seven are K2-detected systems: K2-43, +K2-239, K2-240, which hosts two transiting Neptune-size planets, +has been detected in X-rays (Foster et al. 2022); K2-284, previously +reported having a young age by David et al. (2018), K2-324, K2-354, +and KOI-5879, a flaring M dwarf (Yang & Liu 2019). +6.1 +Individual Noteworthy Systems +GJ 229: The nearby (5.76 pc) M1 dwarf Gliese/GJ 229 has an +ultra-cool (T7-type) dwarf companion (Nakajima et al. 1995). The +primary’s 27.3-day rotation period was determined from ASAS- +SN photometry (Suárez Mascareño et al. 2016) and we estimate +an age of 3.8 ± 0.5 Gyr, with the caveat that the existence of the +companion on a 29 au orbit could have affected the rotation history. +No uncertainty in 𝑃rot was reported but this is likely to be small +given the multi-year baseline of ASAS-SN. +GJ 1214: This nearby M4-type dwarf with a well-studied tran- +siting “sub"-Neptune-size planet on a 1.58-day orbit (Charbonneau +et al. 2009). Based on the 125 ± 5 day rotation period identified in +STELLA photometry by Mallonn et al. (2018), we estimate an age +of 5.9 ± 3.1 Gyr. +LHS-1815: This M1-type dwarf (aka TOI-704) hosts a transit- +ing Earth-size planet on a 3.8-day orbit. The star lies 1.8 kpc above +the Galactic plane and kinematically belongs to the “thick" Galactic +disk population (Gan et al. 2020). We estimate an age of 7.3 ± 1.3 +Gyr, consistent with the expected age of that population. +K2-22: K2-22 is a late K dwarf that hosts what has been pro- +posed to be a “evaporating" planet on a 9-hour orbit that manifests +itself as quasi-periodic dimming due to accompanying dust cloud +(Sanchis-Ojeda et al. 2015). The highest peak in a Lomb-Scargle pe- +riodogram in K2 photometry; is at 7.61±0.26 days but the shape of +the lightcurve (Fig. A1) suggests this is one-half the period (Sanchis- +Ojeda et al. 2015). A period of 15.2 ± 0.5 days yields an estimated +age of 1.1 ± 0.2 Gyr, but this star has an M dwarf companion at a +projected separation of 460 au Sanchis-Ojeda et al. (2015) which +could have affected its rotation history. +Barnard’s Star (GJ 699): This very metal-poor, high peculiar +motion M4-type is classified as intermediate between the Galac- +tic Disk and Halo populations (Gizis 1997); a putative Doppler +RV-detected planet (Ribas et al. 2018) around this star has been +disputed (Lubin et al. 2021). It is not in our current catalog because +its 𝑇eff is marginally cooler than our 3200K cut-off, but, motivated +by its unusual nature and recently confirmed 𝑃rot of 145±15 days +(Toledo-Padrón et al. 2019; Terrien et al. 2022), we compare this +to the cool extremum of the Dungee et al. (2022) M67 gyrochrone, +which reaches 120 days at 3250K (Fig. 3) and at cooler temperatures +is essentially an unconstrained extrapolation. Thus the gyrochronol- +ogy suggests an age older than M67, as expected, but extension of +M dwarf gyrochrones into the fully convective area is needed before +assigning any robust age. +7 +SUMMARY AND DISCUSSION +7.1 +Rotation and Ages of M dwarf Exoplanet Hosts +The value of robust ages for exoplanet studies, and advances in the +gyrochronology of older cool M dwarfs motivated us to catalog +rotation periods among late K and early M-type dwarfs (𝑇eff=3200- +4200K) that host known planets, and to apply empirical, 𝑇eff- +dependent rotation-age relations to estimate ages and their standard +errors. This complements work on calibrated rotation-based ages +among younger PMS stars (Kounkel et al. 2022). We cataloged 249 +stars with rotation periods, 227 of which we are able to estimate ages +with a median error of 20% and mode of 14%, and to an additional 8 +we assign upper limits (Table 6). Our fractional error is significantly +higher than the 5-10% estimated by Otani et al. (2022), probably +because we include additional potential sources of error. Figure 10 +shows ages of candidate or confirmed planets around these stars +and the distribution with semi-major axis, radius, and equilibrium +temperature, as reported in the NASA Exoplanet Archive. +The age distributions of both Kepler and non-Kepler host stars +peak at around 3 Gyr with a steady decline to near zero at 10 +Gyr, the age of the Galactic disk (Fig. 7). The resemblance of the +actual and “mock" populations (the latter with a uniform 0-10 Gyr +age distribution) shown in Figure 8 indicates that the structure of +the distribution, particularly at young ages, is partly due to the +discontinuous and limited coverage of the current gyrochronology +and dispersion of the distribution due to error. The peaks at <3 +Gyr correspond approximately with the location of the calibration +ages, and are likely artefacts due to ages that cannot be assigned +in certain regions of 𝑇eff-𝑃rot space and have only upper limits +assigned. Other effects impacting the derived distribution include +the opposing biases against detection of planets around younger, +more rapidly rotating, and more active stars (Miyakawa et al. 2022), +and against detection of rotational variability among older, slowly +rotating, less active stars (Morris 2020). 2 Several versions of the +local star formation history based on white dwarf cooling ages and +Gaia astrometry also peak at around 3-5 Gyr (Isern 2019; Mor +et al. 2019; Alzate et al. 2021), so the distribution of host star ages +could reflect this. Our error analysis (Sec. 5) shows that one limiting +source of error could prove to be the precision of stellar parameters, +i.e. 𝑇eff and [Fe/H]. The sensitivity to 𝑇eff is due to the steepness +of rotation sequences for very cool dwarfs. 𝑇eff, which is related to +the surface brightness and hence convective vigor of a star, is the +appropriate independent variable for gyrochronology, but must be +inferred from observables. The ultimate limit on accurate values of +𝑇eff precision for M dwarfs is the challenge of establishing a reliable +temperature scale. +7.2 +Caveats and Limitations +We have adopted the values and standard errors of 𝑃rot from the +literature at face value. The possibility that the true period is twice +the published value needs to be considered in cases of stars with +anomalous rapid rotation and young ages. Ground-based observa- +tions can also suffer from aliasing imposed by diurnal, lunar, and +annual window functions. +Our gyrochronology assumes that the narrow rotational se- +quence observed among the late K dwarfs and the warmer M dwarfs +in the 2.7 Gyr-old Ruprecht 147 cluster (Curtis et al. 2020) extends +to 3200K by 2.7 Gyr, and that the 𝑛 = 0.62 power-law spin-down +derived by (Dungee et al. 2022) also applies to cooler M dwarfs +at later times. This assumption could fail if the rotational sequence +among M67 M dwarfs was formed by stalling, rather than a tran- +sition from saturated to un-saturated braking laws. Core-envelope +re-coupling is expected to become weaker and take longer towards +the fully convective boundary, which could mean that a rotational +2 “Stalling" of spin-down due to core-envelope decoupling would result in +broadening of the age distribution, not peak formation. +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +13 +sequence appears much later, or not at all. This depends in part +on the 𝑇eff or mass dependence of the core-envelope coupling time +vs. 𝜏𝑐. This issue can only be resolved by deeper monitoring of +Ruprecht 147 or a cluster of similar age. +Another contributor to our error budget is sensitivity of the +rotation-age relation to non-solar [Fe/H]. In the absence of appropri- +ate calibration, we relied entirely on theoretical models to estimate +this effect. The part due to changes in the structure and moment of +inertia of the star is reasonably well-constrained by observations, +but magnetic field pressure could modulate this. A model that in- +cludes this effect (Feiden 2016) predicts that at a fixed 𝑇eff=3700K, +inclusion of magnetic pressure increases the moment of inertia by +40%, but this is almost entirely due to a change in the mass inferred +for a given 𝑇eff. More uncertain is the metallicity-dependence of +the torque, which, at least in our models, dominates the sensitivity. +We based this scaling on the magnetized wind formulation of Matt +et al. (2012) as reformulated by van Saders & Pinsonneault (2013), +but this was developed for solar-type stars with rotationally-aligned +dipole fields. +Finally, rotation-based ages for binary systems must be care- +fully considered, particularly given the possibility of a third, closer +and unresolved component (Reipurth & Mikkola 2012). While most +published exoplanets have had some sort of screening for binaries, +not all of them cover all the parameter space, and surveys of very +cool KOIs are only now coming to fruition. +7.3 +Outlook +Since 𝑇eff is not an observable and cannot be readily derived with- +out stellar radii, gyrochrones could be established in a common +reddening-corrected color which is also available for stars of inter- +est; ideally, this color should be directly related to 𝑇eff, it should +be relatively [Fe/H]-independent. It should also use redder filters in +which M dwarfs are comparatively bright and reddening is smaller. +These requirements impact the use of Gaia photometry since M +dwarfs are faint in the 𝐵𝑝 synthetic band used to construct 𝐵𝑝 − 𝑅𝑝 +colors. Analysis of color magnitude diagrams of Kepler M dwarfs +using PanSTARRS photometry suggest that 𝑔-𝑌 holds promise (A. +Ali, pers. comm.). +Establishing precise M dwarf metallicities is a work in progress +(e.g., Passegger et al. 2022). More challenging will be to validate +the effect of metallicity on rotation-age relations, which here we +have here treated only via stellar interior models and torque-law +scaling. Tests of the metallicity-scaling of the torque law are des- +perately needed. Metal-poor or metal-rich clusters are relative rare +(Heiter et al. 2014) and thus, statistically, found at greater distances +where observations to establish rotational sequences will be chal- +lenging. The wealth of binaries provided by Gaia (El-Badry et al. +2021) might serve as a road to calibration over a wider range of +stellar parameters (Otani et al. 2022), provided sufficiently wide +examples can be identified and precise ages for the primaries can +be determined by other means. +Gyrochronology of exoplanet host stars is an ongoing effort +and we envision the TIME-Table to be a “living" catalog of very +cool dwarf rotation periods and ages that is periodically updated +and revised with new discoveries and advances in gyrochronology. +New planetary systems are constantly being detected, validated, +or confirmed, particularly by the TESS mission, now in its fifth +year. Rotational variability is being detected by two ongoing space +surveys: TESS and, with much longer baseline but much sparser +cadence, Gaia (Distefano et al. 2022). Ground-based surveys like +ZTF and the Rubin Observatory (Hambleton et al. 2022) can provide +observations of distant field stars and young clusters. Although the +TESS 27-day sector interval severely limits its ability to detect the +rotation of older field stars (Claytor et al. 2022), those stars in and +around the two Continuous Viewing Zones around the ecliptic poles +are observed for multiple sectors and in principle it is possible to +detect longer periods (Hedges et al. 2020; Claytor et al. 2022). +Many rotation periods have been established by analysis of +Doppler RV residuals or indicators of activity in the time-series +high-resolution spectroscopy obtained to detect, confirm, or mea- +sure the masses of planets (Suárez Mascareño et al. 2015). Terrien +et al. (2022) showed that detection of the periodic signal in the +Zeeman broadening of lines can reveal rotation of magnetic ac- +tive regions on the star and yield a rotation period. The Zeeman +effect increases with wavelength-squared, and the proliferation of +high-resolution spectrographs operating in the infrared could lead +to additional rotation periods using this approach. Alternatives to +Lomb-Scargle periodogram analysis which are more robust to spot +evolution such as autocorrelation and Gaussian process regression +(Angus et al. 2018; Nicholson & Aigrain 2022) could be used to +obtain more precise ages. Age-dating could also adopt a Bayesian +approach, with Galactic population age distributions as priors (e.g., +Mor et al. 2019; Cukanovaite et al. 2022). +Last but not least, additional observations of open clusters +for calibration will improve the gyrochronology and lead to more +precise (and hopefully more accurate) ages. In particular, the 𝑇eff +range of existing gyrochrones (including M67) should be extended +through the fully convective boundary to include mid- and late- +type M dwarfs representing hosts stars of particular interest (e.g., +TRAPPIST-1), and, foremost, to establish whether a narrow rota- +tional sequence appears by a few Gyr — without which gyrochronol- +ogy is futile. The small effective area and large pixel size of TESS +greatly limits its utility here, since older clusters are rare and hence +more distant. The Plato mission will offer only limited improve- +ment over TESS (15" vs. 20" pixels). However, the Roman Space +Telescope will have a field of view of 0.28 deg2 with 0.11" pixels. +Ages of calibrator clusters could see refinement from a combi- +nation of Gaia parallaxes, asteroseismology, and high-throughput +spectroscopy (Fu et al. 2022). Otherwise, much of the observations +need to be performed from the ground using wide-field telescopes +with sufficient aperture. Wide-field adaptive optics can alleviate the +issues of source confusion in the fields of more distant clusters. +For example, ground-layer adaptive optics (GLAO) can provide a +factor of 2–3 improvement in spatial resolution compared to seeing- +limited observations while still capturing an entire cluster in one ob- +servation (Rigaut 2002). Such improvements enable observations of +dwarfs to spectral type M7 in the majority of clusters older than 1 +Gyr (Dungee 2022)3. +ACKNOWLEDGEMENTS +This work is dedicated to the memory of F.C.G., for whom the +very stars of heaven were new. E.G. and R.D. were supported by +NSF Astronomy & Astrophysics Research Program Grant 1817215. +We thank Jen van Saders and an anonymous reviewer for helpful +feedback on earlier versions of this manuscript. This paper includes +data collected by the Kepler and TESS missions and obtained from +the MAST data archive at the Space Telescope Science Institute +3 https://scholarspace.manoa.hawaii.edu/collections/67c8d51f-97a4-4d45- +8a69-8baeaacaeaf6 +MNRAS 000, 1–16 (2022) + +14 +E. Gaidos et al. +0 +2 +4 +6 +8 +10 +12 +14 +age [Gyr] +10 +2 +10 +1 +100 +semi-major axis [au] +Earth +Neptune +Jupiter +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +equilibrium temperature +Figure 10. 111 confirmed planets with properties from the NASA Exoplanet +Archive with age estimates from this work. Points are scaled with planet +radius and color-coded by planet equilibrium temperature. Ages have not +been filtered for binarity nor corrected for metallicity. +(STScI). Funding for the Kepler mission is provided by the NASA +Science Mission Directorate. STScI is operated by the Associa- +tion of Universities for Research in Astronomy, Inc., under NASA +contract NAS 5–26555. This paper makes use of data from the +MEarth Project, which is a collaboration between Harvard Univer- +sity and the Smithsonian Astrophysical Observatory. The MEarth +Project acknowledges funding from the David and Lucile Packard +Fellowship for Science and Engineering and the National Science +Foundation under grants AST-0807690, AST-1109468, and AST- +1004488 (Alan T. Waterman Award), and a grant from the John +Templeton Foundation. +DATA AVAILABILITY +All data used in this work are in the public domain and available +through various archives. +REFERENCES +Adams F. C., Bodenheimer P., Laughlin G., 2005, Astronomische +Nachrichten, 326, 913 +Agüeros M. A., et al., 2018, ApJ, 862, 33 +Alzate J. A., Bruzual G., Díaz-González D. J., 2021, MNRAS, 501, 302 +Amado P. J., et al., 2021, A&A, 650, A188 +Angus R., Morton T., Aigrain S., Foreman-Mackey D., Rajpaul V., 2018, +MNRAS, 474, 2094 +Astudillo-Defru N., et al., 2017a, A&A, 602, A88 +Astudillo-Defru N., et al., 2017b, A&A, 605, L11 +Bakos G., Noyes R. W., Kovács G., Stanek K. Z., Sasselov D. D., Domsa I., +2004, PASP, 116, 266 +Balona L. A., Abedigamba O. P., 2016, MNRAS, 461, 497 +Barnes S. A., 2007, ApJ, 669, 1167 +Basri G., Streichenberger T., McWard C., Edmond Lawrence I., Tan J., Lee +M., Melton T., 2022, ApJ, 924, 31 +Batalha N. M., et al., 2013, ApJS, 204, 24 +Belokurov V., et al., 2020, MNRAS, 496, 1922 +Berger T. A., Huber D., van Saders J. L., Gaidos E., Tayar J., Kraus A. L., +2020, AJ, 159, 280 +Berta Z. K., Irwin J., Charbonneau D., Burke C. J., Falco E. E., 2012, AJ, +144, 145 +Biddle L. I., et al., 2014, MNRAS, 443, 1810 +Binks A. S., Jeffries R. D., 2014, MNRAS, 438, L11 +Borucki W. J., et al., 2010, Science, 327, 977 +Boulade O., 1998, Experimental Astronomy, 8, 25 +Boyajian T. S., et al., 2012, ApJ, 757, 112 +Bragaglia A., Fu X., Mucciarelli A., Andreuzzi G., Donati P., 2018, A&A, +619, A176 +Brandt T. D., 2021, ApJS, 254, 42 +Brown T. M., et al., 2013, PASP, 125, 1031 +Burke C. J., et al., 2014, ApJS, 210, 19 +Burt J., Vogt S. S., Butler R. P., Hanson R., Meschiari S., Rivera E. J., Henry +G. W., Laughlin G., 2014, ApJ, 789, 114 +Caballero J. A., et al., 2022, A&A, 665, A120 +Canto Martins B. L., et al., 2020, ApJS, 250, 20 +Casali G., et al., 2020, A&A, 639, A127 +Charbonneau D., et al., 2009, Nature, 462, 891 +Christensen-Dalsgaard J., Aguirre V. S., 2018, in Deeg H. J., Belmonte J. A., +eds, , Handbook of Exoplanets. Springer, p. 184, doi:10.1007/978-3- +319-55333-7_184 +Christy C. T., et al., 2022, arXiv e-prints, p. arXiv:2205.02239 +Cieza L. A., et al., 2009, ApJ, 696, L84 +Clark C. A., van Belle G. T., Ciardi D. R., Lund M. B., Howell S. B., Everett +M. E., Beichman C. A., Winters J. G., 2022, AJ, 163, 232 +Claytor Z. R., van Saders J. L., Santos Â. R. G., García R. A., Mathur S., +Tayar J., Pinsonneault M. H., Shetrone M., 2020a, kiauhoku: Stellar +model grid interpolation, Astrophysics Source Code Library, record +ascl:2011.027 (ascl:2011.027) +Claytor Z. R., van Saders J. L., Santos Â. R. G., García R. A., Mathur S., +Tayar J., Pinsonneault M. H., Shetrone M., 2020b, ApJ, 888, 43 +Claytor Z. R., van Saders J. L., Llama J., Sadowski P., Quach B., Avallone +E. A., 2022, ApJ, 927, 219 +Cloutier R., Doyon R., Menou K., Delfosse X., Dumusque X., Artigau É., +2017, AJ, 153, 9 +Cody A. M., Hillenbrand L. A., 2010, ApJS, 191, 389 +Colomé J., Casteels K., Ribas I., Francisco X., 2010, in Radziwill N. M., +Bridger A., eds, Society of Photo-Optical Instrumentation Engineers +(SPIE) Conference Series Vol. 7740, Software and Cyberinfrastructure +for Astronomy. p. 77403K, doi:10.1117/12.857672 +Coughlin J. L., et al., 2016, ApJS, 224, 12 +Cukanovaite E., Tremblay P. E., Toonen S., Temmink K. D., Manser C. J., +O’Brien M. W., McCleery J., 2022, arXiv e-prints, p. arXiv:2209.13919 +Cummings J. D., Deliyannis C. P., Maderak R. M., Steinhauer A., 2017, AJ, +153, 128 +Cunha D., Figueira P., Santos N. C., Lovis C., Boué G., 2013, A&A, 550, +A75 +Curtis J. L., Wolfgang A., Wright J. T., Brewer J. M., Johnson J. A., 2013, +AJ, 145, 134 +Curtis J. L., Agüeros M. A., Mamajek E. E., Wright J. T., Cummings J. D., +2019a, arXiv e-prints, p. arXiv:1905.10588 +Curtis J. L., Agüeros M. A., Douglas S. T., Meibom S., 2019b, ApJ, 879, 49 +Curtis J. L., et al., 2020, ApJ, 904, 140 +David T. J., et al., 2018, AJ, 156, 302 +Dedrick C. M., et al., 2021, AJ, 161, 86 +Denissenkov P. A., Pinsonneault M., Terndrup D. M., Newsham G., 2010, +ApJ, 716, 1269 +Díaz M. R., et al., 2019, arXiv e-prints, p. arXiv:1911.02012 +Díez Alonso E., et al., 2019, MNRAS, 489, 5928 +Distefano E., et al., 2022, arXiv e-prints, p. arXiv:2206.05500 +Donati J. F., et al., 2020, MNRAS, 498, 5684 +Dotter A., Chaboyer B., Jevremović D., Kostov V., Baron E., Ferguson J. W., +2008, ApJS, 178, 89 +Douglas S. T., Curtis J. L., Agüeros M. A., Cargile P. A., Brewer J. M., +Meibom S., Jansen T., 2019, ApJ, 879, 100 +Dungee R., 2022, PhD thesis, University of Hawaii at Manoa +Dungee R., van Saders J., Gaidos E., Chun M., García R. A., Magnier E. A., +Mathur S., Santos Â. R. G., 2022, ApJ, 938, 118 +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +15 +El-Badry K., Rix H.-W., Heintz T. M., 2021, MNRAS, 506, 2269 +Epstein C. R., Pinsonneault M. H., 2014, ApJ, 780, 159 +Esselstein R., Aigrain S., Vanderburg A., Smith J. C., Meibom S., Van +Saders J., Mathieu R., 2018, ApJ, 859, 167 +Feiden G. A., 2016, A&A, 593, A99 +Feng F., et al., 2019, ApJS, 242, 25 +Fleming D. P., Barnes R., Davenport J. R. A., Luger R., 2019, ApJ, 881, 88 +Foster G., Poppenhaeger K., Ilic N., Schwope A., 2022, A&A, 661, A23 +Fragkou V., Parker Q. A., Zijlstra A. A., Vázquez R., Sabin L., Rechy-Garcia +J. S., 2022, ApJ, 935, L35 +Fritzewski D. J., Barnes S. A., James D. J., Strassmeier K. G., 2020, A&A, +641, A51 +Fu X., et al., 2022, arXiv e-prints, p. arXiv:2207.09121 +Gaia Collaboration et al., 2016, A&A, 595, A1 +Gaia Collaboration et al., 2022, arXiv e-prints, p. arXiv:2208.00211 +Gaidos E., Mann A. W., Kraus A. L., Ireland M., 2016, MNRAS, 457, 2877 +Gallet F., Bouvier J., 2015, A&A, 577, A98 +Gan T., et al., 2020, AJ, 159, 160 +Giacobbe P., et al., 2020, MNRAS, 491, 5216 +Gizis J. E., 1997, AJ, 113, 806 +Godoy-Rivera D., Pinsonneault M. H., Rebull L. M., 2021, ApJS, 257, 46 +González-Álvarez E., et al., 2021, A&A, 649, A157 +Gossage S., Conroy C., Dotter A., Choi J., Rosenfield P., Cargile P., Dolphin +A., 2018, ApJ, 863, 67 +Hambleton K. M., et al., 2022, arXiv e-prints, p. arXiv:2208.04499 +Hardegree-Ullman K. K., Cushing M. C., Muirhead P. S., Christiansen J. L., +2019, AJ, 158, 75 +Hartman J. D., Bakos G. Á., Noyes R. W., Sipőcz B., Kovács G., Mazeh T., +Shporer A., Pál A., 2011, AJ, 141, 166 +Hedges C., Angus R., Barentsen G., Saunders N., Montet B. T., Gully- +Santiago M., 2020, Research Notes of the American Astronomical So- +ciety, 4, 220 +Heiter U., Soubiran C., Netopil M., Paunzen E., 2014, A&A, 561, A93 +Helled R., Morbidelli A., 2021, in Madhusudhan N., ed., , ExoFrontiers; +Big Questions in Exoplanetary Science. Institute of Physics, pp 12–1, +doi:10.1088/2514-3433/abfa8fch12 +Henry G. W., 1999, PASP, 111, 845 +Howard A. W., et al., 2014, ApJ, 794, 51 +Howell S. B., et al., 2014, PASP, 126, 398 +Hsu D. C., Ford E. B., Ragozzine D., Ashby K., 2019, AJ, 158, 109 +Huber D., et al., 2016, ApJS, 224, 2 +Isern J., 2019, ApJ, 878, L11 +Jeffries R. D., Jackson R. J., Briggs K. R., Evans P. A., Pye J. P., 2011, +MNRAS, 411, 2099 +Kane S. R., von Braun K., Henry G. W., Waters M. A., Boyajian T. S., Mann +A. W., 2017, ApJ, 835, 200 +Kemmer J., et al., 2020, A&A, 642, A236 +Kemmer J., et al., 2022, A&A, 659, A17 +Kilic M., Munn J. A., Harris H. C., von Hippel T., Liebert J. W., Williams +K. A., Jeffery E., DeGennaro S., 2017, ApJ, 837, 162 +Kiraga M., Stepien K., 2007, Acta Astron., 57, 149 +Kite E. S., Manga M., Gaidos E., 2009, ApJ, 700, 1732 +Kochanek C. S., et al., 2017, PASP, 129, 104502 +Kounkel M., et al., 2019, AJ, 157, 196 +Kounkel M., Stassun K. G., Bouma L. G., Covey K., Hillenbrand L. A., Lee +Curtis J., 2022, AJ, 164, 137 +Kraus A. L., Ireland M. J., Hillenbrand L. A., Martinache F., 2012, ApJ, +745, 19 +Kraus A. L., Ireland M. J., Huber D., Mann A. W., Dupuy T. J., 2016, AJ, +152, 8 +Lam K. W. F., et al., 2021, Science, 374, 1271 +Lamman C., et al., 2020, AJ, 159, 139 +Lammer H., 2013, Origin and Evolution of Planetary Atmospheres. Springer, +doi:10.1007/978-3-642-32087-3 +Lépine S., Shara M. M., 2005, AJ, 129, 1483 +Lothringer J. D., et al., 2018, AJ, 155, 66 +Lu Y., Curtis J. L., Angus R., David T. J., Hattori S., 2022, arXiv e-prints, +p. arXiv:2210.06604 +Lubin J., et al., 2021, arXiv e-prints, p. arXiv:2105.07005 +Luque R., et al., 2018, A&A, 620, A171 +Luque R., et al., 2019, A&A, 628, A39 +Magaudda E., Stelzer B., Covey K. R., Raetz S., Matt S. P., Scholz A., 2020, +A&A, 638, A20 +Maldonado J., et al., 2021, A&A, 651, A93 +Mallonn M., et al., 2018, A&A, 614, A35 +Mann A. W., Brewer J. M., Gaidos E., Lépine S., Hilton E. J., 2013, AJ, +145, 52 +Mann A. W., Deacon N. R., Gaidos E., Ansdell M., Brewer J. M., Liu M. C., +Magnier E. A., Aller K. M., 2014, AJ, 147, 160 +Mann A. W., Feiden G. A., Gaidos E., Boyajian T., von Braun K., 2015, +ApJ, 804, 64 +Mann A. W., et al., 2019, ApJ, 871, 63 +Masci F. J., et al., 2019, PASP, 131, 018003 +Matt S. P., Pinzón G., Greene T. P., Pudritz R. E., 2012, ApJ, 745, 101 +Matt S. P., Brun A. S., Baraffe I., Bouvier J., Chabrier G., 2015, ApJ, 799, +L23 +Messina S., 2019, A&A, 627, A97 +Miyakawa K., Hirano T., Sato B., Okuzumi S., Gaidos E., 2022, AJ, 164, +209 +Molenda-Żakowicz J., Brogaard K., Niemczura E., Bergemann M., Frasca +A., Arentoft T., Grundahl F., 2014, MNRAS, 445, 2446 +Montes D., et al., 2018, MNRAS, 479, 1332 +Mor R., Robin A. C., Figueras F., Roca-Fàbrega S., Luri X., 2019, A&A, +624, L1 +Morris B. M., 2020, ApJ, 893, 67 +Mulders G. D., Pascucci I., Apai D., 2015, ApJ, 798, 112 +Mullally F., et al., 2015, ApJS, 217, 31 +Nakajima T., Oppenheimer B. R., Kulkarni S. R., Golimowski D. A., +Matthews K., Durrance S. T., 1995, Nature, 378, 463 +Newton E. R., Mondrik N., Irwin J., Winters J. G., Charbonneau D., 2018, +AJ, 156, 217 +Nicholson B. A., Aigrain S., 2022, MNRAS, 515, 5251 +Oelkers R. J., et al., 2018, AJ, 155, 39 +Otani T., von Hippel T., Buzasi D., Oswalt T. D., Stone-Martinez A., Ma- +jewski P., 2022, ApJ, 930, 36 +Pasquini L., Biazzo K., Bonifacio P., Randich S., Bedin L. R., 2008, A&A, +489, 677 +Passegger V. M., et al., 2022, A&A, 658, A194 +Pecaut M. J., Mamajek E. E., 2013, ApJS, 208, 9 +Pepe F., et al., 2000, in Iye M., Moorwood A. F., eds, Society of Photo- +Optical Instrumentation Engineers (SPIE) Conference Series Vol. 4008, +Optical and IR Telescope Instrumentation and Detectors. pp 582–592, +doi:10.1117/12.395516 +Pepe F., et al., 2021, A&A, 645, A96 +Pojmanski G., 2002, Acta Astron., 52, 397 +Pollacco D. L., et al., 2006, PASP, 118, 1407 +Quirrenbach A., et al., 2016, CARMENES: an overview six months after +first light. SPIE, p. 990812, doi:10.1117/12.2231880 +Rebassa-Mansergas A., et al., 2021, MNRAS, 505, 3165 +Rebull L. M., et al., 2016, AJ, 152, 113 +Rebull L. M., Stauffer J. R., Cody A. M., Hillenbrand L. A., David T. J., +Pinsonneault M., 2018, AJ, 155, 196 +Reefe M. A., et al., 2022, AJ, 163, 269 +Reiners A., et al., 2018, A&A, 609, L5 +Reinhold T., Hekker S., 2020, A&A, 635, A43 +Reinhold T., Reiners A., Basri G., 2013, A&A, 560, A4 +Reipurth B., Mikkola S., 2012, Nature, 492, 221 +Ribas I., et al., 2018, Nature, 563, 365 +Richer H. B., Fahlman G. G., Rosvick J., Ibata R., 1998, ApJ, 504, L91 +Ricker G. R., et al., 2014, in Proc. SPIE. p. 914320 (arXiv:1406.0151), +doi:10.1117/12.2063489 +Rigault M., 2018, ztfquery, a Python tool to access ZTF data, Zenodo, +doi:10.5281/zenodo.1345222 +Rigaut F., 2002, in European Southern Observatory Conference and Work- +shop Proceedings. p. 11 +Robertson P., Mahadevan S., Endl M., Roy A., 2014, Science, 345, 440 +MNRAS 000, 1–16 (2022) + +16 +E. Gaidos et al. +Rodríguez-López C., 2019, Frontiers in Astronomy and Space Sciences, 6, +76 +Roquette J., Matt S. P., Winter A. J., Amard L., Stasevic S., 2021, MNRAS, +508, 3710 +Rosotti G. P., Clarke C. J., 2018, MNRAS, 473, 5630 +Rowe J. F., et al., 2015, ApJS, 217, 16 +Sanchis-Ojeda R., et al., 2015, ApJ, 812, 112 +Sandquist E. L., et al., 2021, AJ, 161, 59 +Santos A. R. G., García R. A., Mathur S., Bugnet L., van Saders J. L., +Metcalfe T. S., Simonian G. V. A., Pinsonneault M. H., 2019, ApJS, +244, 21 +Santos A. R. G., Breton S. N., Mathur S., García R. A., 2021, ApJS, 255, 17 +Sarajedini A., Dotter A., Kirkpatrick A., 2009, ApJ, 698, 1872 +Scargle J. D., 1982, ApJ, 263, 835 +Schiavon R. P., Caldwell N., Rose J. A., 2004, AJ, 127, 1513 +Serna J., et al., 2021, ApJ, 923, 177 +Shappee B. J., et al., 2014, ApJ, 788, 48 +Shporer A., et al., 2020, ApJ, 890, L7 +Simonian G. V. A., Pinsonneault M. H., Terndrup D. M., 2019, ApJ, 871, +174 +Skumanich A., 1972, ApJ, 171, 565 +Somers G., Stauffer J., Rebull L., Cody A. M., Pinsonneault M., 2017, ApJ, +850, 134 +Souto D., et al., 2020, ApJ, 890, 133 +Sozzetti A., et al., 2013, in European Physical Journal Web of Conferences. +p. 03006 (arXiv:1303.1275), doi:10.1051/epjconf/20134703006 +Stauffer J., et al., 2016, AJ, 152, 115 +Stock S., et al., 2020, A&A, 643, A112 +Strassmeier K. G., et al., 2004, Astronomische Nachrichten, 325, 527 +Su X.-N., Xie J.-W., Zhou J.-L., Thebault P., 2021, AJ, 162, 272 +Suárez Mascareño A., Rebolo R., González Hernández J. I., Esposito M., +2015, MNRAS, 452, 2745 +Suárez Mascareño A., Rebolo R., González Hernández J. I., 2016, A&A, +595, A12 +Suárez Mascareño A., Rebolo R., González Hernández J. I., Esposito M., +2017, MNRAS, 468, 4772 +Suárez Mascareño A., et al., 2018, A&A, 612, A89 +Suto Y., Sasaki S., Nakagawa Y., Benomar O., 2022, PASJ, 74, 857 +Terrien R. C., et al., 2022, ApJ, 927, L11 +Thompson S. E., et al., 2018, ApJS, 235, 38 +Toledo-Padrón B., et al., 2019, MNRAS, 488, 5145 +Toledo-Padrón B., et al., 2021, A&A, 648, A20 +Toyouchi D., Chiba M., 2018, ApJ, 855, 104 +VandenBerg D. A., Stetson P. B., 2004, PASP, 116, 997 +VanderPlas J. T., 2018, ApJS, 236, 16 +Venuti L., et al., 2017, A&A, 599, A23 +Walkowicz L. M., Basri G. S., 2013, MNRAS, 436, 1883 +Winters J. G., et al., 2019, AJ, 157, 216 +Woźniak P. R., et al., 2004, AJ, 127, 2436 +Wright N. J., Newton E. R., Williams P. K. G., Drake J. J., Yadav R. K., +2018, MNRAS, 479, 2351 +Xiang M., Rix H.-W., 2022, Nature, 603, 599 +Yang H., Liu J., 2019, ApJS, 241, 29 +Ziegler C., et al., 2018, AJ, 156, 83 +Ziegler C., Tokovinin A., Latiolais M., Briceño C., Law N., Mann A. W., +2021, AJ, 162, 192 +van Saders J. L., Pinsonneault M. H., 2013, ApJ, 776, 67 +APPENDIX A: LIGHTCURVE ANALYSIS +Figures A1-A4 show K2 lightcurves of 21 host star in which new +rotational signals were identified or, in the case of K2-345, replace +a previously published value. In the cases of K2-5, 14, 83, 124, +125, 129, 151, 288 B, 315, 322, and 377, a rotation period twice +the period of the signal with peak power was adopted. Figure A5 +shows periodograms and phased lightcurves from ZTF photometry +of four host stars for which new rotation periods are identified and +reported in this work. Fig. A6 shows the ASAS-SN data for four +stars for which significant (𝑝 < 0.01) signals with the same period +appear in both 𝑔- and 𝑉-band photometry. In the case of GJ 486, +the recovered signal at 13.7 days differs markedly from the value of +49.9 ± 5.5 days published by Caballero et al. (2022). +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +17 +56810 +56820 +56830 +56840 +56850 +56860 +56870 +56880 +MJD +0.990 +0.995 +1.000 +1.005 +1.010 +K2-5 C1 +100 +101 +period (days) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +13.69 days +56810 +56820 +56830 +56840 +56850 +56860 +56870 +56880 +MJD +0.998 +1.000 +1.002 +1.004 +K2-14 C1 +100 +101 +period (days) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +21.99 days +56810 +56820 +56830 +56840 +56850 +56860 +56870 +56880 +MJD +0.9900 +0.9925 +0.9950 +0.9975 +1.0000 +1.0025 +1.0050 +1.0075 +K2-22 C1 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +7.61 days +56810 +56820 +56830 +56840 +56850 +56860 +56870 +56880 +MJD +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +K2-45 C1 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +21.12 days +56980 +56990 +57000 +57010 +57020 +57030 +57040 +MJD +0.994 +0.996 +0.998 +1.000 +1.002 +1.004 +1.006 +K2-69 C3 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +22.58 days +57070 +57080 +57090 +57100 +57110 +57120 +57130 +MJD +0.9925 +0.9950 +0.9975 +1.0000 +1.0025 +1.0050 +1.0075 +K2-83 C4 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +15.66 days +Figure A1. De-trended K2 PDCSAP lightcurves of M dwarf host stars with significant periodic signals. The left panels contain the lightcurves. The right +panels show the Lomb-Scargle periodograms with the horizontal green line marking the false alarm probability 𝑝 = 0.001, the red point marking the period +of highest power, and the vertical dashed lines marking upper and lower harmonics of that period. For K2-22, twice the peak period was adopted. +MNRAS 000, 1–16 (2022) + +18 +E. Gaidos et al. +57140 +57150 +57160 +57170 +57180 +57190 +57200 +57210 +MJD +0.990 +0.995 +1.000 +1.005 +1.010 +K2-124 C5 +100 +101 +period (days) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +19.23 days +57220 +57230 +57240 +57250 +57260 +57270 +57280 +57290 +MJD +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +1.020 +K2-125 C6 +100 +101 +period (days) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +25.67 days +57310 +57320 +57330 +57340 +57350 +57360 +57370 +57380 +MJD +0.98 +0.99 +1.00 +1.01 +1.02 +K2-129 C7 +100 +101 +period (days) +0.00 +0.05 +0.10 +0.15 +0.20 +23.21 days +57400 +57410 +57420 +57430 +57440 +57450 +57460 +57470 +MJD +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +K2-151 C8 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +23.64 days +57830 +57840 +57850 +57860 +57870 +57880 +57890 +57900 +MJD +0.9985 +0.9990 +0.9995 +1.0000 +1.0005 +1.0010 +1.0015 +K2-155 C13 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +24.66 days +57990 +58000 +58010 +58020 +58030 +58040 +58050 +58060 +58070 +MJD +0.990 +0.995 +1.000 +1.005 +1.010 +K2-286 C15 +100 +101 +period (days) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +28.91 days +Figure A2. Additional K2 lightcurves. See Fig. A1 for explanation. +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +19 +57070 +57080 +57090 +57100 +57110 +57120 +57130 +MJD +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +K2-288B C4 +100 +101 +period (days) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +22.35 days +57990 +58000 +58010 +58020 +58030 +58040 +58050 +58060 +58070 +MJD +0.9900 +0.9925 +0.9950 +0.9975 +1.0000 +1.0025 +1.0050 +1.0075 +K2-315 C15 +100 +101 +period (days) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +29.57 days +57910 +57920 +57930 +57940 +57950 +57960 +57970 +57980 +MJD +0.990 +0.995 +1.000 +1.005 +1.010 +K2-322 C14 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +8.04 days +57740 +57750 +57760 +57770 +57780 +57790 +57800 +57810 +MJD +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +K2-326 C12 +100 +101 +period (days) +0.0 +0.2 +0.4 +0.6 +0.8 +22.56 days +58100 +58110 +58120 +58130 +58140 +58150 +58160 +58170 +MJD +0.96 +0.98 +1.00 +1.02 +1.04 +K2-345 C16 +100 +101 +period (days) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +19.22 days +57220 +57230 +57240 +57250 +57260 +57270 +57280 +57290 +MJD +0.996 +0.998 +1.000 +1.002 +1.004 +K2-377 C6 +100 +101 +period (days) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +19.57 days +Figure A3. Additional K2 lightcurves. See Fig. A1 for explanation. For K2-322, twice the peak period was adopted. +MNRAS 000, 1–16 (2022) + +20 +E. Gaidos et al. +57400 +57410 +57420 +57430 +57440 +57450 +57460 +57470 +MJD +0.985 +0.990 +0.995 +1.000 +1.005 +1.010 +1.015 +K2-387 C8 +100 +101 +period (days) +0.0 +0.2 +0.4 +0.6 +0.8 +34.31 days +57910 +57920 +57930 +57940 +57950 +57960 +57970 +57980 +MJD +0.996 +0.998 +1.000 +1.002 +1.004 +K2-404 C14 +100 +101 +period (days) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +34.36 days +56810 +56820 +56830 +56840 +56850 +56860 +56870 +56880 +MJD +0.98 +0.99 +1.00 +1.01 +1.02 +EPIC 201170410 C1 +100 +101 +period (days) +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +20.18 days +Figure A4. Additional K2 lightcurves. See Fig. A1 for explanation. +MNRAS 000, 1–16 (2022) + +Cool Exoplanet Host Star Ages +21 +101 +102 +period (days) +0.00 +0.02 +0.04 +0.06 +0.08 +K2-26 zg +85.88 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +15.200 +15.225 +15.250 +15.275 +15.300 +15.325 +15.350 +101 +102 +period (days) +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +K2-153 zr +39.14 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +14.26 +14.28 +14.30 +14.32 +14.34 +14.36 +101 +102 +period (days) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +TOI-519 zr +84.31 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +16.10 +16.12 +16.14 +16.16 +16.18 +101 +102 +period (days) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +TOI-1693 zg +38.01 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +13.475 +13.500 +13.525 +13.550 +13.575 +13.600 +13.625 +Figure A5. Periodograms and phased ZTF lightcurves of four M dwarf exoplanet hosts with significant (𝑝 < 0.01, horizontal green line) signals (red dots) +that passed our visual inspection and are considered candidate rotational signals. The host star and the band-pass are indicated in the The blue solid and dashed +lines are the lunar synodic period and its aliases with the annual observing window function. Note that the phased lightcurves are repeated. +MNRAS 000, 1–16 (2022) + +22 +E. Gaidos et al. +101 +102 +period (days) +0.000 +0.005 +0.010 +0.015 +0.020 +GJ 27.1 g +138.71 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +0.2 +0.1 +0.0 +0.1 +0.2 +101 +102 +period (days) +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +GJ 486 g +13.72 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +1.0 +0.5 +0.0 +0.5 +1.0 +101 +102 +period (days) +0.000 +0.005 +0.010 +0.015 +0.020 +TOI-833 g +15.17 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +101 +102 +period (days) +0.00 +0.02 +0.04 +0.06 +WASP-43 g +15.69 days +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +phase +0.075 +0.050 +0.025 +0.000 +0.025 +0.050 +0.075 +Figure A6. Periodograms and phased ASAS-SN 𝑔-band lightcurves of four M dwarf exoplanet hosts with significant (𝑝 < 0.01, horizontal green line) signals +(red dots) that also had equivalent significant signals in 𝑉 -band, passed our visual inspection, and are considered candidate rotational signals. The blue solid +and dashed lines are the lunar synodic period and its aliases with the annual observing window function. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Feiden7 1Department of Earth Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' University of Hawai’i at M¯anoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' HI 96822,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' USA 2Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 1180 Wien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Austria 3Institute for Particle Physics & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ETH Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 8093 Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Switzerland 4Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' University of Hawai’i at M¯anoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Honolulu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' HI 96822 USA 5Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' University of Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 211 Bryant Space Science Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gainesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' FL 32611 USA 6Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' University of Hawai’i at Hilo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Hilo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' HI 96720 USA 7Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' University of North Georgia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Dahlonega,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' GA 30597 USA Submitted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' accepted ABSTRACT Age is a stellar parameter that is both fundamental and difficult to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Among middle- aged M dwarfs, the most prolific hosts of close-in and detectable exoplanets, gyrochronology is the most promising method to assign ages, but requires calibration by rotation-temperature sequences (gyrochrones) in clusters of known ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We curated a catalog of 249 late K- and M-type (𝑇eff=3200-4200K) exoplanet host stars with established rotation periods, and applied empirical, temperature-dependent rotation-age relations based on relevant published gyrochrones, including one derived from observations of the 4 Gyr-old open cluster M67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We estimated ages for 227 of these stars, and upper limits for 8 others, excluding 14 which are too rapidly rotating or are otherwise outside the valid parameter range of our gyrochronology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We estimated uncertainties based on observed scatter in rotation periods in young clusters, error in the gyrochrones, and uncertainties in temperature and non-solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For those stars with measured metallicities, we provide but do not incorporate a correction for the effects of deviation from solar-metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The age distribution of our sample declines to near zero at 10 Gyr, the age of the Galactic disk, with the handful of outliers explainable by large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Continued addition or extension of cluster rotation sequences to more thoroughly calibrate the gyrochronology in time and temperature space, more precise and robust measurement of rotation periods, and more accurate stellar parameter measurements will enable continued improvements in the age estimates of these important exoplanet host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Key words: exoplanets – stars: evolution – stars: late-type – stars: low-mass – stars: rotation – planetary systems 1 INTRODUCTION Over the past three decades, thousands of planets have been discov- ered around other stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Exoplanet surveys have revealed that M- type dwarfs, the least massive but most numerous stars, host more planets on close-orbits than their solar-mass counterparts (Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Hardegree-Ullman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' These include Earth-size, rocky planets that orbit within the compact hab- itable zone of these intrinsically faint stars, and which are more feasible to study, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' with JWST, because of the host stars’ lower mass, radius, and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Precise characterization of the host star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', radius, mass, luminosity, metallicity) is essential to obtain properties of its plan- ets, but is challenging for very low-mass stars for which methods tuned to the Sun do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For this reason, empirical approaches ★ Contact e-mail: gaidos@hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='edu have proven useful for M dwarfs, enabled by the advent of the Gaia astrometry mission, space- and ground-based photometric surveys, and advances and expansion in spectroscopic instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For example, interferometry can directly measure the angular radii of very nearby M dwarfs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' pairing with trigonometric parallaxes yields physical radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Combined with a bolometric luminosity from a flux- calibrated spectral energy distribution (SED), this allows the effec- tive temperature 𝑇eff to be derived using the Stefan-Boltzmann law (Boyajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Spectra of stars with a range of established 𝑇eff can then serve as templates to estimate the 𝑇eff of more distant stars using spectra and, combining with SEDs, their radii (Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Metallicities can be calibrated using binaries where the solar-type companion has an established metallicity (relative to the Sun) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Montes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Souto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Age is a fundamental property of planets but is difficult to accu- rately estimate for most systems (Christensen-Dalsgaard & Aguirre © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='12109v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='EP] 28 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Planets and their atmospheres are expected to evolve under the influence of their host star and their own internal thermodynam- ics and compositional change (Kite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Lammer 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For temperate, Earth-like planets, changes in atmospheric composition will be the backdrop against which biosignatures will be searched for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Observations of disk lifetimes, planet formation theory, and isotope-ages of Solar System bodies indicate that the age of a star should be no more than a few tens of Myr older than that of its planets (Helled & Morbidelli 2021), but ages of most host stars remain poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Planets are difficult to detect around young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Miyakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022) and most planet hosts are older and no longer members of (relatively) well-dated clusters and young-moving groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ages of isolated field stars have been estimated by (1) compari- son of stellar parameters to stellar models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', in a color-magnitude diagram);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2) asteroseismic measurement of increasing density due to the conversion of H into He and heavy elements in stellar in- teriors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (3) the abundance of lithium, which is destroyed in stellar interiors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (4) metallicity and the overall age-metallicity relation of the Galactic disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (5) increase in the peculiar motion of stars with time with respect to the overall orbital motion of the Galactic disk as a result of perturbations from molecular clouds and other stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' and (6) rotation and rotation-driven magnetic activity that decline as angular momentum (AM) is lost through a magnetized wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' But M dwarfs are resistant to most age-dating techniques: They evolve imperceptibly on the main sequence (MS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2005) and the pulsations that are the grist of asteroseismology are below current detection thresholds (Rodríguez-López 2019, and regardless the stars’ densities do not change) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M dwarfs consume their lithium within ∼50 million years (Binks & Jeffries 2014) and thus this proxy cannot be used at later ages, and metallicity and peculiar motions are only meaningful in a statistical sense for stellar populations, not individual stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' the age-metallicity relation appears flat for disk stars (Rebassa-Mansergas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021) and age-abundance ratio relations do not appear to apply universally (Casali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This leaves gyrochronology, the application of relations be- tween age and rotation (and its proxies) brought about by stellar spin-down, as a viable method to age-date main-sequence M dwarfs in the field (Barnes 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Such stars are potentially well-suited for this approach because they have a smaller or no inner radiative core which can rotate quasi-independently of the convective envelope, and instead could have behavior similar to simple “solid-body" ro- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' However, spin-down driven by magnetic activity scales with the Rossby number 𝑅𝑜 ≡ 𝑃rot/𝜏𝑐, where 𝜏𝑐 is the local turnover time in the convective envelope, and the longer 𝜏𝑐 of M dwarfs compared to solar-type stars means their rotational evolution will be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Using co-eval bainries, Otani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) tested several color-dependent spin-down models for internal consistency, and de- rived internal errors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', only those arising from errors in 𝑃rot and color) of 5-10% for relatively young early M-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Pioneering ground-based observations of open clusters of co- eval stars, followed by the revolutionary wide-field surveys of the Kepler and TESS space telescopes, document the formation of tight sequences in rotation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' color or 𝑇eff diagrams that extend to cooler temperatures and longer 𝜏𝑐 with time (Gallet & Bouvier 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The formation of the sequence among solar-mass stars is thought to be the result of a change in the braking law (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', the rotation-rate dependence of the torque) as stars transition from a “saturated" (less rotation rate-dependent) phase to an “unsaturated" (more rotation rate-dependent) phase of stellar activity at a critical value of 𝑅𝑜 (Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' After a star enters this regime the strong rotation rate dependence effectively erases the effect of initial conditions, allowing gyrochronologic relations to be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Among cooler dwarf stars the situation is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The outer convective envelope—the part of the star that is both observ- able and feels the decelerating torque from the magnetized wind—is more substantial, and the coupling timescale between the the en- velopes and the radiative core is longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Among K dwarfs, this can lead to early differential rotation, with the core spinning faster than the envelope, and later “stalling" as the core transfers AM to the envelope and the observed spin-down temporarily slows or halts (Denissenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019b, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Stalling could contribute to the formation of a rotational sequence, but also delays the epoch at which gyrochronology is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Among yet cooler M dwarfs the radiative core is smaller or absent, the coupling timescale is expected to get longer, and the magnitude of the stalling could diminish/disappear (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The mechanism responsible for core-envelope coupling has not been established, nor has a theoretical model that is quantitatively consistent with the observations and has predictive power over a range of 𝑇eff been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Moreover, it is not certain if braking laws developed for solar-type stars apply to M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For these reasons, observations of stars in clusters of known ages are im- perative for identifying rotational sequences (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑇eff), constructing empirical rotation-age relations, and calibrating successful models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' But M dwarfs are intrinsically faint and their rotation evolves more slowly, and thus deeper observations of older (and statistically more distant) clusters are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M dwarfs in these clusters are beyond the range of current space telescopes both in terms of signal (limited by the modest telescope aperture) and spatial resolution (limited by the large pixel size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K2 monitoring of the oldest nearby cluster (Ruprecht 147, ≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013) captured rotation peri- odsfor only a handful of M dwarfs near the K-M boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Similar observations of the nearest old cluster (M67, ≈4 Gyr Richer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' VandenBerg & Stetson 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Schiavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Sarajedini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2009) failed to yield useful results (Esselstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ground-based observatories can go deeper and with high res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) carried out a Sloan 𝑖-band monitor- ing campaign of late K and early M dwarf members of M67 with the MegaCAM wide-field camera at the prime focus of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6-m Canada France Hawaii Telescope (CFHT) (Boulade 1998), obtain- ing 294 rotation periods and identifying a rotational sequence that ranged from ≈25 days at 4200K to 125 days at 3200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) found that the “warm" end of the M67 sequence could be ex- plained by the overlapping “cool" end of the Ruprecht 147 sequence identified by Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020), plus Skumanich-like power-law spin-down 𝑃 ∝ 𝑡𝑛 Skumanich (1972) with an index 𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This indicates that (a) the rotation sequence of M dwarfs extends close to the fully convective boundary (near 𝑇eff=3200K) by no later than 4 Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (b) spin-down among middle-aged M dwarfs seems to obey a relatively simple braking law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Both of these findings bode well for the gyrochronology of very cool dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In this work, we curate a catalog of rotation periods of late K and early M-type dwarfs known to host validated or confirmed planets1, and apply empirical rotation-age relations based on the M67 gyrochrone of Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) and previously published gyrochrones (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020) to estimate ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The rotation pe- riods of many host stars have been established either using the 1 A “validated" planet is one for which known false positive scenarios are highly unlikely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' a “confirmed" planet has been detected by a second, independent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 3 same space-based photometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kepler, K2, TESS) used to iden- tify their transiting planets, or by data obtained from the ground- or space as part of the validation/confirmation of candidate planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' There are also collections of rotation periods of field stars (including planet hosts) based on data from Kepler (Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019, 2021), K2 (Reinhold & Hekker 2020), TESS (Canto Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020), and ground-based surveys (Oelkers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Newton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Christy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We also identify additional candidate rotational signatures directly in the photometric datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We emphasize that some rotation periods are tentative and that very cool dwarf gy- rochronology is a work in progress and makes assumptions which will be borne out or refuted by future observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2 SOURCES OF ROTATION PERIODS We identified all host stars of validated or confirmed exoplanets with 𝑇eff of 3200–4200 K in the NASA Exoplanet Archive as of August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This included 112 Kepler host stars or Kepler Objects of Interest (KOIs) having rotation periods 𝑃rot in Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' From the list of the non-Kepler host stars we removed evolved (giant), T Tauri, and pre-main sequence (PMS) stars, as rotation of these stars obviously does not follow the gyrochronology of dwarfs, as well as members of star-forming regions and young moving groups that have ages estimated by other techniques, leaving 215 non-Kepler stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rotation periods for these stars were obtained from the litera- ture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' these were determined using ground- or space-based photome- try, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' by WASP (Pollacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2006), MEarth (Berta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012), ASAS-SN (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014), K2(Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014), and TESS (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014) and/or time-series spectroscopy of indicators of active regions and magnetic fields, notably by the HARPS (Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2000) and CARMENES Doppler RV surveys for exoplanets (Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We revisited the K2 data by matching all stars against the EPIC catalog (Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016) and downloaded all Pre-search Data Con- ditioning Simple Aperture Photometry (PDCSAP) lightcurves from the MAST archive (including many K2-detected transiting exoplanet host stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The lightcurves were further de-trended with a best-fit second-order polynomial before a Lomb-Scargle analysis (Scargle 1982) to search for signals with periods of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4-40 days, an interval chosen to avoid the pervasive 6-hr thruster firing signal and for the typical lightcurve to span at least two periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 115 lightcurves of 95 EPIC stars contained peaks exceeding a 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='001 false-positive level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Of these 20 were not previously published, and in one other case (K2-345) we replaced the Reinhold & Hekker (2020) value as being obviously erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We did not revise other Reinhold & Hekker (2020) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In 11 cases (K2-5, 14, 83, 124, 125, 129, 151, 288B, 315, 322, 377) we judged that the peak was an upper harmonic and doubled the period and its error based on inspection of the lightcurve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The de-trended lightcurves and periodograms are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A1-A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We retrieved lightcurves from the Zwicky Transient Facility (ZTF, Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019) using the Python wrapper of the InfraRed Science Archive (IRSA) API query (Rigault 2018) and a matching criteria of 5".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We obtained 319 ZTF lightcurves for 102 stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' each star has up to three (𝑔𝑟𝑖) lightcurves and sometimes the 5" search cone contained more than one star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We constructed Lomb-Scargle periodograms of the 145 lightcurves with at least 100 points (the maximum was 1092) and identified 25 signals among 22 stars with peaks with a false alarm probability < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (We were less stringent than for K2 because of the availability of data in multiple filters for comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For three stars with two periodic signals (in different filters), none had matching periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Seven signals were rejected based on similarity to the lunar synodic period (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 day) or its seasonal alias (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Marginal, long term (> 100-day) signals seen in the lightcurves of K2-123 and K2-124 were rejected based on detection of shorter rotation periods by K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ZTF periods for two other systems (TOIs-2136 and 3174) were already reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We assigned tentative periods to four other stars based on these data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' none of these periods are close to the 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5-day lunar synodic period or its annual alias, or harmonics of 1-day (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' All other periodograms contained no clearly visible peaks or a forest of peaks of roughly equal (in)significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We also retrieved 𝑔- and 𝑉-band lightcurves from the All-Sky Automated Search for Super-Novae (ASAS-SN Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017) on which we performed a similar analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We identified 16 and 15 host stars with significant (𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='01) periodic signals in the 𝑔- and 𝑉-band data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Among these are four stars that have matching 𝑔- and 𝑉-band periods that are not at the lunar synodic period, its aliases, or 1-day harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The 𝑔-band photometry for these is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the case of GJ 486, the detected signals at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 days differ markedly from the published 𝑃rot of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 days (Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022) (which we retain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We included 𝑃rot values determined from time-series measure- ments of spectroscopic indicators of stellar activity such as Ca ii HK and H𝛼, but we excluded estimates based on correlations with indi- cators of stellar activity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', the overall level of Ca ii HK emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Astudillo-Defru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3 STELLAR PARAMETERS We retrieved values for 𝑇eff and metallicities ([Fe/H]) of KOIs from the literature via the NASA Exoplanet Archive tables (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Mullally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For each star, we used the most recent KOI table available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' However, the present character- ization of cool exoplanet host stars is markedly heterogeneous, with many stars lacking published metallicities, and multiple values for the same star can differ by much more than the published uncertain- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For those stars without effective temperature solutions from the NASA Exoplanet Archive, we used their absolute 𝐾𝑠 magnitude to calculate an effective temperature with the Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2015) empirical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We identified stars in multiple systems using the literature, as well as stars with Gaia astrometric renormalized unit weight error (RUWE) values >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 indicative of unresolved multiplicity (Be- lokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We flagged but did not exclude these systems, cautioning that binaries that are unresolved in the data used to ob- tain rotational signals could be assigned incorrect rotation periods, and a single-star gyrochronology is expected to be more erroneous or fail in sufficiently close binaries (see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We searched for additional, unpublished stellar companions re- solved by Gaia (separations ≳1′′) and identifiable based on similar parallaxes and proper motions in the DR3 catalog (Gaia Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This was performed by calculating a Bayesian probability that a candidate companion’s astrometry is the same as a given star, relative to the probability that this occurs in a “background" population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We identified five stars with FAP < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='01, but all are previously known binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We searched the Hipparcos-Gaia (EDR3) Catalog of Acceler- ations (Brandt 2021) and found 5 matches, a paucity that reflects the magnitude limit of the Hipparcos catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Of these, only two MNRAS 000, 1–16 (2022) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3200 3400 3600 3800 4000 4200 Teff (K) 101 102 period (days) Galactic Disk age Rossby number = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 lunar synodic half Kepler quarter Other binaries Kepler Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rotation periods of late-K and early M-type hosts of known exoplanets from the literature and this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Members of multi-star systems are indicated in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The horizontal dashed and dotted lines mark one half the Kepler rotation interval (close to one half the K2 campaign interval) and the lunar synodic period, near which ground-based detection of 𝑃rot values are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The magenta curve is the 𝑃rot above which a star’s Rossby number exceeds the critical value (based on convective turnover times from the Dartmouth magnetic model) and activity leaves the “saturated" phase, and the blue curve is the 𝑃rot predicted for stars with the age of the Galactic disk using the Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) gyrochrone and power-law evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Errors in 𝑇eff are not shown but are typically 75K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' have 𝜒2 fits approaching but not reaching a formal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3% false-alarm probability threshold of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8: HD 238090 (𝜒2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1) is the primary of a mid-M-type companion at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6′′ (224 au) that is unlikely to be the source of any acceleration, and HIP 71135 (𝜒2=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3) which has no known stellar companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The RV-detected planets are inferred to have (sub)-Neptune-like masses (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020), and could not produce detectable acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Finally, we searched the catalog of the Robo-AO M-dwarf Multiplicity Survey (Lamman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020), which identified candidate companions with separations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1-4′′ of nearby M dwarfs from the Lépine & Shara (2005) catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We identified 20 overlapping stars, none of which had AO-identified companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Figure 1 plots 𝑃rot vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑇eff for the catalog of host stars, with Ke- pler- and non-Kepler hosts marked by different points and members of known binaries shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The horizontal dashed line marks 90 days, the cadence at which Kepler performed a role maneuver that introduced systematics in lightcurves, only slightly longer than the 80-day interval which the spacecraft could observe a field dur- ing the K2 mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Recovery of rotation periods longer than these intervals is inhibited by these systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The horizontal dotted line is the lunar synodic period of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 days near which the ground- based detection of rotation periods is inhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The magenta curve is the critical 𝑃rot at which 𝑅𝑜 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 using the Jeffries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2011) prescription for 𝜏𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Nearly all stars are above this line and are thus in the “unsaturated" regime of dynamo-driven activity, although not necessarily yet in Skumanich-like spin-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The blue curve is the expected 𝑃rot for 10 Gyr, the approximate age of the Galactic disk (Kilic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017, but see Xiang & Rix (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4 AGE ESTIMATION We estimated the age of each star with an established 𝑃rot (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2 by comparing it to available empirical cluster gyrochrones that (partially) include the 𝑇eff range of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', that of M67 Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022), but also that of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr-old Ruprecht 147 (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020), the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 Gyr-old NGC 752 (Agüeros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018), the 1 Gyr-old NGC 6811 (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019a), and the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='67 Gyr-old Praesepe (Douglas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019), as filtered for binaries and fit with polynomials by Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) fit the M67 rotation sequence with a fourth-order polynomial with 𝑇eff over 3200-4200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑃4Gyr = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='66 × 10−10 · (𝑇eff − 4000)4 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='25 × 10−7 · (𝑇eff − 4000)3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='69 × 10−4 · (𝑇eff − 4000)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='016 · (𝑇eff − 4000) + 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9, (1) 𝑇eff of M67 members were estimated from their Pan-STARRS 𝑟 − 𝑖 colors using a polynomial relation based on synthetic photometry of nearby M dwarf standards (Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020) fit the other rotational sequences as polynomials with Gaia 𝐵𝑃 − 𝑅𝑃 color (see their Appendix B), which we converted to 𝑇eff using the empirical MS of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We estimated ages by simple power-law interpolation between gyrochrone calibration points (linear interpolation in a log-log plot of period vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Calibration points were calculated from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 1 and Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020) for a given stellar 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Figure 2 plots the derived period vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' age tracks for a range of representative 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The flattening of the tracks at ages younger than Ruprecht 147 could be “stalling" of spin-down as a faster rotating radiative core adds AM to the convective envelope (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The bunching of the hotter tracks reflects the weak dependence of 𝑃rot on 𝑇eff among co-eval K dwarfs (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A calibration point was used only if (1) 𝑇eff falls within the range of a gyrochrone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2) for the particular 𝑇eff and gyrochrone rotation period, the star would not be in the “saturated" phase of activity (a condition for a rotational sequence);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' and (3) the star would not be on the PMS and still affected by contraction and spin-up at that gyrochrone age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The condition for saturated activity is a Rossby number 𝑅𝑜 =𝑃rot/𝜏cwhere 𝜏𝑐 is the local convective turnover time, to be less than a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We adopted the relation of 𝜏𝑐 with luminosity of Jeffries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2011) and critical 𝑅𝑜 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' PMS durations were taken from the Dartmouth standard models of stellar evolution (Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If the only available calibration point was that of M67 (4 Gyr) then we calculate the age of the star using the Skumanich-like spin-down law that Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) found by comparing the rotational sequence of M67 with that of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr-old Ruprecht 147 (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020) 𝑡 = 𝑡0 (𝑃/𝑃0)1/𝑛 , (2) with 𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' However, if the age derived in this manner was <2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', that of Ruprecht 147) we took this to be an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If additional calibration points were available we derived an age as above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' if that was <4 Gyr we then consequently derived ages by power-law interpolation between successfully younger pairs of neighboring calibration points until the derived age lay between the ages of the gyrochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' However, if the age determined from the 4 Gyr and next youngest calibration points was >4 Gyr, we adopted that age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This allowed for the occasional pathological cases where the Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) gyrochrone produced an age that was slightly <4 Gyr, but the gyrochrone pair produced an age slightly >4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If no self-consistent age was produced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', the age was MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 5 1 2 3 5 10 age (Gyr) 10 20 30 50 100 200 rotation period (days) M67 Rup 147 NGC 752 NGC 6811 Praesepe Teff 3200 3300 3400 3500 3600 3700 3800 3900 4000 4100 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rotation period vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' age tracks constructed from five cluster gy- rochrones (vertically labelled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The dashed parts of the trajectory are based on the Skumanich-like relation derived by Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022), with extrap- olation beyond 4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Dotted lines are the periods above which the Rossby number exceeds the critical value for unsaturated activity (≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13) at each 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' younger than the youngest calibration point) we adopted the age of the youngest valid gyrochrone as an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Monte Carlo (MC) realizations of these calculations were per- formed to estimate the uncertainty in the age, incorporating error in rotation period, stellar parameters, the gyrochronology, and initial conditions (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The mean and standard error of the age distributions were adopted as the nominal age and its uncertainty reported in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If the MC realizations produced only upper limits, the 95 percentile value of the distribution was reported as an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If neither values nor upper limits were produced, or the resulting age would place the star on the PMS, no age was assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5 AGE ERROR ANALYSIS Error in gyrochronologic age assignment arises from (1) the formal uncertainty in 𝑃rot, as well as systematic errors in 𝑃rot not included in the formal error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', aliasing and confusion with harmonics);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2) the formal error in the gyrochrones as well as uncertainty in the ages of the calibrator clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (3) errors in 𝑇eff used to apply the calibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (4) variation in initial conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e the angular momentum or rotation rate at an early time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' and (5) stellar rotational evolution that deviates from the assumed behavior due to differences in the internal structure of a star or the wind torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Epstein & Pinsonneault (2014) quantified the effect of uncer- tainty in 𝑃rot (#1 above) and scatter in rotation period at a fixed age (#4) by projecting “initial" rotation conditions found in the ∼500 Myr-old cluster M37 forward in time with a rotational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' They found that the age uncertainty among middle-aged M dwarf stars is very large, dominated by the scatter in initial conditions, and as a result the age precision is very poor (factors of two or more between minimum and maximum ages) because, essentially, these stars have not yet spun down to form a rotational sequence, but this depends on the details of the model, including the assumed braking law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Here we use Monte Carlo methods to calculate probabilistic distributions of ages — an approach used by Otani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) — and take the standard deviation as the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Where data are available we determine variation empirically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' otherwise we rely on models of stellar interior and rotational evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 Period Error We report and use the formal uncertainties in rotation period either from the literature, or from our periodogram analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the latter case, Gaussian functions are set to the periodogram peak and the standard deviation is taken to be the Gaussian width 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This uncer- tainty can arise from the finite observation baseline relative to the rotation period, particularly for older stars with longer rotation peri- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Significant evolution in the spots causing rotational variability can occur over a few rotation periods (Basri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), causing drift in the phase of the overall photometric signal and broadening of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Differential rotation, combined with changes in the latitude of spots, will also broaden a periodogram peak or even pro- duce distinct neighboring peaks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Reinhold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Balona & Abedigamba 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The first (upper) harmonic might be confused for the true pe- riod as a result of the distribution of star-spots (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Suto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), causing the star to appear much younger than it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ground- based observations may incur much larger systematic error as a result of aliasing caused by sampling bias on nocturnal, lunar syn- odic, or seasonal timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This can distribute the power in single periodic signal into multiple weaker peaks in a periodogram (Van- derPlas 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' These peaks can either be “failure modes" if the peak is far from the true period, or if close to the true period and unresolved due to limiting sampling, broaden the peak and resulting uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Given the diverse data sets and sources, we point out but do not attempt to quantify such errors in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 Gyrochrone error Due to finite sample size and error in 𝑇eff and 𝑃rot, the gyrochrones themselves have uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' These are not typically published, but since most of the ages in TIME-Table rely heavily on the 4 Gyr-old M67 gyrochrone, we determined uncertainties in the best- fit polynomial coefficients using Monte-Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This is shown as the dashed black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The ages of the clusters used to calibrate the gyrochronology themselves have significant standard errors and systematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' full consideration of these is beyond the scope of this work but should be included when considering rotation-based ages in an absolute sense (Sandquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 Error in effective temperature Effective temperature 𝑇eff is usually used as the independent vari- able of a gyrochronology since it is set by radiated energy per unit area and is thus related to the eddy velocity and magnetodynamo strength in the convective envelope of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Precise and accurate determination of 𝑇eff is a classic problem in stellar astrophysics, and a particularly acute issue for the coolest stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Recent calibra- tions of the temperature scale of M dwarfs based on interferometric measurement of stellar radii, (Gaia) parallaxes, and the Stefan- Boltzmann law (Boyajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015) permit precision as good as 50 K, but most host stars have 𝑇eff with pre- cision no better than 75 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The M67 gyrochrone of Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) was fit to the 𝑔-𝑖 colors of stars, and converted to 𝑇eff using the temperature scale of Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This combined with the slope of the M67 gyrochrone will produce formal uncertainties MNRAS 000, 1–16 (2022) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3200 3400 3600 3800 4000 4200 Teff (K) 0 20 40 60 80 100 120 P4Gyr (days) 1 gyrochrone from 75K error Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Intrinsic dispersion in the 4 Gyr-old M67 gyrochrone (dashed black lines) established by Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) and dispersion produced by errors in 𝑇eff of 75K (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' of 0-17% in age, with the largest value at the cool (3200K) end, decreasing to zero near 3900K, and rising to 12% at 4200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 Initial Rotation A major contributor to age error is departure of a star’s behavior from a rotation-age relation due to variation in initial conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e angular momentum/rotation rate at an early age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A star that is initially spinning slower/faster than the mean of the population used to construct the gyrochronology will have an estimated age that is erroneously older/younger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Surveys of star-forming regions and very young clusters show that low-mass members emerge from their disk-hosting phase with a wide range of rotation rates at a given mass (Cody & Hillenbrand 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rebull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Venuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rebull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Kounkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Serna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021), perhaps due to variations of disk lifetime brought on by differences in the tidal and ultraviolet environment of stars (Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Roquette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Since it is not possible to know the initial rotation of an individual star, this variation can induce significant uncertainty in derived ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Epstein & Pinsonneault (2014) used rotational evolution models to show that this greatly limited the utility of rotation-based ages among cooler, older dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The scatter in initial rotation rates can be estimated from ob- servations of the PMS members of very young clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If AM loss through winds is small over the PMS interval, the fractional dis- persion in rotation rate at a given mass should be conserved even as the stars contract and spin up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The fractional dispersion is also conserved during the saturated phase of magnetic activity because the torque is proportional to rotation rate and the spin-down is ex- ponential with a rate-independent time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the 130 Myr-old Pleiades, no well-defined rotational sequence exists for our range of 𝑇eff/colors but there is a strong trend of decreasing 𝑃rot (6–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 days) with decreasing 𝑇eff (Rebull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' An iterative fit to this over the equivalent de-reddened 𝑉 − 𝐾 color range (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='22–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='29) yields a fractional dispersion of 45% (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 18 in Stauffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2016)), but this value does not reflect the numerous outliers, some of which may be binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Likewise, the rotation periods of members of the ∼150 Myr-old cluster NGC 2516 (Fritzewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020) within this same color range have a scatter of ≳50%, about one order of magnitude larger than what is observed at later epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This large scatter in initial (ZAMS) rotation rates 𝑃0 will be compressed once the stars decelerate to the unsaturated regime at 𝑃crit ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13𝜏𝑐 (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018) where the torque becomes highly rotation rate-dependent and period-time trajectories flatten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the saturated regime, 𝑃(𝑡) = 𝑃0𝑒𝑡/𝑇 , where 𝑇 is the spin-down timescale (Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015), thus a star with a ZAMS rotation period that differs by Δ𝑃0 ≪ 𝑃0, will reach 𝑃crit at a time that differs by Δ𝑡 = −𝑇 log � 𝑃0 + Δ𝑃0 𝑃0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (3) Because subsequent rotational evolution proceeds from the con- dition 𝑃 = 𝑃crit, independent of 𝑃0, the star will experience the same rotation history, but delayed by Δ𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Thus this is the error in gyrochronological age induced by Δ𝑃0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑇 depends on the brak- ing torque parameter and moment of inertia of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Somers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2017) found that stars in mass bins of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='25 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='40M⊙ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='40 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='60M⊙ lose 25% and 39%, respectively, of their AM in the ≈ 115 Myr interval between the age of the Upper Scorpius star-forming region and the Pleiades cluster, implying a spin-down timescale of 225-400 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Somers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2017) also found a disper- sion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='21 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='44 dex in the specific (mass-normalized) AM of Pleiades stars for stars in these respective mass bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Application of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3 produces an age error of about 200 Myr, which for a nominal 4 Gyr-old star is 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Finally, the intrinsic dispersion in the rotation sequences of cluster stars can be used to empirically estimate age error due to rotation rate dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For stars undergoing power-law spin-down (𝑃rot∝ 𝑡𝜒, Skumanich 1972), the dispersion in age consistent with a given 𝑃rot is related to the dispersion in period for a given age (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', in the co-eval population of a cluster): Δ𝑡 𝑡 = 1 𝜒 Δ𝑃 𝑃 (4) We measured the outlier-excluded dispersion Δ𝑃 around the best- fit rotational sequence of M67 to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 days, but this is consistent with the 10% measurement errors alone, and the intrinsic dispersion could be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The dispersion in other, younger (≲1 Gyr) clusters like those observed by K2 can be better determined, but with the caveat that Skumanich-like spin-down only applies to times much later than the era of saturated magnetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A well-defined rotational sequence is apparent in the Praesepe cluster (upper right- hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4), estimated to be 600 Myr-old (Gossage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We fit a second-order polynomial with iterative 3𝜎 outlier rejection to 3500-4200K members with rotation periods cataloged by Douglas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (Cooler stars have yet to converge to an identifiable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=') The scatter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='97 days (N=121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The same analysis applied to the much sparser Hyades sample yields the same dispersion: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='95 days (N=13), a fractional dispersion of about 5% (bottom left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The data on M dwarfs in other, older clusters are much sparser: Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2019a) estimated a dispersion of ±10% for the coolest stars in NGC 6819 (≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 Gyr) and Ruprecht 147 (≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr) but of these, only five stars have Gaia 𝐵𝑝 − 𝑅𝑝 colors that correspond to our 𝑇eff range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Using the Godoy-Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2021) catalog of members of M37 (≈470 Myr Fragkou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022) and NGC 6811 (≈950 Myr) with rotation periods we estimate dispersions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8 days (n=20) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 days (n=12) respectively (upper left and bottom right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The existence of a rotation sequence among M37 M dwarfs is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Some of the observed dispersion in these clusters is probably the result of errors in 𝑇eff: for a standard error of 75 K, this is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5-1 MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 7 day, depending on cluster and 𝑇eff (purple dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Thus the actual period dispersion could be substantially lower than the observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We adopted a conservative fractional dispersion in 𝑃0 of 5%, with the caveat that current data on the establishment of a rotational sequence at these epochs only extends to a spectral type of M2 (𝑇eff∼ 3550 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For 𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='62, this corresponds to an age dispersion of 8% (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4), which we incorporate in our MC realizations of age estimates by adopting a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' These calculations assume that a rotational sequence has developed by the epoch of interest for the relevant 𝑇eff, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', that stars have spun down into the unsaturated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This is not the outcome for M dwarfs in the models of Epstein & Pinsonneault (2014), which explains their large predicted age errors, but it is the case at least by 4 Gyr for this 𝑇eff range (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 Metallicity The rotational evolution of cool dwarf stars is expected to be metallicity-dependent through the effects of opacity and mean atomic weight on the interior structure and dynamics of the star, which in turn govern the moment of inertia, and the scale of con- vection that drives the dynamo responsible for star’s magnetic field, activity, and mass loss through a wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Moreover, at a fixed 𝑇eff, a more metal-rich MS star will have a higher mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The [Fe/H] of the nearby stellar clusters used to construct gyrochronologies is close to solar: M67, Ruprecht 147, and NGC 6811 are within uncertainties of solar (Pasquini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Bragaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Molenda- Żakowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014), while both Praesepe and the Hyades have [Fe/H] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='15 (Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' However, stars in the solar neighborhood have metallicities ranging from −1 to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 (Toyouchi & Chiba 2018), and the M dwarf stars observed by Kepler have a [Fe/H] distribution that is approximately Gaussian with a mean of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='09 dex and standard deviation of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='22 dex (Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We predict the effect of non-solar metallicity on the duration of the stellar PMS phase and the subsequent spin-down on the MS, which we model as exponential “saturated" spin-down from an initial rotation period 𝑃0 to the critical value 𝑃crit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13𝜏𝑐 at which the Rossby number exceeds a critical value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018), and unsaturated, Skumanich-like power-law spin-down thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The error in an age based on solar-metallicity gyrochrones induced by a non-solar metallicity is the sum of the variation in these two intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We used the Dartmouth standard (non-magnetic) models to compute these for a range of masses and [Fe/H] in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 dex intervals from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The MS saturated spin-down interval is taken to be: 𝑇sat = 𝐼 𝑛 log 𝑃crit 𝑃0 , (5) where 𝐼 is the moment of inertia (nearly constant for MS M dwarfs) and Γ is the (constant) torque parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Both 𝐼 and 𝑃crit are metallicity-dependent and were calculated using the Dartmouth models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We adopted the scaling relationship for the torque parame- ter from van Saders & Pinsonneault (2013), which is based on Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2012): Γ ∝ 𝑅3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 ∗ 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='56 ∗ 𝑀−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='22 ∗ 𝑝0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='44 phot (6) where 𝑝phot is the pressure at the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑝phot will be propor- tional to gravity and inversely proportional to the specific opacity 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The mass inferred for a given MS 𝑇eff will also vary with [Fe/H] be- cause the radius and hence luminosity changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Using the 𝑀∗−𝑀𝐾𝑠 (mass-luminosity) relations of Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2019), and exploiting the fact that the bolometric correction for the 𝐾𝑠-band is only weakly dependent on [Fe/H] (Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015) and thus will be approx- imately fixed at a given 𝑇eff, 𝐿∗ ≈ 𝑀2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 ∗ in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' With that relation and the Stefan-Boltzmann law, at a given mass and 𝑇eff, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 6 becomes: Γ ∝ 𝑅4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='53 ∗ 𝜅−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='44 (7) In the interiors of cool stars where bound-free opacity dominates, the opacity will scale linearly with the metal abundance 𝑍 ∝ 10[𝐹𝑒/𝐻].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We calculated for Γ for the solar-metallicity case by finding the age at which Skumanich-like rotational evolution marched backward from the M67 gyrochrone (Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022) gives 𝑃rot=𝑃crit and then setting Γ so that exponential spin-down from 𝑃rot=𝑃0 also reach 𝑃crit at this age, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e: Γ = 𝐼0 4Gyr � 𝑃4 𝑃crit �1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='62 log 𝑃crit 𝑃0 (8) The PMS interval of M dwarfs is not readily defined since these stars gradually approach the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Since we are concerned only with the effect of stellar contraction on spin-up, we define the interval at which the timescale for contraction and spin-up (taken to be the logarithmic change in the momentum of inertia with time) greatly exceeds the timescale for spin-down by saturated magnetic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Here, we adopted “greatly" to be 10× but our estimates are not sensitive to the exact figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5 plots the variation in PMS duration rel- ative to the solar-metallicity cases vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' [Fe/H] for the same mass/𝑇eff cases as the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5 plots variation of MS 𝑇sat relative to the solar-metallicity value vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' [Fe/H] for 40 different mass tracks with MS 𝑇eff falling within 3200-4200K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' As one metric of the age error due to non-solar [Fe/H] we added the PMS and saturated spin-down durations and performed linear re- gression with [Fe/H] over the range of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3, where most M dwarfs fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This slope of this regression vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑇eff is the light- colored curve in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The sensitivity of the age estimates to [Fe/H] increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 Gyr/dex at 3200K, peak- ing at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 Gyr/dex at around 3900K, and declining in the K dwarf regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This behavior is almost entirely a consequence of the metallicity dependence of the braking torque (van Saders & Pin- sonneault 2013), with a lesser contribution from changes in mass with metallicity at a fixed 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Although the power-law index of the Skumanich-like spin down observed on long time-scales is not considered metallicity- dependent, metallicity-dependent torque could still impart addi- tional deviation from predictions based on solar-metallicity gy- rochrones during the transition to purely power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' To quantify this, we performed calculations of rotation evolution using the models of Claytor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020b), which use the torque scaling of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We used the stellar model interpolation and Markov Chain Monte Carlo (MCMC) tools in kiauhoku (Claytor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020a) to make 𝑃rot-based age estimates of 4 Gyr-old model stars with a given 𝑇eff and varying [Fe/H], but assuming solar [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We per- formed a linear regression of the inferred age minus the “true" age (4 Gyr) versus [Fe/H] for different values of 𝑇eff, and the slope is plotted as the black curve in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (These calculations do not go below 3500 K because of incompleteness in the model grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=') This curve has the same overall shape as our curve of PMS+MS age sensitivity (light colored curve) but is generally larger in magnitude, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Based on a comparison of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5a, we approxi- mately incorporate the metallicity dependence of spin-down during the transition to pure Skumanich-like behavior by doubling the off- MNRAS 000, 1–16 (2022) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3000 3250 3500 3750 4000 4250 0 5 10 15 20 25 30 M37 (470 Myr) 3000 3250 3500 3750 4000 4250 0 5 10 15 20 25 30 Praesepe (600 Myr) 3000 3250 3500 3750 4000 4250 effective temperature (K) 0 5 10 15 20 25 30 rotation period (days) Hyades (700 Myr) 3000 3250 3500 3750 4000 4250 0 5 10 15 20 25 30 NGC 6811 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 Gyr) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Fits of the cool dwarf rotational sequences of four open clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (The existence of a rotation sequence in M37 M dwarfs is unclear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Black points indicate stars used in iterative, outlier-rejection fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The solid red line is the fit, the dashed red lines are the ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5𝜎 rejection boundaries, and the dotted magenta lines show the extent of the scatter induced solely by an error of 𝑇eff of 75K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The blue line is the critical rotation period for 𝑅𝑜 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 using the relation between convective turnover time and luminosity of Jeffries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2011), with luminosities from the Dartmouth stellar evolution models (Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Feiden 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Stars below this line will have saturated magnetic fields and experience exponential spin-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' set during the MS saturated and adding it to the PMS offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This is the heavy colored curve in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We multiply the slope by a typical uncertainty of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 dex in [Fe/H] and add this (up to ±150 Myr) to our error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We also use kiauhoku to calculate individual [Fe/H]-dependent corrections for the age of each star with known metallicity and 𝑇eff>3500K that can be added to age estimated from our solar-metallicity gyrochronology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 6 RESULTS: PLANET HOST STAR AGES Table 6 provides the 𝑇eff and [Fe/H] (if available) that were used for the gyrochrone calculations, the rotation period, the method and instruments used to obtain it and the reference, and the es- timated age and uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' If a metallicity is available, we also provide, but do not incorporate, a kiauhoku-calculated value for the [Fe/H]-dependent correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Only the first 50 entries are shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' the complete machine-readable table is provided on Zenodo (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7578269).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Figure 6 compares the empirical ages of host stars to ages using generated with the kiauhoku model (Claytor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' There is good agreement among the warmer stars in the sample, but a clear trend of older kiauhoku-based es- timates for cooler 𝑇eff, where the models have not been calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The systematic offset of model-derived ages for the coolest stars further illustrates the need for calibrators across the full range of temperature and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The distribution of ages assigned to Monte Carlo realizations is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7, where we have plotted the KOIs and all other host stars with separate curves as distinct in terms of sensitivity MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 [Fe/H] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 age [Gyr] variation in PMS duration 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 [Fe/H] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 age [Gyr] variation in MS time to Pcrit 3200 3400 3600 3800 4000 4200 Teff [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 age [Gyr] per dex age sensitivity to [Fe/H] 3200 3400 3600 3800 4000 4200 Teff Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Difference in actual age relative to gyrochrone age due to non-solar metallicity due to variation in the PMS duration (a) and MS interval required for spin-down sufficiently for Ro to exceed the critical value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 and the star to leave the saturated phase of activity and follow power-law Skumanich- like spin-down (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Positive values means that a star will be older than its gyrochronologic age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Each color corresponds to a different mass track in Dartmouth standard model calculations, converted to 𝑇eff on the main- sequence using the empirical relation of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The light colored line in the bottom panel is the slope of the summed intervals vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' [Fe/H] obtained at each value of 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The black curve is the slope calculated from a full model of metallicity-dependent spin-down over a representative age of 4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The heavy colored line is the PMS interval plus twice the MS interval used as an approximation for the actual sensitivity that reproduces the shape and magnitude of the model simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' and systematics as well as (potentially) stellar populations, and compare these to the isochrone-based distribution for all Kepler stars from Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For upper limits, ages were drawn from a uniform distribution from zero to the upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We did not exclude known binary stars from these distributions since the effect of binaries depends on semi-major axis in a manner that is still being actively investigated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Messina 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' To help discern between actual structure and systematics in the age distribution, we created a mock stellar population with a uniform age distribution, 𝑇eff drawn with replacement from the actual catalog, and 𝑃rot calculated with a simple model of the spin- 0 2 4 6 8 10 12 14 model age 0 2 4 6 8 10 12 14 empirical age 3200 3400 3600 3800 4000 4200 equilibrium temperature Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Empirical ages of host stars based on the rotation-age relations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' the estimates using the kiauhoku model (Claytor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 age (Gyr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='20 probability density other Kepler Kepler (Berger+2020) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Distribution of estimated ages for KOIs and all other known M dwarf host stars, accounting for the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The grey curve is the isochrone-based age distribution for all Kepler stars from Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' down of stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The model assumed solid-body rotation and power-law spin-down with index 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='62 for 𝑅𝑜 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', at MS ages 𝑡 > 𝑡crit where the condition 𝑃rot > 𝑃crit, as defined before, is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For main-sequence ages 𝑡 < 𝑡crit, stars undergo exponential spin-down with a time constant set such that at 𝑡 = 0, 𝑃rot is equal to an initial value 𝑃0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The initial period is derived from the specific momentum distribution of among ∼10 Myr-old M dwarf members of the Upper Scorpius star-forming region (Somers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017), and the moments of rotational inertia from the Dartmouth stellar evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Main sequence ages were a random draw from a uniform 0-10 Gyr distribution, minus the PMS duration as taken from Dartmouth solar-metallicity models (Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (The 𝑃rot of PMS stars is fixed at 𝑃0, but this choice is not important since these stars were subsequently excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=') We added the Gaussian-distributed error of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4% to the periods, the median of the distribution of actual error, and then derived the ages and MNRAS 000, 1–16 (2022) 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 age (Gyr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='175 probability density actual uniform Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Distribution of actual ages vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' those inferred from a “mock" population of stars with the same 𝑇eff distribution as the exoplanet host stars, but a uniform 0-10 Gyr age distribution (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' errors with the same routines used for the actual exoplanet host star catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The actual and mock distributions are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' There is a marked deficit and marked structure at <3 Gyr in the age distribution of stars in both the Kepler and non-Kepler samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7), but this feature also appears in the simulation of a uniform distribution of age (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 8), showing that the inferred age distribu- tion is heavily affected by systematics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' the incomplete working gyrochronology over the entire 𝑇eff range at young ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Monte Carlo realizations that fall in these gaps are assigned upper limits and thus not correctly represented in this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Both the in- ferred Kepler and non-Kepler distributions decline with age, and more rapidly than that inferred from the mock uniform-age popula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A declining rotation-based age distribution was also inferred for solar-type Kepler host stars and is in part a bias caused by the difficulty of detecting the lower amplitude, longer period rotational signals that are more prevalent around older stars (Walkowicz & Basri 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This pattern is also mimicked by an age distribution of Kepler target stars based on isochrone analysis (Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020, grey curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' this is also biased towards younger (and more massive stars) that evolve more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The Kepler distribution peaks at older ages than the non-Kepler sample, which could be due to the greater sensitivity and longer monitoring interval of the prime mission, but perhaps also because Kepler was observing a field centered at 𝑏 = 13 deg containing a slightly older population further above the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The Kepler distribution terminates at 10 Gyr, about the age of the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The distribution of non-Kepler host stars has a tail that extends well beyond 10 Gyr but this is largely due to host stars with significant uncertainties in 𝑃rot, along with a handful of binaries (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Binaries: Twenty-six of the 249 planet host stars are known to have stellar companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This 11% fraction is much lower than 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4% among field M dwarfs (Winters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Since exo- planet hosts are comparatively well-studied among cool field stars, this is very unlikely due to limited characterization of these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Instead, it probably reflects survey/detection bias where binary stars are avoided in exoplanet surveys because it is usually more difficult to detect planets around them (Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), and because contamina- 0 2 4 6 8 10 12 14 age (Gyr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='35 probability density singles binaries Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Distribution of estimated ages for single vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' binary/multiple host stars, accounting for the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The irregularity of the binary distri- bution is sampling noise: only 26 systems constitute the binary sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' tion of the host star signal by other stars is detrimental to precise measurement of RVs (Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The rotational history of stars in multiple systems can differ greatly from that of single stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Stars with stellar companions are more likely to be rapidly rotating relative to single stars of the same age/mass (Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Simonian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Very close (≪ 1 au) binaries transfer orbital AM to rotational AM via tidal torques (Fleming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' At moderate separations (∼100 au) a stellar companion will truncate a disk and shorten its viscous lifetime (Cieza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Kraus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rosotti & Clarke 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This removes a sink of stellar angular momentum, allowing the star to spin-up unimpeeded during pre-main sequence contraction (Messina 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We computed ages based on rotation without regard to mul- tiplicity, but warn that in the case of binaries, such ages could be seriously in error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In this sample, at least, the distribution of rotation periods does not appear remarkably different from single stars (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 1) nor does the age distribution of known binaries appear remark- able (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 9), although it is greatly limited by the small sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This could be because the smaller fraction of binaries that do appear in the catalog tend to be very wide, and the effects on rotation and hence age are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Anomalously old stars: Four host stars (GJ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1, GJ 667 C, HD 238090, and HIP 70849) are assigned problematic ages that are > 2𝜎 older than 10 Gyr, the nominal age of the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' These are unlikely to be Galactic halo or former globular cluster members because unusual abundances and peculiar motion characteristic of such stars would have been noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Three of the stars are in binaries, in which stellar companions could directly or indirectly affect the rotation evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The fourth, GJ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1, has a 140±10 day rotation period estimated from ASAS-SN photometry, far longer than that expected for an early-type M dwarf in the Galactic disk, and the age is obviously unphysical: 35 ± 9 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The star’s metallicity has not been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Potentially the rotation period is an artifact, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', confusion with another star (the survey’s resolution is 15").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Young stars: No PMS-ages were assigned to our stars, which is expected since we removed all known disk-hosting and PMS stars from our catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Fourteen stars can only be assigned upper limits for ages, and of these 8 are younger than 1 Gyr at the 95-percentile MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 11 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' TIME-Table: Catalog of Cool Host Stars with Established Rotation Periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 Name 𝑇eff Fe/H Period (unc) Method2 (Instruments3) Reference Age (unc) Corr4 Note5 [K] [dex] [days] Gyr Gyr EPIC 201170410 3650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='16 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6) P (K2) this work 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='58) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 binary EPIC 211822797 3856 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='88 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='72) P (K2) Reinhold & Hekker (2020) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='24) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='03 G 264-012 3326 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0) PS (TE/ME/AN/CA) Amado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2021) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='66 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='26) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='12 G 9-40 3713 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='04 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='08 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='46) P (K2) Reinhold & Hekker (2020) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='35 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='57) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 GJ 1132 3270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='12 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5) P (ME) Cloutier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2017) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='31 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='23) —– GJ 1148 3304 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='16 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 P (HA) Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2011) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='44 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='51 no error GJ 1214 3252 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='29 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0) P (ST) Mallonn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2018) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='91 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0) —– GJ 1252 3458 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0) P (WA) Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2020) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='61 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='34) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='58 GJ 1265 4052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='04 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='17 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='27) P (K2) Reinhold & Hekker (2020) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='44 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='53) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='11 GJ 15 A 3607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='32 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='82 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='56) PS (Fa/HI) Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2014) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='87 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='66 binary GJ 176 3680 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='14 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='92 P (AS) Kiraga & Stepien (2007) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 (0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4) S (HA) Maldonado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2021) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='35 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='82) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='18 Gl 49 3740 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7) S (HA) Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2018) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='41) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='07 binary 1The full table is available as a machine-readable table (DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7578269) 2P = photometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S = spectroscopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3AN=All-Sky Automated Survey for Super-Novae (ASAS-SN, Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2017), AP=APACHE (Sozzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' AS=All Sky Automated Survey (ASAS, Pojmanski 2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' CA=CARMENES (Quirrenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016), ES=ESPRESSO (Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' FA=Fairborn (Henry 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' HA=HARPS (Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' HN=Hungarian Automated Telescope Network (HAT-Net, Bakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K2=K2 (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014), KE=Kepler (Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' LC=Las Cumbres Observatory Global Telescope (LCO, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ME=MEarth (Berta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2012), NS=Northern Sky Variability Survey (NSVS, Woźniak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' OS=Observatorio de Sierra Nevada;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' SP=SPIRou (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ST=STELLA (Strassmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' TE=TESS (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' TJ=Telescope Joan Oró (TJO, Colomé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' WA=Wide Angle Search for Planets (WASP, Pollacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4Metallicity-dependent age correction to be added to value for solar-metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5If no period error is provided, an uncertainty of 1 day is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' level and not known to be binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The rotation period of one of these, TOI-620, is tentative and the star is also a suspected binary (Reefe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The other seven are K2-detected systems: K2-43, K2-239, K2-240, which hosts two transiting Neptune-size planets, has been detected in X-rays (Foster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K2-284, previously reported having a young age by David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2018), K2-324, K2-354, and KOI-5879, a flaring M dwarf (Yang & Liu 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 Individual Noteworthy Systems GJ 229: The nearby (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='76 pc) M1 dwarf Gliese/GJ 229 has an ultra-cool (T7-type) dwarf companion (Nakajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The primary’s 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3-day rotation period was determined from ASAS- SN photometry (Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2016) and we estimate an age of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 Gyr, with the caveat that the existence of the companion on a 29 au orbit could have affected the rotation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' No uncertainty in 𝑃rot was reported but this is likely to be small given the multi-year baseline of ASAS-SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' GJ 1214: This nearby M4-type dwarf with a well-studied tran- siting “sub"-Neptune-size planet on a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='58-day orbit (Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Based on the 125 ± 5 day rotation period identified in STELLA photometry by Mallonn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2018), we estimate an age of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' LHS-1815: This M1-type dwarf (aka TOI-704) hosts a transit- ing Earth-size planet on a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8-day orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The star lies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8 kpc above the Galactic plane and kinematically belongs to the “thick" Galactic disk population (Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We estimate an age of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 Gyr, consistent with the expected age of that population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K2-22: K2-22 is a late K dwarf that hosts what has been pro- posed to be a “evaporating" planet on a 9-hour orbit that manifests itself as quasi-periodic dimming due to accompanying dust cloud (Sanchis-Ojeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The highest peak in a Lomb-Scargle pe- riodogram in K2 photometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' is at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='26 days but the shape of the lightcurve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A1) suggests this is one-half the period (Sanchis- Ojeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A period of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 days yields an estimated age of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 Gyr, but this star has an M dwarf companion at a projected separation of 460 au Sanchis-Ojeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2015) which could have affected its rotation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Barnard’s Star (GJ 699): This very metal-poor, high peculiar motion M4-type is classified as intermediate between the Galac- tic Disk and Halo populations (Gizis 1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' a putative Doppler RV-detected planet (Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018) around this star has been disputed (Lubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' It is not in our current catalog because its 𝑇eff is marginally cooler than our 3200K cut-off, but, motivated by its unusual nature and recently confirmed 𝑃rot of 145±15 days (Toledo-Padrón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Terrien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), we compare this to the cool extremum of the Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) M67 gyrochrone, which reaches 120 days at 3250K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 3) and at cooler temperatures is essentially an unconstrained extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Thus the gyrochronol- ogy suggests an age older than M67, as expected, but extension of M dwarf gyrochrones into the fully convective area is needed before assigning any robust age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7 SUMMARY AND DISCUSSION 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1 Rotation and Ages of M dwarf Exoplanet Hosts The value of robust ages for exoplanet studies, and advances in the gyrochronology of older cool M dwarfs motivated us to catalog rotation periods among late K and early M-type dwarfs (𝑇eff=3200- 4200K) that host known planets, and to apply empirical, 𝑇eff- dependent rotation-age relations to estimate ages and their standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This complements work on calibrated rotation-based ages among younger PMS stars (Kounkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We cataloged 249 stars with rotation periods, 227 of which we are able to estimate ages with a median error of 20% and mode of 14%, and to an additional 8 we assign upper limits (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Our fractional error is significantly higher than the 5-10% estimated by Otani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022), probably because we include additional potential sources of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Figure 10 shows ages of candidate or confirmed planets around these stars and the distribution with semi-major axis, radius, and equilibrium temperature, as reported in the NASA Exoplanet Archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The age distributions of both Kepler and non-Kepler host stars peak at around 3 Gyr with a steady decline to near zero at 10 Gyr, the age of the Galactic disk (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The resemblance of the actual and “mock" populations (the latter with a uniform 0-10 Gyr age distribution) shown in Figure 8 indicates that the structure of the distribution, particularly at young ages, is partly due to the discontinuous and limited coverage of the current gyrochronology and dispersion of the distribution due to error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The peaks at <3 Gyr correspond approximately with the location of the calibration ages, and are likely artefacts due to ages that cannot be assigned in certain regions of 𝑇eff-𝑃rot space and have only upper limits assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Other effects impacting the derived distribution include the opposing biases against detection of planets around younger, more rapidly rotating, and more active stars (Miyakawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), and against detection of rotational variability among older, slowly rotating, less active stars (Morris 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2 Several versions of the local star formation history based on white dwarf cooling ages and Gaia astrometry also peak at around 3-5 Gyr (Isern 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Mor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Alzate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021), so the distribution of host star ages could reflect this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Our error analysis (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 5) shows that one limiting source of error could prove to be the precision of stellar parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑇eff and [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The sensitivity to 𝑇eff is due to the steepness of rotation sequences for very cool dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝑇eff, which is related to the surface brightness and hence convective vigor of a star, is the appropriate independent variable for gyrochronology, but must be inferred from observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The ultimate limit on accurate values of 𝑇eff precision for M dwarfs is the challenge of establishing a reliable temperature scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 Caveats and Limitations We have adopted the values and standard errors of 𝑃rot from the literature at face value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The possibility that the true period is twice the published value needs to be considered in cases of stars with anomalous rapid rotation and young ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ground-based observa- tions can also suffer from aliasing imposed by diurnal, lunar, and annual window functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Our gyrochronology assumes that the narrow rotational se- quence observed among the late K dwarfs and the warmer M dwarfs in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr-old Ruprecht 147 cluster (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020) extends to 3200K by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 Gyr, and that the 𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='62 power-law spin-down derived by (Dungee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022) also applies to cooler M dwarfs at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This assumption could fail if the rotational sequence among M67 M dwarfs was formed by stalling, rather than a tran- sition from saturated to un-saturated braking laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Core-envelope re-coupling is expected to become weaker and take longer towards the fully convective boundary, which could mean that a rotational 2 “Stalling" of spin-down due to core-envelope decoupling would result in broadening of the age distribution, not peak formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 13 sequence appears much later, or not at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This depends in part on the 𝑇eff or mass dependence of the core-envelope coupling time vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 𝜏𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This issue can only be resolved by deeper monitoring of Ruprecht 147 or a cluster of similar age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Another contributor to our error budget is sensitivity of the rotation-age relation to non-solar [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the absence of appropri- ate calibration, we relied entirely on theoretical models to estimate this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The part due to changes in the structure and moment of inertia of the star is reasonably well-constrained by observations, but magnetic field pressure could modulate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A model that in- cludes this effect (Feiden 2016) predicts that at a fixed 𝑇eff=3700K, inclusion of magnetic pressure increases the moment of inertia by 40%, but this is almost entirely due to a change in the mass inferred for a given 𝑇eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' More uncertain is the metallicity-dependence of the torque, which, at least in our models, dominates the sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We based this scaling on the magnetized wind formulation of Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2012) as reformulated by van Saders & Pinsonneault (2013), but this was developed for solar-type stars with rotationally-aligned dipole fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Finally, rotation-based ages for binary systems must be care- fully considered, particularly given the possibility of a third, closer and unresolved component (Reipurth & Mikkola 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' While most published exoplanets have had some sort of screening for binaries, not all of them cover all the parameter space, and surveys of very cool KOIs are only now coming to fruition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='3 Outlook Since 𝑇eff is not an observable and cannot be readily derived with- out stellar radii, gyrochrones could be established in a common reddening-corrected color which is also available for stars of inter- est;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ideally, this color should be directly related to 𝑇eff, it should be relatively [Fe/H]-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' It should also use redder filters in which M dwarfs are comparatively bright and reddening is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' These requirements impact the use of Gaia photometry since M dwarfs are faint in the 𝐵𝑝 synthetic band used to construct 𝐵𝑝 − 𝑅𝑝 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Analysis of color magnitude diagrams of Kepler M dwarfs using PanSTARRS photometry suggest that 𝑔-𝑌 holds promise (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ali, pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Establishing precise M dwarf metallicities is a work in progress (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Passegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' More challenging will be to validate the effect of metallicity on rotation-age relations, which here we have here treated only via stellar interior models and torque-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Tests of the metallicity-scaling of the torque law are des- perately needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Metal-poor or metal-rich clusters are relative rare (Heiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2014) and thus, statistically, found at greater distances where observations to establish rotational sequences will be chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The wealth of binaries provided by Gaia (El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2021) might serve as a road to calibration over a wider range of stellar parameters (Otani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), provided sufficiently wide examples can be identified and precise ages for the primaries can be determined by other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gyrochronology of exoplanet host stars is an ongoing effort and we envision the TIME-Table to be a “living" catalog of very cool dwarf rotation periods and ages that is periodically updated and revised with new discoveries and advances in gyrochronology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' New planetary systems are constantly being detected, validated, or confirmed, particularly by the TESS mission, now in its fifth year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rotational variability is being detected by two ongoing space surveys: TESS and, with much longer baseline but much sparser cadence, Gaia (Distefano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ground-based surveys like ZTF and the Rubin Observatory (Hambleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022) can provide observations of distant field stars and young clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Although the TESS 27-day sector interval severely limits its ability to detect the rotation of older field stars (Claytor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022), those stars in and around the two Continuous Viewing Zones around the ecliptic poles are observed for multiple sectors and in principle it is possible to detect longer periods (Hedges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Claytor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Many rotation periods have been established by analysis of Doppler RV residuals or indicators of activity in the time-series high-resolution spectroscopy obtained to detect, confirm, or mea- sure the masses of planets (Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Terrien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022) showed that detection of the periodic signal in the Zeeman broadening of lines can reveal rotation of magnetic ac- tive regions on the star and yield a rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The Zeeman effect increases with wavelength-squared, and the proliferation of high-resolution spectrographs operating in the infrared could lead to additional rotation periods using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Alternatives to Lomb-Scargle periodogram analysis which are more robust to spot evolution such as autocorrelation and Gaussian process regression (Angus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Nicholson & Aigrain 2022) could be used to obtain more precise ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Age-dating could also adopt a Bayesian approach, with Galactic population age distributions as priors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Cukanovaite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Last but not least, additional observations of open clusters for calibration will improve the gyrochronology and lead to more precise (and hopefully more accurate) ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In particular, the 𝑇eff range of existing gyrochrones (including M67) should be extended through the fully convective boundary to include mid- and late- type M dwarfs representing hosts stars of particular interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', TRAPPIST-1), and, foremost, to establish whether a narrow rota- tional sequence appears by a few Gyr — without which gyrochronol- ogy is futile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The small effective area and large pixel size of TESS greatly limits its utility here, since older clusters are rare and hence more distant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The Plato mission will offer only limited improve- ment over TESS (15" vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 20" pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' However, the Roman Space Telescope will have a field of view of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='28 deg2 with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='11" pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ages of calibrator clusters could see refinement from a combi- nation of Gaia parallaxes, asteroseismology, and high-throughput spectroscopy (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Otherwise, much of the observations need to be performed from the ground using wide-field telescopes with sufficient aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Wide-field adaptive optics can alleviate the issues of source confusion in the fields of more distant clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For example, ground-layer adaptive optics (GLAO) can provide a factor of 2–3 improvement in spatial resolution compared to seeing- limited observations while still capturing an entire cluster in one ob- servation (Rigaut 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Such improvements enable observations of dwarfs to spectral type M7 in the majority of clusters older than 1 Gyr (Dungee 2022)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is dedicated to the memory of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', for whom the very stars of heaven were new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' were supported by NSF Astronomy & Astrophysics Research Program Grant 1817215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' We thank Jen van Saders and an anonymous reviewer for helpful feedback on earlier versions of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This paper includes data collected by the Kepler and TESS missions and obtained from the MAST data archive at the Space Telescope Science Institute 3 https://scholarspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='manoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='edu/collections/67c8d51f-97a4-4d45- 8a69-8baeaacaeaf6 MNRAS 000, 1–16 (2022) 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 age [Gyr] 10 2 10 1 100 semi-major axis [au] Earth Neptune Jupiter 250 500 750 1000 1250 1500 1750 2000 equilibrium temperature Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 111 confirmed planets with properties from the NASA Exoplanet Archive with age estimates from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Points are scaled with planet radius and color-coded by planet equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Ages have not been filtered for binarity nor corrected for metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (STScI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Funding for the Kepler mission is provided by the NASA Science Mission Directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' STScI is operated by the Associa- tion of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', under NASA contract NAS 5–26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This paper makes use of data from the MEarth Project, which is a collaboration between Harvard Univer- sity and the Smithsonian Astrophysical Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The MEarth Project acknowledges funding from the David and Lucile Packard Fellowship for Science and Engineering and the National Science Foundation under grants AST-0807690, AST-1109468, and AST- 1004488 (Alan T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Waterman Award), and a grant from the John Templeton Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' DATA AVAILABILITY All data used in this work are in the public domain and available through various archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' REFERENCES Adams F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bodenheimer P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Laughlin G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2005, Astronomische Nachrichten, 326, 913 Agüeros M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ApJ, 862, 33 Alzate J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bruzual G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Díaz-González D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, MNRAS, 501, 302 Amado P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, A&A, 650, A188 Angus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Morton T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Aigrain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Foreman-Mackey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rajpaul V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, MNRAS, 474, 2094 Astudillo-Defru N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017a, A&A, 602, A88 Astudillo-Defru N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017b, A&A, 605, L11 Bakos G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Noyes R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kovács G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stanek K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Sasselov D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Domsa I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2004, PASP, 116, 266 Balona L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Abedigamba O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, MNRAS, 461, 497 Barnes S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2007, ApJ, 669, 1167 Basri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Streichenberger T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', McWard C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Edmond Lawrence I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Tan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Lee M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Melton T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, ApJ, 924, 31 Batalha N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, ApJS, 204, 24 Belokurov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, MNRAS, 496, 1922 Berger T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Huber D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Tayar J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kraus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, AJ, 159, 280 Berta Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Irwin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Charbonneau D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Burke C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Falco E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2012, AJ, 144, 145 Biddle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, MNRAS, 443, 1810 Binks A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Jeffries R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, MNRAS, 438, L11 Borucki W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2010, Science, 327, 977 Boulade O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1998, Experimental Astronomy, 8, 25 Boyajian T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2012, ApJ, 757, 112 Bragaglia A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Fu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mucciarelli A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Andreuzzi G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Donati P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, A&A, 619, A176 Brandt T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, ApJS, 254, 42 Brown T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, PASP, 125, 1031 Burke C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, ApJS, 210, 19 Burt J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Vogt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Butler R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hanson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Meschiari S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rivera E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Henry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Laughlin G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, ApJ, 789, 114 Caballero J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, A&A, 665, A120 Canto Martins B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, ApJS, 250, 20 Casali G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, A&A, 639, A127 Charbonneau D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2009, Nature, 462, 891 Christensen-Dalsgaard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Aguirre V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, in Deeg H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Belmonte J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', eds, , Handbook of Exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Springer, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 184, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1007/978-3- 319-55333-7_184 Christy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='02239 Cieza L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2009, ApJ, 696, L84 Clark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Belle G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ciardi D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Lund M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Howell S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Everett M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Beichman C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Winters J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, AJ, 163, 232 Claytor Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Santos Â.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', García R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mathur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Tayar J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Shetrone M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020a, kiauhoku: Stellar model grid interpolation, Astrophysics Source Code Library, record ascl:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='027 (ascl:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='027) Claytor Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Santos Â.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', García R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mathur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Tayar J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Shetrone M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020b, ApJ, 888, 43 Claytor Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Llama J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Sadowski P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Quach B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Avallone E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, ApJ, 927, 219 Cloutier R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Doyon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Menou K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Delfosse X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Dumusque X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Artigau É.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, AJ, 153, 9 Cody A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hillenbrand L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2010, ApJS, 191, 389 Colomé J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Casteels K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ribas I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Francisco X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2010, in Radziwill N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bridger A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 7740, Software and Cyberinfrastructure for Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 77403K, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='857672 Coughlin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, ApJS, 224, 12 Cukanovaite E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Tremblay P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Toonen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Temmink K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Manser C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', O’Brien M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', McCleery J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='13919 Cummings J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Deliyannis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Maderak R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Steinhauer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, AJ, 153, 128 Cunha D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Figueira P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Santos N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Lovis C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Boué G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, A&A, 550, A75 Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Wolfgang A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Wright J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Brewer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Johnson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, AJ, 145, 134 Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Agüeros M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mamajek E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Wright J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Cummings J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019a, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10588 Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Agüeros M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Douglas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Meibom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019b, ApJ, 879, 49 Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, ApJ, 904, 140 David T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, AJ, 156, 302 Dedrick C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, AJ, 161, 86 Denissenkov P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Terndrup D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Newsham G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2010, ApJ, 716, 1269 Díaz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='02012 Díez Alonso E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, MNRAS, 489, 5928 Distefano E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05500 Donati J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, MNRAS, 498, 5684 Dotter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Chaboyer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Jevremović D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kostov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Baron E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ferguson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2008, ApJS, 178, 89 Douglas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Agüeros M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Cargile P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Brewer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Meibom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Jansen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJ, 879, 100 Dungee R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, PhD thesis, University of Hawaii at Manoa Dungee R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Chun M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', García R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Magnier E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mathur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Santos Â.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, ApJ, 938, 118 MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 15 El-Badry K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Heintz T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, MNRAS, 506, 2269 Epstein C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, ApJ, 780, 159 Esselstein R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Aigrain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Vanderburg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Smith J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Meibom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mathieu R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ApJ, 859, 167 Feiden G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, A&A, 593, A99 Feng F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJS, 242, 25 Fleming D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Barnes R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Davenport J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Luger R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJ, 881, 88 Foster G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Poppenhaeger K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ilic N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Schwope A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, A&A, 661, A23 Fragkou V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Parker Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Zijlstra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Vázquez R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Sabin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rechy-Garcia J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, ApJ, 935, L35 Fritzewski D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Barnes S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', James D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Strassmeier K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, A&A, 641, A51 Fu X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='09121 Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, A&A, 595, A1 Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00211 Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kraus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ireland M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, MNRAS, 457, 2877 Gallet F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bouvier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, A&A, 577, A98 Gan T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, AJ, 159, 160 Giacobbe P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, MNRAS, 491, 5216 Gizis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1997, AJ, 113, 806 Godoy-Rivera D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rebull L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, ApJS, 257, 46 González-Álvarez E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, A&A, 649, A157 Gossage S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Conroy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Dotter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Choi J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rosenfield P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Cargile P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Dolphin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ApJ, 863, 67 Hambleton K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='04499 Hardegree-Ullman K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Cushing M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Muirhead P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Christiansen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, AJ, 158, 75 Hartman J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bakos G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Noyes R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Sipőcz B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kovács G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mazeh T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Shporer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pál A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2011, AJ, 141, 166 Hedges C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Angus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Barentsen G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Saunders N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Montet B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gully- Santiago M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, Research Notes of the American Astronomical So- ciety, 4, 220 Heiter U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Soubiran C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Netopil M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Paunzen E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, A&A, 561, A93 Helled R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Morbidelli A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, in Madhusudhan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', , ExoFrontiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Big Questions in Exoplanetary Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Institute of Physics, pp 12–1, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1088/2514-3433/abfa8fch12 Henry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1999, PASP, 111, 845 Howard A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, ApJ, 794, 51 Howell S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, PASP, 126, 398 Hsu D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ford E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ragozzine D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ashby K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, AJ, 158, 109 Huber D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, ApJS, 224, 2 Isern J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJ, 878, L11 Jeffries R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Jackson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Briggs K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Evans P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pye J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2011, MNRAS, 411, 2099 Kane S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', von Braun K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Henry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Waters M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Boyajian T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, ApJ, 835, 200 Kemmer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, A&A, 642, A236 Kemmer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, A&A, 659, A17 Kilic M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Munn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Harris H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', von Hippel T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Liebert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Williams K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Jeffery E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', DeGennaro S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, ApJ, 837, 162 Kiraga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stepien K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2007, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 57, 149 Kite E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Manga M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2009, ApJ, 700, 1732 Kochanek C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, PASP, 129, 104502 Kounkel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, AJ, 157, 196 Kounkel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stassun K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bouma L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Covey K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hillenbrand L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Lee Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, AJ, 164, 137 Kraus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ireland M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hillenbrand L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Martinache F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2012, ApJ, 745, 19 Kraus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ireland M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Huber D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Dupuy T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, AJ, 152, 8 Lam K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, Science, 374, 1271 Lamman C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, AJ, 159, 139 Lammer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, Origin and Evolution of Planetary Atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Springer, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1007/978-3-642-32087-3 Lépine S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Shara M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2005, AJ, 129, 1483 Lothringer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, AJ, 155, 66 Lu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Curtis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Angus R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', David T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hattori S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='06604 Lubin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='07005 Luque R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, A&A, 620, A171 Luque R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, A&A, 628, A39 Magaudda E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stelzer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Covey K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Raetz S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Matt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Scholz A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, A&A, 638, A20 Maldonado J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, A&A, 651, A93 Mallonn M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, A&A, 614, A35 Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Brewer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Lépine S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hilton E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, AJ, 145, 52 Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Deacon N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ansdell M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Brewer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Liu M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Magnier E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Aller K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, AJ, 147, 160 Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Feiden G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Boyajian T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', von Braun K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, ApJ, 804, 64 Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJ, 871, 63 Masci F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, PASP, 131, 018003 Matt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinzón G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Greene T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pudritz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2012, ApJ, 745, 101 Matt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Brun A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Baraffe I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bouvier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Chabrier G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, ApJ, 799, L23 Messina S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, A&A, 627, A97 Miyakawa K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hirano T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Sato B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Okuzumi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Gaidos E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, AJ, 164, 209 Molenda-Żakowicz J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Brogaard K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Niemczura E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bergemann M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Frasca A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Arentoft T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Grundahl F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, MNRAS, 445, 2446 Montes D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, MNRAS, 479, 1332 Mor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Robin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Figueras F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Roca-Fàbrega S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Luri X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, A&A, 624, L1 Morris B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, ApJ, 893, 67 Mulders G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pascucci I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Apai D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, ApJ, 798, 112 Mullally F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, ApJS, 217, 31 Nakajima T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Oppenheimer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kulkarni S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Golimowski D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Matthews K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Durrance S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1995, Nature, 378, 463 Newton E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mondrik N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Irwin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Winters J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Charbonneau D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, AJ, 156, 217 Nicholson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Aigrain S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, MNRAS, 515, 5251 Oelkers R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, AJ, 155, 39 Otani T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', von Hippel T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Buzasi D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Oswalt T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stone-Martinez A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ma- jewski P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, ApJ, 930, 36 Pasquini L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Biazzo K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bonifacio P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Randich S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bedin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2008, A&A, 489, 677 Passegger V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, A&A, 658, A194 Pecaut M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mamajek E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, ApJS, 208, 9 Pepe F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2000, in Iye M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Moorwood A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', eds, Society of Photo- Optical Instrumentation Engineers (SPIE) Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 4008, Optical and IR Telescope Instrumentation and Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' pp 582–592, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='395516 Pepe F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, A&A, 645, A96 Pojmanski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2002, Acta Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 52, 397 Pollacco D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2006, PASP, 118, 1407 Quirrenbach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, CARMENES: an overview six months after first light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' SPIE, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 990812, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2231880 Rebassa-Mansergas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, MNRAS, 505, 3165 Rebull L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, AJ, 152, 113 Rebull L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stauffer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Cody A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hillenbrand L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', David T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, AJ, 155, 196 Reefe M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, AJ, 163, 269 Reiners A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, A&A, 609, L5 Reinhold T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Hekker S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, A&A, 635, A43 Reinhold T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Reiners A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Basri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, A&A, 560, A4 Reipurth B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mikkola S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2012, Nature, 492, 221 Ribas I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, Nature, 563, 365 Richer H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Fahlman G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rosvick J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Ibata R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1998, ApJ, 504, L91 Ricker G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' SPIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 914320 (arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0151), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2063489 Rigault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ztfquery, a Python tool to access ZTF data, Zenodo, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1345222 Rigaut F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2002, in European Southern Observatory Conference and Work- shop Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 11 Robertson P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mahadevan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Endl M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Roy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, Science, 345, 440 MNRAS 000, 1–16 (2022) 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Rodríguez-López C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, Frontiers in Astronomy and Space Sciences, 6, 76 Roquette J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Matt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Winter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Amard L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stasevic S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, MNRAS, 508, 3710 Rosotti G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Clarke C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, MNRAS, 473, 5630 Rowe J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, ApJS, 217, 16 Sanchis-Ojeda R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, ApJ, 812, 112 Sandquist E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, AJ, 161, 59 Santos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', García R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mathur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Bugnet L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Metcalfe T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Simonian G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJS, 244, 21 Santos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Breton S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mathur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', García R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, ApJS, 255, 17 Sarajedini A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Dotter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Kirkpatrick A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2009, ApJ, 698, 1872 Scargle J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1982, ApJ, 263, 835 Schiavon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Caldwell N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rose J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2004, AJ, 127, 1513 Serna J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, ApJ, 923, 177 Shappee B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2014, ApJ, 788, 48 Shporer A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, ApJ, 890, L7 Simonian G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Terndrup D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJ, 871, 174 Skumanich A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 1972, ApJ, 171, 565 Somers G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stauffer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rebull L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Cody A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, ApJ, 850, 134 Souto D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, ApJ, 890, 133 Sozzetti A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, in European Physical Journal Web of Conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 03006 (arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1275), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='1051/epjconf/20134703006 Stauffer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, AJ, 152, 115 Stock S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2020, A&A, 643, A112 Strassmeier K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2004, Astronomische Nachrichten, 325, 527 Su X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Xie J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Zhou J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Thebault P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, AJ, 162, 272 Suárez Mascareño A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rebolo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', González Hernández J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Esposito M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2015, MNRAS, 452, 2745 Suárez Mascareño A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rebolo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', González Hernández J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2016, A&A, 595, A12 Suárez Mascareño A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rebolo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', González Hernández J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Esposito M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, MNRAS, 468, 4772 Suárez Mascareño A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, A&A, 612, A89 Suto Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Sasaki S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Nakagawa Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Benomar O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, PASJ, 74, 857 Terrien R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, ApJ, 927, L11 Thompson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ApJS, 235, 38 Toledo-Padrón B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, MNRAS, 488, 5145 Toledo-Padrón B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, A&A, 648, A20 Toyouchi D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Chiba M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ApJ, 855, 104 VandenBerg D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Stetson P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2004, PASP, 116, 997 VanderPlas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, ApJS, 236, 16 Venuti L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2017, A&A, 599, A23 Walkowicz L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Basri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, MNRAS, 436, 1883 Winters J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, AJ, 157, 216 Woźniak P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2004, AJ, 127, 2436 Wright N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Newton E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Williams P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Drake J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Yadav R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, MNRAS, 479, 2351 Xiang M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Rix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2022, Nature, 603, 599 Yang H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Liu J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2019, ApJS, 241, 29 Ziegler C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2018, AJ, 156, 83 Ziegler C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Tokovinin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Latiolais M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Briceño C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Law N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Mann A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2021, AJ, 162, 192 van Saders J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', Pinsonneault M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=', 2013, ApJ, 776, 67 APPENDIX A: LIGHTCURVE ANALYSIS Figures A1-A4 show K2 lightcurves of 21 host star in which new rotational signals were identified or, in the case of K2-345, replace a previously published value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the cases of K2-5, 14, 83, 124, 125, 129, 151, 288 B, 315, 322, and 377, a rotation period twice the period of the signal with peak power was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Figure A5 shows periodograms and phased lightcurves from ZTF photometry of four host stars for which new rotation periods are identified and reported in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A6 shows the ASAS-SN data for four stars for which significant (𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='01) signals with the same period appear in both 𝑔- and 𝑉-band photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' In the case of GJ 486, the recovered signal at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='7 days differs markedly from the value of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='5 days published by Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 17 56810 56820 56830 56840 56850 56860 56870 56880 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For K2-22, twice the peak period was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 57140 57150 57160 57170 57180 57190 57200 57210 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='995 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='35 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='91 days Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Additional K2 lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A1 for explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) Cool Exoplanet Host Star Ages 19 57070 57080 57090 57100 57110 57120 57130 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='015 K2-288B C4 100 101 period (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='35 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='35 days 57990 58000 58010 58020 58030 58040 58050 58060 58070 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='9975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0050 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0075 K2-315 C15 100 101 period (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='25 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='57 days 57910 57920 57930 57940 57950 57960 57970 57980 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='010 K2-322 C14 100 101 period (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 0.' metadata={'source': 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+page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='04 K2-345 C16 100 101 period (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='22 days 57220 57230 57240 57250 57260 57270 57280 57290 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='004 K2-377 C6 100 101 period (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='150 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='57 days Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Additional K2 lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' A1 for explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' For K2-322, twice the peak period was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 57400 57410 57420 57430 57440 57450 57460 57470 MJD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='990 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='600 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='625 Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Periodograms and phased ZTF lightcurves of four M dwarf exoplanet hosts with significant (𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='01, horizontal green line) signals (red dots) that passed our visual inspection and are considered candidate rotational signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The host star and the band-pass are indicated in the The blue solid and dashed lines are the lunar synodic period and its aliases with the annual observing window function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Note that the phased lightcurves are repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' MNRAS 000, 1–16 (2022) 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Gaidos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' 101 102 period (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='075 Figure A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' Periodograms and phased ASAS-SN 𝑔-band lightcurves of four M dwarf exoplanet hosts with significant (𝑝 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content='01, horizontal green line) signals (red dots) that also had equivalent significant signals in 𝑉 -band, passed our visual inspection, and are considered candidate rotational signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFLT4oBgHgl3EQfjC8Y/content/2301.12109v1.pdf'} +page_content=' The blue solid and dashed lines are the lunar synodic period and its aliases with the annual observing window function.' metadata={'source': 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